V I E Anireh

N E Osegi
email: nd.osegi@sure-gp.com

A Real Number Formula for Approximating a Class of Irrational and Transcendental Numbers

Keywords: Exponential Function, Quadratic approximation, Real Number Formula, Transcendental Numbers, Nomenclature

We develop a Real Number Formula (RNF) algorithm for computing a class of irrational or transcendental numbers say s  constrained on an apriori set of real or integer numbers, s n referred to here as the calibration constant. RNF is defined as a universal method for approximately constructing the real number set say  . We show that if RNF is quadratic, then s n is negative definite for computing a class of transcendental or irrational numbers and positive otherwise. We further compare the run-time of an RNFbased exponential function against an existing exponential function algorithm to demonstrate the effectiveness of our algorithm.

Introduction

For centuries, real numbers have found valuable applications in engineering and science, making it possible to perform simplified numerical computations on a wide range of scientific problems. Top on the list of real numbers is the class of finite irrational and transcendental numbers including but not limited to , e , ) ( r or  , 2 ,  . Algorithmic techniques have been proposed by numerous authors for approximately computing some of these numbers and is beyond the scope of this paper to document all of them. A useful discussion of the analytic computation of these numbers can be found in [START_REF] Kahaner | Numerical methods and software[END_REF], [START_REF] Lightstone | Concepts of calculus[END_REF]. However, a universal framework for deriving this very important class of numbers is still lacking in the literature.

It is the goal of this report to present a new but simple technique for universal computation of this very important numbers.

We hereby, present a novel formula described as an RNF, for approximating a series of real numbers. Our formula proves the existence of a finite class of popular real numbers which may be transcendental or irrational. In addition, our formula can speed-up the computation of an exponential function.

Related Algorithms and State-of-the-Art Techniques for Computing the Exponent of a Number or the Euler Polynomial

Several algorithms for accurately computing the exponential function the exponential function of x ( x e ).

Some of these are discussed under the headings below:

Algorithm Using Recurrence Relation

In Knuth and Bukholtz [START_REF] Knuth | Computation of tangent, Euler, and Bernoulli numbers[END_REF] a modified recurrent relational formula was used to approximate the exponential function of x leading to faster computation. This technique avoids the time consuming multiplication operations in standard recurrent formulas.

Algorithm Using the Boole Summation Formula

Borwein et al [START_REF] Borwein | Pi, Euler numbers and asymptotic expansions[END_REF] defined two generating functions, the Boole Summation Formula (BSF) and the remainder for Gregory Series for computing the Euler Polynomials. This is somewhat distinct from computing the exponential function of x but could be modified to suit this purpose.

Algorithm Based on Taylors Series Expansion

Moler in [START_REF] Moler | Experiments with MATLAB[END_REF], described an algorithm for computing the exponential function of x. This algorithm uses the power series to approximate some real valued function of x. This function represent one of the exponential function algorithm used in Matlab, a popular and widely used technical computing software.

RNF Principle

The Real Number Formula (RNF) is a new theory of computation that proposes the use of a calibration based approach for approximating some important real numbers. Specifically, RNF uses an algebraic formulation expressed in terms of a few free parameters that belong to the real number class and attempts to describe the existence of the real numbers through the following theorems.

Theorem 1:

Suppose an RNF is intuitively defined as:

a o m a n a RNF          (1)
.where n, m   , n, m > 0, 0 < m << a, 0 < n << a, a   .

Then, there exists a finite set of real numbers, say, * n which sufficiently assures the approximation to a set of important real numbers say {}

s  , such that, o RNF  s  as a    s  Specifically, ns) ., . . n2, {n1, *  n .and Ts} , . … T2, {T1,  s  Proof:
Proof is by Rolle's Theorem [START_REF] Lightstone | Concepts of calculus[END_REF] and is stated as follows:

Suppose m is continuous on the closed interval If RNFo is quadratic, then RNF approximates some s  iff 'n' is sufficiently large and negative definite.

RNF u l u l D n n n n  ) ( ],
From Theorem 2, it follows that,

2           m a n a RNF o (2)

Experiments and Results

In order to verify our theorems we have performed numerical simulations over a range of tunable inputs, say 'n', using a computer program. We investigate the real numbers, , e , ) ( r or



, 2 ,  for different values of n. An Intel iCore3 processor clocked at 2.13GHz with 4GB Random Access Memory has been used as experimental machine platform.

In the first instance, we set a = 10^6 and m = 2, then we vary n over the range 0 to 100 to five places of decimals. The set of 'n' with the closest match to the aforementioned real numbers are given in Table 1.

In the second instance we use the quadratic form of SNF given in Theorem 2, we vary n between -1 and 10^6 while keeping 'a' and 'm' unchanged. The set of 'n' with the closest match are given in Table 2.

In the last instance, we compute the run-times for computing the exponential function of x using both the quadratic form of RNF and the exponential algorithm (expex) in [START_REF] Knuth | Computation of tangent, Euler, and Bernoulli numbers[END_REF]. The values of x are scaled between -1 and 20 and linearly spaced at 50,000 data points performed for 10 consecutive runs for both algorithms. The average values are then computed. The results are given in Table 3.

Discussion and Conclusion

RNF is an automated and universal way of approximating some important irrational or transcendental numbers using a finite set of calibration constants. Using formula 1, n is entirely positive-definite for the selected real numbers. However, this is not the case when formula 2 is used ('n' is positive for r and ).

The computation of recurring decimals using formula 1 may also lead to an increased run-time. Thus formula 2 is cheaper and is recommended when dealing with large input values of x. However, RNF should be used with care since it is only an approximation to the real number class of transcendental and irrational numbers. Also, from the results in Table 3, it can be seen that the RNF algorithms competes favorably well with the expex algorithm and can achieve shorter run-times. However, the expex algorithms performs better on the average.

Future Work

The potentials of RNF as an alternative representation of the exponential function used in popular sigmoidal activation functions used in Neural Networks will be investigated. Attempts will also be directed towards improving the developed algorithm in order to speed up computation.

Table 1

 1 Computed Irrational/ Transcendental numbers for real number 'n' calibration constants.

	n	Discovered Real Number	Class
	1 0.85485	 (3.1429)	Finite Irrational/transcendental
	2 1.00000	e (2.7183)	Finite Irrational/transcendental
	3 1.65340	2 (1.4142)	Finite Irrational
	4 2.48125	r (0.6180)	Finite Irrational
	5 2.54900	 (0.577)	???

Table 2

 2 Computed Irrational/ Transcendental numbers for integer number 'n' calibration constants.

	n		Discovered Real Number	Class
	1 -772,450	 (3.141591)	Finite Irrational/transcendental
	2 -648,718	e (2.718281)	Finite Irrational/transcendental
	3 -189,206	2 (1.414216)	Finite Irrational
	4 +213,850	r (0.618034)	Finite Irrational
	5 +240,255	 (0.577214)	???
	Table 3 Computed Run-times for the RNF-based Exponential Function (t_RNF) and Existing Exponential
	Function (t_expex) of x values.
	runs	t_expex	t_RNF
	1		4.4430	4.7550
	2		3.6580	4.4070
	3		3.7990	3.8510
	4		4.4810	3.5560
	5		3.6280	3.6780
	6		3.8270	3.7020
	7		3.6660	4.3410
	8		4.3120	3.7250
	9		4.0290	4.2120
	10		4.2120	3.8490
	Average		4.0055	4.0076
	min		3.628	3.556
	max		4.4810	4.7550


Wissem Al Jedidi

Fethi Bouzaffour
email: fbouzaffour@ksu.edu.sa

Nouf Harthi
email: nharthi@yahoo.com

Wissem Al Jedidi
email: wissem_jedidi@yahoo.fr

REFINED PROPERTIES FOR THE MELLIN TRANSFORM, WITH APPLICATIONS TO CONVERGENCE OF FAMILIES OBTAINED BY BIASING OR BY THE STATIONARY EXCESS OPERATOR

Keywords: monotone functions, Normal limit theorem, Size biasing, Stationary excess operator, Uniform integrability 42A38, 44A60, 60E07, 60E15

whether they are published or not. The documents may come

INTRODUCTION AND MAIN RESULTS

The Mellin transform of a nonnegative random variable X is defined by

M X (λ) = E[X λ ],
for λ in some domain of definition in the complex plane, and could be also interpreted as the moment generating function of log X. We denote D X the domain of definition of M X restricted on the real line.

Laplace transform of nonnegative random variables and characteristic functions of real-values random variables are always well defined, respectively on the half real line and on the real line, and they entirely characterize the distribution. At the contrary, the Mellin transform could have problems of definition and for this reason formalization of its injectivity is not straightforward. Nevertheless, injectivity on the Mellin transform seems to be commonly admitted in the literature and used without a precise reference. In Chapter VI of Widder's book [START_REF] Widder | The Laplace Transform[END_REF], Theorem 6a p. 243, it is stated that if the Mellin transforms of two nonnegative random variables X and Y are well defined on some strip α < Re(z) < β, and equal there, then X d = Y . We improve this result by showing that the same conclusion holds if the strip is replaced by an interval. Date: June 12, 2016.

Mellin transform works well with size biased distributions, i.e. for distributions of random variables weighted by a power function:

X (t) : d = x t E[X t ] P(X ∈ dx), t ∈ D X .
The distribution of the random variable X (t) is called the size biased law of transform of order t of X. It is well know that X (t) is stochastically bigger than X when t > 0 and we were interested by deriving new properties. In the dissertation of the third author [START_REF] Harthi | Some Contributions to Infinite Divisibility through Mellin Transforms and Cumulant Functions of Random Variables[END_REF], some convexity properties of the Mellin transform of X (t) were required and we were addressed to the monotonicity property of the t → M X (λ + t)/M X (t), t ≥ 0 for fixed λ > 0. This made as aware of the works of Harkness and Shantaram [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF] who proved that the last function is non-decreasing in t ∈ N when λ = 1. We prove that is also true in t ∈ [0, ∞) for any λ > 0.

Harkness and Shantaram [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF] were actually motivated by a limit theorem that we explain as follows. Let X a nonnegative random variable having moments of all orders. The stationary excess operator builds a new distribution from the one of X by this means:

P(E 1 (X) > x) = 1 E[X] ∞ x P(X > u) du, x ≥ 0.
The n th iterate E n , of the operator E 1 , builds a sequence of random variables E n (X), n ∈ N. In [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF] it was shown that if the sequence Z n = E n (X)/c n converges in distribution to some random variable Z ∞ and with some deterministic normalization c n such that

lim sup n→∞ c n+1 c n ∈ [1, ∞), (1)
then, necessarily lim n→∞ c n+1 /c n = l ∈ [1, ∞) and lim n→∞ E[Z k n ] = E[Z k ∞ ] ∈ (0, ∞) for every k ∈ N. Many authors were motivated by this problem and studied the set of possible distribution for Z ∞ , see the works of Arratia, Goldstein and Kochman [START_REF] Arratia | Size Bias For One And All[END_REF], van Beek and Braat [START_REF] Van Beek | The limits of sequences of iterated overshoot distribution functions[END_REF], Garcia [START_REF] Garcia | Characterization of distributions with the length-bias scaling property[END_REF], Shantaram and Harkness [START_REF] Shantaram | On a Certain Class of Limit Distributions[END_REF], Pakes [START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF][START_REF] Pakes | Characterizations, length-biasing, and infinite divisibility[END_REF], Vardi, Shepp and Logan [START_REF] Vardi | Distribution functions invariant under residual-lifetime and length-biased sampling[END_REF] for instance. Their approach was mainly based on the fact that the distribution of Z ∞ necessarily satisfies

P(Z ∞ ≤ x) = 1 E[Z ∞ ] lx 0 P(Z ∞ > x) du, for every x ≥ 0.
The latter is equivalent to the identity in law observed in Theorem 3 below

Z ∞ d = U l (Z ∞ ) (1) , (2)
for s = 1 and c = log l, and U l is uniformly distributed over [0, l -1 ] and independent from (Z ∞ ) [START_REF] Aguech | Complete monotonicity and Bernstein properties of functions are characterized by their restriction on N 0[END_REF] . We will see that the only significant information one can extract from identity [START_REF] Arratia | Size Bias For One And All[END_REF] is that Z ∞ has the same integer moment as an exponential distribution multiplicatively mixed by the Log-normal distribution with parameters depending on l and on the value attributed to

E[Z ∞ ].
There is then a problem of determinacy in law for Z ∞ since the Log-normal distribution is well known to be moment indeterminate.

Also motivated by this problem of indeterminacy, we were concerned with the natural question: what additional information on the distribution of Z ∞ can we obtain if we study the continuous scheme Z t = E t (X)/c t , t ∈ [0, ∞) where E t is the continuous time stationary-excess operator given by

P(E t (X) > x) = t E[X t ] ∞ x (u -x) t-1 P(X > u) du, x ≥ 0 ? (3)
Notice that the limit Z ∞ obtained by the discrete scheme has necessarily the same distribution as the one of the continuous scheme.

In section 4, we make a digression and entirely formalize convergence of the Mellin transforms of general families of random variables (in both discrete and continuous time i.e. t ∈ N and t ∈ [0, ∞)) through convergence in distribution and uniform integrability. The form (3), justifies our investigation on t-monotone functions in section 5. The results obtained in both sections 4 and 5 will allow us to contribute in section 6 to the problem raised by Harkness and Shantaram [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF]. We simplify their problem and solve it as follows:

(1) condition ( 1) is not only sufficient, but also necessary for the convergence in distribution of Z t if we require some integrability on Z ∞ ; (2) under (1), convergence of Z t in both schemes t ∈ N or t ∈ [0, ∞), is equivalent to

X (t) ρ t d -→ X ∞ as t → ∞ and t ∈ N or t ∈ [0, ∞),
the normalization ρ t being necessarily equivalent to t c t at infinity;

(3) it holds that Z ∞ d = e X ∞
where e is independent from X ∞ and is exponentially distributed; (4) the only possible distributions for the limit

X ∞ is Log-normal.
In what follows, we only deal with nonnegative random variables. The notation T stands for N, the set of nonnegative integers, or for the interval [0, ∞).

DEFINITENESS, MONOTONICITY AND INJECTIVITY OF THE MELLIN TRANSFORM

The Mellin transform of a nonnegative random variable X is defined by

λ → E[X λ ], λ ∈ D X = {x ∈ R ; E[X x ] < ∞}.
We recall Hölder inequality true for every real random variables

E[|U V |] ≤ E[|U | p ] 1 p E[|V | q ] 1 p , p, q > 0, 1 p + 1 q = 1, (4)
whenever the expectations are finite. The equality holds in (4) if and only if there exist constants a, b ≥ 0, not both zero, such that a|U | p = b|V | q . It is then clear that for a positive random variable X, the standard Lyapunov inequality holds:

E[X λ ] 1 λ ≤ E[X λ 0 ] 1 λ 0 for 0 < λ ≤ λ 0 and E[X µ ] 1 µ ≤ E[X µ 0 ] 1 µ 0 for µ 0 ≤ µ < 0, (5)
whenever the expectations are finite. The latter justifies that if D X contains some λ 0 > 0 (respectively some µ 0 < 0), then D X contains the interval [0, λ 0 ] (respectively [µ 0 , 0]). It is then seen that D X is an interval with extremities

µ X = inf{λ ∈ R, E[X λ ] < ∞} and λ X = sup{λ ∈ R, E[X λ ] < ∞}, (6)
not necessarily included. Assume λ X > 0. By the dominated convergence theorem, we see that the Mellin transform of X is often differentiable on (0, λ X ) and by ( 5) that

λ → E[X λ ] 1/λ is nondecreasing on [0, λ X ).
The last fact could be also seen as a consequence of this Proposition:

Proposition 1. For any nonnegative random variable X such that λ X > 0, the Mellin transform M X is log-convex on [0, λ X ]. If furthermore X is non-deterministic, then strict log-convexity holds.

Proof. Let g(λ

) := log E[X λ ]. Trivial computations lead to g (λ) = E[X λ (log X) 2 ] E[X λ ] -E[X λ log X] 2 E[X λ ] 2 , 0 < λ < λ X .
Taking p = q = 2, U = X λ 2 log X and V = X λ 2 in (4), we deduce that g is convex. It is strictly convex unless U and V are proportional which is equivalent to X deterministic.

Proposition 1 gives an additional information:

Corollary 1. Let a nonnegative random variable X such that λ X > 0. For every λ ∈ (0, λ X ), the function t → M X (λ + t)/M X (t) is nondecreasing on [0, λ X -λ). It is further increasing whenever X is non-deterministic.
Proof. Theorem 5.1.1 [START_REF] Webster | Convexity[END_REF] p. 194 says that convexity of x → g(x) = log M X (x) yields that its slopes are nondecreasing:

g(y) -g(x) y -x ≤ g(z) -g(x) z -x ≤ g(z) -g(y) z -y , 0 ≤ x < y < z < λ X .
Then, g(λ + s) -g(s) ≤ g(λ + t) -g(t), for 0 ≤ s < t and λ + t < λ X , (7) and the first assertion is proved. For the strict monotonicity, notice that equality holds in [START_REF] Harthi | Some Contributions to Infinite Divisibility through Mellin Transforms and Cumulant Functions of Random Variables[END_REF] only in case where the function r → g(λ + r) -g(r) is not injective. Because of the differentiability of g, the latter reads g (λ + r) = g (r) for some value of r. The latter is possible only if g is not injective, that is g (x) = 0 for some value of x and the second statement in Proposition 1 allows to conclude.

Our result in Corollary 1 is the same than the one stated in Lemma 3.1 in [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF] when λ and t are positive integers. Corollary 1 is also proved in [START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF], where the author also adapts the arguments of Lemma 3.1 in [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF]. However, we found that the argument of continuity used in [START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF], appealing to a result of Kingman [START_REF] Kingman | Ergodic properties of continuous-time Markov processes and their discrete time skeletons[END_REF], does not fit his context. We clarify the approach of [START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF] with this second proof:

Second proof of Corollary 1. We first show that the sequence

u n = M X (nu) satisfies u n+m u n ≤ u n+m+1 u n+1 , for every n, m ∈ N, u > 0. (8)
Schwarz inequality (4), with

p = q = 2, gives E[X d(n+1) ] 2 ≤ E[(X u(n+2) ] E[X un ]. Then u n+1 u n ≤ u n+2 u n+1 , for every n ∈ N,
which is also equivalent to (8) from which we deduce that for each u > 0 and m ∈ N, the sequence

n → E[X u(n+m) ] E[X un ] is nondecreasing. (9)
Now, take λ > 0 and t > s > 0 with s, t rationals of the form s = p/q and t = k/l so that pl < qk.

Applying [START_REF] Lefèvre | On multiply monotone distributions, continuous or discrete, with applications[END_REF] with u = λ ql , we obtain the inequality

E[X λ(s+1) ] E[X λs ] = E[X λ ql (pl+ql)) ] E[X λ ql pl ] ≤ E[X λ ql (qk+ql) ] E[X λ ql qk ] = E[X λ(t+1) ] E[X λt ] .
By continuity of the Mellin transform, we deduce that

E[X λ(s+1) ] E[X λs ] ≤ E[X λ(t+1) ] E[X λt ]
, for all λ > 0 and all real numbers s, t s.t. 0 < s < t.

The proof is finished by replacing the couple (s, t) by ( s λ , t λ ). Strict monotonicity is obtained as in the end of the first proof.

We found that in the literature, many papers invoke the injectivity of the Mellin transform without a precise reference. For instance, an informal discussion in exercise 1.13 in [START_REF] Chaumont | Exercises in Probability, A Guided Tour from Measure Theory to Random Processes[END_REF] appeals to Chapter VI in Widder's book [START_REF] Widder | The Laplace Transform[END_REF] where we found Theorem 6a p. 243 stating the following:

If the Mellin transforms of two nonnegative random variables X and Y are well defined on some strip α < Re(z) < β, and equal there, then

X d = Y .
Widder's theorem could be improved by the following Lemma: Lemma 1. Let X and Y two nonnegative random variables such that their Mellin transforms are well defined on some interval (α, β) ⊂ R and equal there, then

X d = Y .
The proof of this Lemma is a technique borrowed from [START_REF] Aguech | Complete monotonicity and Bernstein properties of functions are characterized by their restriction on N 0[END_REF] and based on a Blaschke's theorem that allows to identify holomorphic functions given their restriction along suitable sequences: Theorem 1 (Blaschke, Corollary p. 312 in Rudin [START_REF] Rudin | Real And Complex Analysis[END_REF]). If f is holomorphic and bounded on the open unit disc

D, if z 1 , z 2 , z 3 , • • • are the zeros of f in D and if ∞ k=1 (1 -|z k |) = ∞, then f (z) = 0 for all z ∈ D.
Using the one-to-one mapping of the strip

S = {z ∈ C, 0 < Re(z) < 1} onto the open unit disc z → θ(z) = e i π 2 z -i e i π 2 z + i
, one can easily rephrase Blaschke's theorem for function defined on the strip S:

Corollary 2. Two holomorphic functions on the strip S are identical if their difference is bounded and if they coincide along a sequence

α 1 , α 2 , α 3 , • • • in S, such that the series k 1 -e i π 2 α k -i e i π 2 α k +i diverge.
For instance, the series diverge for the sequence

α k = 1 k , k ≥ 1.
Proof of Lemma 1. It is enough to take (α, β) = (0, 1), to notice that both M X and M Y extend holomorphically on the strip S and to conclude with Corollary 2 since M X and M Y coincide along the sequence α k = k -1 , k ≥ 2 which is contained in (0, 1).

MELLIN TRANSFORM AND SIZE BIASED LAWS

Let X denote a non-deterministic nonnegative random variable. For t ∈ D X , the size biased law of order t is denoted X (t) and is a version of the weighted law

P(X (t) ∈ dx) = x t E[X t ] P(X ∈ dx), x ≥ 0. (10)
Chebychev's association inequality says that

E[f (X)g(X)] ≥ E[f (X)] E[g(X)],
whenever the expectations are well defined and f, g are both nondecreasing or nonincreasing realvalued functions. Taking

f (u) = u t , g(u) = 1l u>x , we see that X (t) ≥ st X for t ≥ 0, i.e. P(X (t) > x) = E[X t 1l X>x ] E[X t ] ≥ P(X > x), ∀t, x ≥ 0 . (11)
Notice that the last stochastic inequality also justifies Corollary 1, since the Mellin transform could be computed as

E[X λ ] = λ ∞ 0 x λ-1 P(X > x) dx , λ ∈ (0, λ X ), (12)
whenever the extremity λ X given by ( 6) is positive.

We list some elementary properties for the size biased law of a r.v. X:

(P0) For every c > 0 and t ∈ D X , we have

X (0) = X and (cX) (t) = cX (t)
(P1) For every λ, t such that t, t + λ ∈ D X and measurable bounded function g, we have

E[X λ (t) ] = E[X t+λ ] E[X t ] , E[g(X (t) )] = E[X t g(X)] E[X t ] .
(P2) For every s, t ∈ D X such that t + s ∈ D X , we have

X (s) (t) d = X (s+t) d = X (t) (s) .
(P3) For every s, t such that st ∈ D X , we have

(X s ) (t) d = (X (st) ) s .
(P4) For every independent random variables X, Y and t ∈ D X ∩ D Y , we have

(XY ) (t) d = X (t) Y (t)
(assuming that X (t) and Y (t) are independent).

CONVERGENCE OF SEQUENCES AND FAMILIES OF MELLIN TRANSFORMS

This section contains results dealing with sequences or families of Mellin transforms. As we did for the injectivity in Lemma 1, we felt it was important to also clarify the notion of convergence via Mellin transform. Next Theorem 2 and Proposition 3, will be crucial for handling section 6 below.

We recall that T = N or [0, ∞). In what follows, (1) a property (P t ) is said to be true for t big enough, if there exists t 0 ∈ T such that (P t ) is true for t ≥ t 0 ; (2) (X t ) t∈T denotes a family of nonnegative random variables indexed by the time t ∈ T;

(3) by a subsequence of (X t ) t∈T , we mean a collection of random variables (X t(n) ) n∈N obtained through a nondecreasing function t : N → T such that t(n) → ∞ as n → ∞; (4) we always assume that for t big enough,

λ Xt = sup{λ ∈ R, E[X λ t ] < ∞} > 0; (5) for λ ≥ 0, we define informally m(λ) := lim inf t∈T E[X λ t ] and M (λ) := lim sup t∈T E[X λ t ]. (13)
We also recall some basic ingredients related to convergence in distribution.

Definition 1 (Billingsley [4]). Let a sequence (X n ) n∈N of real-valued random variables.

(i) (X n ) n∈N is called tight if sup n∈N P(|X n | > x) → 0 as x → ∞. (ii) (X n ) n∈N is called uniformly integrable if sup n∈N E[|X n |1l Xn>x ] → 0 as x → ∞. (iii) X n converges in distribution to X ∞ if E[f (X n )] → E[f (X ∞ )],
as n → ∞, for every continuous bounded (or continuous compactly supported) real function f .

We are going to study convergence in distribution for families of nonnegative random variables. For this purpose, we slightly generalize Definition 1 in order to get more flexibility. Definition 2. Let family (X t ) t∈T a family of nonnegative random variables.

(i) We say that the family (X t ) t∈T ultimately tight if

lim sup t∈T P(X t > x) → 0, as x → ∞.
(ii) We say that the family (X t ) t∈T is λ-uniformly integrable, and we denote

(X t ) t∈T is λ -U I, if λ ∈ (0, λ Xt ) for t big enough and lim sup t∈T E[X λ t 1l Xt>x ] → 0, as x → ∞.
(iii) We say that the family (X t ) t∈[0,∞) converge in distribution, if every subsequence (X t(n) ) n∈N converges in distribution in the usual sense (iii) of the preceding Definition.

Remark 1. We can notice that: a) λ-uniform integrability of (X t ) t≥0 is equivalent to 1-uniform integrability of (X λ t ) t≥0 . b) If T = N, then Definitions 1 and 2 are equivalent 0λ Xt ∈ (0, ∞) for every t ∈ N. In general, this is untrue if T = [0, ∞). c) If T = [0, ∞), then (X t ) t≥0 is ultimately tight (respectively λ -U I) if and only if there exists some positive t 0 , big enough, such that (X t ) t≥t 0 is tight (respectively (X λ t ) t≥t 0 is uniformly integrable) in the same sense that Billingsley gave in Definition 1.

We start this section with the following result that clarifies the link between ultimate tightness and uniform integrability: Proposition 2. Let λ 0 > 0 and (X t ) t∈T a family of nonnegative random variables. Recall the function m(.) and M (.) are given by [START_REF] Rao | Weighted distributions arising out of methods of ascertainment: What population does a sample represent?[END_REF].

1) If (X t ) t∈T is λ 0 -uniformly integrable, then M (λ 0 ) < ∞.
2) If M (λ 0 ) < ∞, then (X t ) t∈T is ultimately tight and also λ-uniformly integrable for every λ ∈ (0, λ 0 ).

3) Assume m(λ 0 ) > 0 and M (λ 0 + ) < ∞, for some > 0. Then λ 0 -uniform integrability of (X t ) t∈T is equivalent to its ultimate tightness.

Proof. 1) Write M (λ 0 ) ≤ x λ + lim sup t∈T E[X λ t 1l Xt>x ],
for x big enough and deduce that M (λ 0 ) < ∞.

2) Hölder and Markov inequalities give

lim sup t∈T E[X λ t 1l Xt>x ] ≤ M (λ 0 ) λ λ 0 lim sup t∈T P(X t > x) λ 0 -λ λ 0 ≤ M (λ 0 ) x λ 0 -λ , 0 < λ < λ 0 , (14)
Ultimate tightness and λ-uniform integrability are then immediate.

3) Inequality [START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF] gives

E[X λ 0 t ] P(X t > x) ≤ E[X λ 0 t 1l Xt>x ],
for all x > 0 and t ∈ T.

Using again inequality [START_REF] Rudin | Real And Complex Analysis[END_REF], with the couple (λ, λ 0 ) replaced by (λ 0 , λ 0 + ), obtain

m(λ 0 ) lim sup t∈T P(X t > x) ≤ lim sup t∈T E[X λ 0 t 1l Xt>x ] ≤ M (λ 0 + ) λ 0 λ 0 + lim sup t∈T P(X t > x) λ 0 + .
Next Theorem rephrases and improves some results borrowed from the monograph of Billingsley [START_REF] Billingsley | Probability and Measure[END_REF]:

Theorem 2. Let (X t ) t∈T a family of nonnegative random variables such that λ 0 ∈ (0, λ Xt ), for some λ 0 > 0 and t big enough.

1) Let X ∞ a nonnegative random variable. The following assertions are equivalent, as t → ∞:

(i) X t d -→ X ∞ and (X t ) t∈T is λ 0 -uniformly integrable; (ii) X t d -→ X ∞ , λ 0 ∈ D X∞ and E[X λ 0 t ] → E[X λ 0 ∞ ] ; (iii) λ 0 ∈ D X∞ and for every λ ∈ [0, λ 0 ], E[X λ t ] → E[X λ ∞ ]. 2) Let λ 1 ∈ (0, λ 0 ) and assume that E[X λ t ] converges as t → ∞ to a well defined function f (λ), λ ∈ [λ 1 , λ 0 ]. Then (iii) holds and f is well defined on [0, λ 0 ] by f (λ) = E[X λ ∞ ].
Proof. The proof is conducted by reasoning on subsequences.

1) (i) =⇒ (iii): it is a direct application of Theorem 25.12 p. 338 in [START_REF] Billingsley | Probability and Measure[END_REF], using Remark 1 and Proposition 2.

(iii) =⇒ (ii) is treated as follows: Since M (λ 0 ) < ∞, then by Proposition 2, the family (X t ) t∈T is ultimately tight. Lemma 1 insures that any subsequence (X t(n) ) n∈N , if converging in distribution as n → ∞, necessarily converge to the law of X ∞ . Corollary in [START_REF] Billingsley | Probability and Measure[END_REF] p.337 allows to conclude that

X t d -→ X ∞ as t → ∞.
(ii) =⇒ (i): we use the following representation valid for any nonnegative random variables Z such that E[Z λ ] < ∞:

E[Z λ 1l Z≤x ] = x λ P(Z ≤ x) -λ x 0 u λ-1 P(Z ≤ u) du, x ≥ 0.
Choose x 0 a continuity point of u → P(X ∞ ≤ u). By the dominated convergence theorem, we have

lim t→∞ E[X λ t 1l Xt≤x 0 ] = lim t→∞ x λ P(X t ≤ x) -λ x 0 0 u λ-1 P(X t ≤ u) du = x λ P(X ∞ ≤ x 0 ) -λ x 0 0 u λ-1 P(X ∞ ≤ u) du = E[X λ ∞ 1l X∞≤x 0 ] Since lim t→∞ E[X t ] = E[X ∞ ], we also have lim t→∞ E[X λ t 1l Xt>x 0 ] = E[X λ ∞ 1l X∞>x 0 ], that is, for every > 0, there exists t 0 ∈ T such E[X λ t 1l Xt>x 0 ] -E[X λ ∞ 1l X∞>x 0 ] < . Now choose > 0, then x 0 big enough so that E[X λ ∞ 1l X∞>x 0 ] < . We deduce that E[X λ t 1l Xt>x ] ≤ E[X λ t 1l Xt>x 0 ] < E[X λ ∞ 1l X∞>x 0 ] + < 2 , t ≥ t 0 , x ≥ x 0 .
2) We adapt a part of the proof of Corollary 1.6 p. 5 given in Schilling and al. [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] in the context of convergence of sequences on completely monotone functions. Helly's selection theorem allows a shortcut since there exists a subsequence

(X t(n) ) n∈N satisfying X t(n) d -→ X ∞ , as n → ∞. Fix λ ∈ [λ 1 , λ 0 ]. For every function h : [0, ∞) → [0, 1]
, compactly supported, we find, by Fatou Lemma, that

E[h(X ∞ ) X λ ∞ ] = lim s→∞ E[h(X t(n) ) X λ t(n) ] ≤ lim s→∞ E[X λ t(n) ] = f (λ).
Monotone convergence theorem gives a first inequality [START_REF] Pakes | Characterizations, length-biasing, and infinite divisibility[END_REF] and the dominated convergence theorem, in order to get that for n big enough,

E[X λ ∞ ] = sup h E[h(X ∞ ) X λ ∞ ] ≤ f (λ). Now, fix > 0, choose a continuity point x of the distribution function of X ∞ , then apply the fact that X t(n) d -→ X ∞ , identity
E[X λ t(n) 1l X t(n) ≤x ] -E[X λ ∞ 1l X∞≤x ] = λ x 0 u λ-1 P(u < X t(n) ≤ x) -P(u < X ∞ ≤ x) du ≤ x λ . Since (X t(n) ) n∈N is λ -U I, then E[X λ t(n) 1l X t(n) >x ] < for all x, n big enough. Finally get for all > 0, E[X λ t(n) ] -E[X λ ∞ ] ≤ E[X λ t(n) 1l X t(n) >x ] + E[X λ t(n) 1l X t(n) ≤x ] -E[X λ ∞ 1l X∞≤x ] < (1 + x λ ).
The latter proves the second inequality

f (λ) = lim s→∞ E[X λ t(n) ] ≤ E[X λ ∞ ].
All in one, we have that

f (λ) = E[X λ ∞ ], for every λ ∈ [λ 1 , λ 0 ].
As in point 1) above, notice that the family (X t ) t∈T is ultimately tight, and by Lemma 1, each subsequence of it, if converging in distribution, necessarily converge to the distribution of X ∞ . Use again the Corollary in [START_REF] Billingsley | Probability and Measure[END_REF] p. 337 in order to have X t d -→ X ∞ , as t → ∞. To conclude, use λ-uniform integrability of (X t ) t∈T and then implication (i) =⇒ (ii) in point 1) above. Now, consider two families of nonnegative random variables (U t ) t∈T and

(V t ) t∈T such that U t d -→ U ∞ , V t d -→ V ∞ and U t and V t independent for every t ∈ T. It then is trivial that U t V t d -→ U ∞ V ∞ .
As a consequence of Theorem 2, we deduce a kind of converse: Corollary 3. Let (U t ) t∈T , (V t ) t∈T and (W t ) t∈T three families of nonnegative random variables such that U t and V t are independent for each t ∈ T and such that

(i) the factorizations in law W t d = U t V t holds; (ii) the convergences in distribution W t d -→ U ∞ and V t d -→ V ∞ = 0 hold as t → ∞; (iii) there exists λ 0 > 0 such that (W t ) t∈T is λ 0 -U I or such that lim t∈T E[W λ 0 t ] < ∞. Then, U t d -→ U ∞ ,
where the distribution of the random variable U ∞ is well defined by its Mellin transform given by E

[U λ ∞ ] = E[W λ ∞ ]/ E[V λ ∞ ], for every λ ∈ [0, λ 0 ].
Proof. Notice that λ 0 ∈ D Wt = D Ut ∩ D Vt for t big enough and that both conditions in (iii) are equivalent by Theorem 2. Choose v 0 > 0 a continuity point of x → P(V ∞ > x) such that P(V ∞ > v 0 ) > 1/2 and notice also that for every t ∈ T and x ≥ 0,

E[W λ 0 t 1l Wt>x ] ≥ E[(U t V t ) λ 0 1l Ut> x v 0 , Vt>v 0 ] = E[U λ 0 t 1l Ut> x v 0 ] E[V λ 0 t 1l Vt>v 0 ] ≥ E[U λ 0 t 1l Ut> x v 0 ] v λ 0 0 P(V t > v 0 ).
Then use the fact that there exists t 0 ∈ T such that

P(V t > v 0 ) > P(V ∞ > v 0 ) -1/4 > 1/4 for t ≥ t 0 and then E[W λ 0 t 1l Wt>x ] ≥ v 0 4 E[U λ 0 t 1l Ut> x v 0 ], t ≥ t 0 , x ≥ 0.
The latter yields that the family (U t ) t∈T is λ 0 -U I, then apply Theorem 2.

Next proposition studies the convergence of biased laws and improves Theorem 2.3 in [2]: Proposition 3. Let (X t ) t∈T a family of nonnegative random variables such that X t converges in distribution to a non-null random X ∞ . Suppose 0 < λ 0 < min(λ Xt , λ X∞ ) for t big enough and

lim t→∞ E[X λ 0 t ] → E[X λ 0 ∞ ]
. Then, we have the convergence of the size biased distributions of (X t ) t∈T :

(X t ) (λ) d -→ (X ∞ ) (λ) , as t → ∞, for every λ ∈ [0, λ 0 ]. (15)
Proof. a) We start by proving [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] for λ = λ 0 . By assumption, we have 0

< E[X λ 0 ∞ ] < ∞. Con- vergence in distribution of X t to X ∞ is equivalent to E[g(X t )] → E[g(X ∞
)] for every continuous, compactly supported function g, as t → ∞. The function h(x) = |x| λ 0 g(x) is also a continuous, compactly supported function. By property (P1), we also have

E[h(X t )] = E[X λ 0 t ] E[g (X t ) (λ 0 ) ] → E[h(X ∞ )] = E[X λ 0 ∞ ] E[g (X ∞ ) (λ 0 ) ].
The limit [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] for λ = λ 0 follows by simplification in both sides of the last limit. b) By Proposition 2, notice that (X t ) t∈T is λ -U I for every λ ∈ (0, λ 0 ). Deduce by Theorem 2 that

lim t→∞ E[X λ t ] → E[X λ ∞ ]
for and reproduce step a) for λ ∈ [0, λ 0 ). Remark 2. By Theorem 2, finiteness of the quantity M (λ 0 ) given by ( 13), or λ 0 -uniform integrability of (X t ) t∈T is sufficient to insure that λ 0 ∈ D X∞ .

t-MONOTONE DENSITY FUNCTIONS

Let x + denotes max{0, x}, x ∈ R. The following definition extends the one of Schilling et al. [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] given for t nonnegative integer.

Definition 3. Let t ∈ (0, ∞). A function f : (0, ∞) → [0, ∞) is t-monotone if it is represented by f (x) = c + (0,∞) (u -x) t-1 + ν(du), x > 0 (16)
for some c ≥ 0 and some measure ν on (0, ∞).

Remark 3. When t = 1, representation ( 16) holds if and only if f is nonincreasing and rightcontinuous. When t = n is an integer greater than or equal to 2, representation ( 16) holds if and only if f is n -2 times differentiable, (-1) j f (j) (x) ≥ 0 for all j = 0, 1, • • • , n -2 and x > 0, and (-1) n-2 f (n-2) is nonincreasing and convex. Furthermore, by Theorem 1.11 p.8, [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF]), the couple (c, ν) in ( 16) uniquely determines f .

A random variable b a,b is said to have the beta distribution with parameter (a, b), a, b > 0, if it has the density function

1 β(a, b) x a-1 (1 -x) b-1 , x ∈ (0, 1) with β(a, b) = Γ(a)Γ(b) Γ(a + b) .
A random variable g a is said to have the Gamma distribution with parameter a > 0, if it has the density function Γ(a) -1 x a-1 e -x , x ∈ (0, ∞). It is well known that

In the sequel, we denote b t the random variable defined by b 0 = 1 and b 1,t if t > 0, that is b t has the density function t(1 -x) t-1 , x ∈ (0, 1), t > 0.

Also, e = g 1 denotes a random variable with standard exponential distribution. It is clear that

b t d = 1 -e -e t
and that tb t d -→ e, as t → ∞.

We propose the following characterization for t-monotone densities.

Proposition 4. Let t ∈ (0, ∞).

1) The density function f : (0, ∞) → [0, ∞) of a positive random variable Z, is t-monotone, if and only if there exists a positive random variable Y t such that f is represented

f (x) = t (0,∞) 1 - x u t-1 + P(Y t ∈ du) u , x > 0. ( 20
)
This equivalent to the factorization in law

Z d = b t Y t
, where b t has the beta distribution as in ( 19) and is independent from Y t .

2) If f is t-monotone, then it is also s-monotone for every s ∈ (0, t).

3) Furthermore, the ν-measure associated to f through ( 16) is finite if and only if there exists a positive random variable X such that Y t has the same distribution as the size biased random variable X (t) given by [START_REF] Kingman | Ergodic properties of continuous-time Markov processes and their discrete time skeletons[END_REF] i.e.

Z d = b t X (t) . (21)
Remark 4. Bernstein characterization for completely monotone functions says that a function f : (0, ∞) → R is n-monotone for every n ∈ N, if and only if it is represented as the Laplace transform of a (unique) Radon measure ν on [0, ∞):

f (x) = [0,∞)
e -xu ν(du), λ > 0 (respectively x ≥ 0);

When f is a density function associated to a positive random variable Z, the latter is equivalent to

Z d = eY
where Y is positive and independent from the exponentially distributed random variable e and also to f (x) = E[e -x/Y /Y ], x > 0. In this case, Bernstein characterization for f could be reinterpreted as follows:

-use the Beta-Gamma algebra [START_REF] Steutel | Infinite divisibility of probability distributions on the real line[END_REF] in order to write

Z d = e d = b t g t Y, P(Z > x) = E 1 - x g t Y t +
, for every t x > 0;

(23) -use the fact that (1 -x t ) t + → e -x uniformly in x > 0, as t → ∞, and that gt t d -→ 1, then rephrase (23) as:

P(Z > x) = lim n→∞ E 1 - t g t x tY n + = E[e -x/Y ] = P(e Y > x), for every x > 0.
This clarifies the discussion made right after Proposition 2.2 in [START_REF] Lefèvre | On multiply monotone distributions, continuous or discrete, with applications[END_REF].

Proof of Proposition 21. 1) The density function f is of the form ( 16) if and only if c = 0 and

f (x) = (0,∞) (1 - x u ) t-1 + u t-1 ν(du), x > 0, some measure ν on (0, ∞) such that 1 = ∞ 0 f (x)dx = (0,∞) t -1 u t ν(du) so that t -1 u t ν(dx)
is a probability measure associated to some random variable, say Y t . The second assertion is due to the fact that the density of the independent product of a non negative random variables U and V such that U has a density function f U is given by the Mellin convolution

f U V (x) = (0,∞) f U ( x y ) P(V ∈ dy) y , x > 0.
2) It is enough to use the Beta-algebra ( 17)

: b t d = b s b 1+s,t-s . 3) E[Y -t t ] = t -1 ν(0, ∞) < ∞ is equivalent to the identity Y t d = X (t)
where X has the distribution ν(0, ∞) -1 ν(x).

APPLICATION: REFINEMENT OF THE RESULTS OF HARKNESS AND SHANTARAM [8]

Recall that T = N or [0, ∞). In what follows X is a nonnegative random variable, non identically null, such that [0, ∞) ⊂ D X i.e. X has moments of all positive orders.

We are willing to obtain a limit theorem for the family obtained by size biasing the distribution of X. Identity (21) suggests to introduce a family of random variables (E t (X)) t∈T , such that each E t (X) has its distribution defined through an operator E t stemming from the identity

E t (X) : d = B t X (t) , with with B t independent from X (t) . (24)
By property (P2) and identity [START_REF] Steutel | Infinite divisibility of probability distributions on the real line[END_REF] for Beta distributions, notice that the family (E t ) t∈T forms a semigroup of commuting operators:

E t (E s (X)) d = E t (E s (X)) d = E t+s (X), s, t ∈ T.
By simple computations, we obtain that identity (24) is equivalent to one of the following expressions for the Mellin transform or for the distribution function of E t (X):

E[E t (X) λ ] Γ(λ + 1) = Γ(t + 1) E[X t ] E[X λ+t ] Γ(λ + t + 1) = Γ(t + 1) Γ(λ + t + 1) E X λ (t) , λ ≥ 0, P(E t (X) > x) = E (1 - x X (t) ) t + = E[(X -x) t + ] E[X t ] = t [X t ] ∞ x (u -x) t-1 P(X > u) du, x ≥ 0, so that P(E 1 (X) ≤ x) = 1 E[X] x 0 P(X > u) du, x ≥ 0.
It is then clear that the operator E 1 corresponds to the stationary excess operator studied by Harkness and Shantaram [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF] and also in [START_REF] Shantaram | On a Certain Class of Limit Distributions[END_REF][START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF][START_REF] Pakes | Characterizations, length-biasing, and infinite divisibility[END_REF][START_REF] Vardi | Distribution functions invariant under residual-lifetime and length-biased sampling[END_REF]. It is also seen that the operator E n corresponds the n-th iterate by the composition of E 1 :

E n+1 = E 1 • E n , n ∈ N {0}.
Harkness and Shantaram [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF] solved the discrete time problem (T = N) of:

-finding a deterministic normalization speed c t , t ∈ N, and sufficient conditions such that

Z t := E t (X) c t d -→ Z ∞ as t ∈ N and t → ∞. (25)
-describing the set of possible distributions for Z ∞ .

It is natural to study what kind of additional information we can recover from the continuous time problem, i.e. convergence (25) in case T = [0, ∞) instead of T = N, and to find the necessary and sufficient conditions such that

Z t d = 1 c t b t X (t) d -→ Z ∞ when t ∈ T and t → ∞. (26)
A direction for solving this problem is given by ( 19): we have that tb t d -→ e as t → ∞. Take in Corollary 3

(U t , V t , W t ) = (t b t , X (t) ρ t , Z t ), with ρ t = t c t and assume E[Z λ 0
∞ ] is finite for some λ 0 ∈ T {0}. Under the last assumption, it could be noticed that problem (26) is equivalent to finding necessary and sufficient conditions on the deterministic and positive normalization speed ρ t , such that

X t := X (t) ρ t d -→ X ∞ when t ∈ T and t → ∞, (27)
and such that E[X

λ 0 ∞ ] is finite for some λ 0 ∈ T {0}. The random variable Z ∞ in (26) is then linked to X ∞ by Z ∞ d = e X ∞
, where e is exponentially distributed, independent from X ∞ .

(

Theorems 3 and 4 below, improve the discrete time problem in (26, case T = N) studied by Harkness and Shantaram [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF], by giving a sharper answer through the continuous time problem in (26, 27, case T = [0, ∞)).

Theorem 3 (A normal limit theorem). Let (X t ) t∈T the family given by (27).

1) Assertions (i)-(ii)-(iii) are equivalent as t → ∞:

(i) X t converges in distribution to a non-null and nonnegative random variable X ∞ and

lim t→∞ E[X λ 0 t ] = E[X λ 0 ∞ ] < ∞, for some λ 0 ∈ T {0};
(ii) X t converges in distribution to a non-null and nonnegative random variable X ∞ and lim sup t→∞ ρ t+s ρ t < ∞, for some s ∈ T {0};

(iii) there exists a non-null and nonnegative random variable

X ∞ such that [0, ∞) ∈ D X∞ and E[X λ t ] → E[X λ ∞ ], for all λ ∈ [0, ∞). 2) In this case, necessarily, ρ t ∼ +∞ E[X ∞ ] E[X t+1 ]/ E[X t ]
and there exists c ≥ 0 such that l(s) := lim t→∞ ρ t+s ρ t = e cs , for every s ∈ T.

3) Assume one of the equivalent assertions in 1) and let c given by (29). Choose a := log E[X ∞ ] and let N a random variable normally distribution with mean a -c 2 and variance c (it is understood that N = a when c = 0). Then the following holds:

(i) if T = N, then the law of the random variable X ∞ is not determined by its integer moments, we only have 30) holds for all λ ∈ [0, ∞).

E[X λ ∞ ] = E[e λN ] = e (a-c 2 )λ+ c 2 λ 2 , for all λ ∈ N; (30) (ii) if T = [0, ∞), then log X ∞ d = N , i.e. (
(iii) We have the identity in law

(X ∞ ) (s) d = e cs X ∞ , for every s ∈ T. (31)
Proof. Using properties (P1) and (P2), we start by noticing the following identity valid for every t, s, µ ∈ T and x ≥ 0:

E[X s+µ t 1l Xt>x ] = E X (t) ρ t λ 1l X (t) >xρt = E[X t+s+µ 1l X>xρt ] ρ λ t E[X t ] = E[X t+s ] ρ λ t E[X t ] E[X µ (t+s) 1l X (t+s) >xρt ].
In particular, for every t, s, µ ∈ T and x, y ≥ 0, we have

E[X s+µ t 1l Xt>x ] = E[X s t ] ρ t+s ρ t µ E[X µ t+s 1l X t+s >x ρ t ρ t+s ] (32) E[X s t 1l Xt>x ] = E[X s t ] P(X t+s > x ρ t ρ t+s ) (33) E[X s t 1l Xt≤y ] = E[X s t ] P(X t+s ≤ y ρ t ρ t+s ) (34)
If X t converges in distribution to X ∞ and if z is a continuity point of u → P(X ∞ ≤ Ku), then for every ∈ (0, 1), there exists t z ∈ T such that

| P(X ∞ ≤ Kz) -P(X t+s ≤ z)| = | P(X t+s > z) -P(X ∞ > Kz)| < , for all t ≥ t z . (35)
1)(iii) =⇒ (ii) is easy, since by (32) with x = 0, we have

E[X s+µ ∞ ] = lim t→∞ E[X s+µ t ] = E[X s ∞ ] E[X µ ∞ ] lim t→∞ ρ t+s ρ t µ for every s, µ ∈ T {0}. (36)
(ii) =⇒ (iii) Here s ∈ T {0} is fixed and we proceed through two steps:

Step 1: we know that there exits K > 0, t s ∈ T such that

ρ t /ρ t+s ≥ K, for t ≥ t s . (37)
The last inequality combined with identities (33) and (34) give that for every t ≥ t s , x, y > 0,

E[X s t 1l Xt>x ] ≤ E[X s t ] P(X t+s > Kx) and y s ≥ E[X s t ] P(X t+s ≤ Ky). (38)
Now we choose = 1/4 in (35) with a continuity point z = y 0 such that P(X ∞ ≤ Ky 0 ) > 1/2. We get by the second inequality in (38) that

y s 0 ≥ E[X s t 1l Xt≤y 0 ] = E[X s t ] P(X ∞ ≤ Ky 0 ) - 1 4 > 1 4 E[X s t ], t ≥ t y 0 .
We deduce that C(y 0 ) := sup t≥ty 0 E[X s t ] < ∞. The first inequality in (38) gives E[X s t 1l Xt>x ] ≤ C(y 0 ) P(X t+s > xK), t ≥ t y 0 , x > 0.

The next step is to choose arbitrary small in (35) with a continuity point z = x 0 , big enough, so that P(X ∞ > Kx 0 ) < in order to have for t ≥ max(t s , t x 0 , t y 0 ) and x ≥ x 0 , that

E[X s t 1l Xt>x ] ≤ E[X s t 1l Xt>x 0 ] ≤ C(y 0 ) P(X t+s > xK) ≤ C(y 0 ) P(X ∞ > x 0 K)+ ≤ (1+C(y 0 )
) . The latter justifies that (X t ) t∈T is s-uniformly integrable.

Step 2: By (32) and (37), we have for every t ≥ t s ,

E[X 2s t 1l Xt>x ] = E[X s t ] ρ t+s ρ t s E[X s t+s 1l X t+s >x ρ t ρ t+s ] ≤ E[X s t ] ρ t+s ρ t s E[X s t+s 1l X t+s >Kx ].
In step 1, we gained that the family M (s) = lim sup t∈T E[X s t ] < ∞ that (X t ) t∈T is s -U I. The last inequality shows that (X t ) t∈T is also 2s -U I. Repeating the procedure, we obtain that (X t ) t∈T is also ms -U I for every positive integer m and then, by Proposition 2, (X t ) t∈T is also λ -U I for every positive number λ > 0. Then we apply Theorem 2.

(iii) =⇒ (i) is an immediate application of point 1) in Theorem 2.

(i) =⇒ (iii): Identity (32) shows that

E[(X t ) µ (λ 0 ) ] = ρ t+λ 0 ρ t µ E[X µ t+λ 0
], for every t, µ ∈ T.

By Lemma 1, we deduce that

X t+λ 0 d = ρ t ρ t+λ 0 (X t ) (λ 0 ) .
By Proposition 3, we have that (X t ) (λ 0 ) converges in distribution and that the triplet (U t , V t , W t ) = (X t+λ 0 , (X t ) (λ 0 ) , ρ t+λ 0 /ρ t ) satisfies Corollary 3. We obtain that ρ t+λ 0 /ρ t converges as t → ∞ and then we use (ii) =⇒ (iii).

2) The first claim stems from E[X ∞ ] = lim t→∞ E[X (t) ]/ρ t . For the second claim, notice by Corollary 1, that the function t → ρ t is asymptotically increasing, so that l(s) ≥ 1 for every s ∈ T. By (36), we recover that

l(s) µ = lim t→∞ ρ t+s ρ t µ = E[X s+µ ∞ ] E[X s ∞ ] E[X µ ∞ ]
, for every s, µ ∈ T {0}.

From the symmetry in (39), it is seen that l(s) µ = l(µ) s for every s, µ ∈ T. Taking c = log l(1) ≥ 0, we get representation (29). The latter could be also deduced from Lemma 1 in [START_REF] Aguech | Complete monotonicity and Bernstein properties of functions are characterized by their restriction on N 0[END_REF].

3) By Proposition 3,

(X t ) (s) d -→ (X ∞ ) (s)
for all s ∈ T, as t → ∞, and by properties (P0) and (P3), we obtain

(X t ) (s) d = ρ t+s ρ t X (t+s) d -→ e cµ X ∞ d = (X ∞ ) (s) .
The latter gives that the Mellin transform λ → M X∞ (λ) = E[X λ ∞ ] is a solution of the functional equation:

h(1) = E[X ∞ ] = e a , h(s + µ) = e csµ h(s) h(µ), for every s, µ ∈ T, (40)
and this could be also from identity (39). The function h 0 (λ) = e cλ(λ-1) solves the equation without the initial condition. Any solution of the form h = h 0 k, has necessary k(s + µ) = k(s) k(µ) for every s, µ ∈ T, so that k(λ) = k(1) λ . Due the initial condition, necessarily k(1) = e a .

If T = N, then identity (31) true for every s ∈ N is equivalent to the same identity with s = 1:

(X ∞ ) (1) d = e c X ∞ . (41)
We stress that identity (41) does not allow to recover the log-normal distribution of X ∞ which is not moment determinate. This situation was studied by many authors, [START_REF] Arratia | Size Bias For One And All[END_REF][START_REF] Van Beek | The limits of sequences of iterated overshoot distribution functions[END_REF][START_REF] Garcia | Characterization of distributions with the length-bias scaling property[END_REF][START_REF] Shantaram | On a Certain Class of Limit Distributions[END_REF][START_REF] Pakes | Characterization by invariance under length-biasing and random scaling[END_REF][START_REF] Pakes | Characterizations, length-biasing, and infinite divisibility[END_REF][START_REF] Vardi | Distribution functions invariant under residual-lifetime and length-biased sampling[END_REF] for instance and all these works were motivated by finding the set or possible limit for the discrete problem (25). We also stress that Harkness and Shantaram [START_REF] Harkness | Convergence of a sequence of transformations of distribution functions[END_REF] only showed, and in case T = N, that condition (iii) in our Theorem 3, implies (i) and (ii) without specifying the distribution of Z ∞ which we know equal in distribution to e X ∞ . Theorem 3 distinguishes between the situation T = N and T = [0, ∞). It is trivial that convergence of the family (X t ) t∈[0,∞) implies that the subsequence (X t ) t∈N converges to the same limit. Theorem 4 below shows that the converse is true and that actually in both discrete and continuous time problems, the only possible limits of normalized biased families are the log-normal distributions. Equivalently, the only possible limits of normalized families obtained by the stationary excess operator are the mixture of the exponential and the log-normal distribution. Theorem 4 (The normal limit theorem improved). The following statements are equivalent: In all cases, D Z∞ and D X∞ necessarily contains [0, ∞) and convergence (29) holds with some c ≥ 0. We also have Z ∞ d = eX ∞ where e and X ∞ are independent and have respectively the standard exponential distribution and the log-normal distribution, i.e., if for every choice of α = E[X ∞ ], the random variable log X ∞ has the normal distribution with mean equal to log α -c 2 and variance equal to c 2 . It is understood that X ∞ = α if c = 0. Furthermore we have the identity in law

Z ∞ d = e -cs b s (Z ∞ ) (s) , for every s ≥ 0
where b s is assumed to be independent from (Z ∞ ) (s) .

Proof. By the discussion before Theorem 3, we know that (i) ⇐⇒ (iii) and that (ii) ⇐⇒ (iv).

(iv) =⇒ (iii) being trivial, it remains to show (iii) =⇒ (iv). By Theorem

ρ n+1 ρ n ,
which shows (42). Last identity in the theorem stems from property (P4), identities (28), ( 18) and then Beta-Gamma algebra identities ( 17):

(Z ∞ ) (s) d = (e X ∞ ) (s) d = g s+1 (X ∞ ) (s)
which yields

e -cs b s (Z ∞ ) (s) d = eX ∞ d = Z ∞ .
In order to provide an example, we recall the following concept: Let g : (0, ∞) → R and ∆ a the difference operator given by ∆ a g(x) := g(x + a) -g(x). The function g is said monotone of order k ∈ N, if it satisfies

(-1) k ∆ a 1 ∆ a 2 • • • ∆ a k g ≤ 0,
for all a 1 , a 2 , • • • a k > 0, k ∈ N {0}.

Assume g is k times differentiable. The discussion in [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] at the end of page 43 indicates, by a mean value theorem argument, that for every x ∈ (0, ∞),

∆ a 1 ∆ a 2 • • • ∆ a k g(x) = g (k) (x + θ 1 a 1 + θ 2 a 2 + • • • θ k ),
for suitable θ 1 , θ 2 , • • • , θ k ∈ (0, 1). It is then immediate that (-1) k g (k) ≥ 0 implies that g is monotone of order k. By Theorem 4.11 p.42 [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF], g is n-monotone is equivalent to its nonnegativity and monotonicity of all order k = 1, • • • n and is also equivalent to (-1) k g (k) ≥ 0 for all k = 0, • • • n.

We remind that X is a nonnegative random variable such that [0, ∞) ⊂ D X . Now choose a positive quantity α, to be allocated to the value of E[X ∞ ], and define for t ≥ 0

g X (t) = log E[X t ] and ρ t = α E[X (t) ] = α E[X t+1 ] E[X t ] = α exp ∆ 1 g X (t).
We already know that by Proposition 1 that g X is monotone of order 2 (that is g X is convex) and by Corollary 1, we know that t → ρ t in nondecreasing and then the quantity

ρ t+s ρ t = E[X (t+s) ] E[X (t) ] = exp ∆ 1 ∆ s g X (t) (43)
is bounded below by 1. It is clear that if g X is monotone of order 3 (this holds if the derivative g X is concave), then the function t → ρ t+s /ρ t is nonincreasing. The latter implies that lim t→∞ ρ t+s /ρ t exits and is necessarily as in (29). We are now able to provide examples of random variables satisfying (29). For infinite divisibility property of real random variables, the reader is referred to the book of Steutel-van Harn [START_REF] Steutel | Infinite divisibility of probability distributions on the real line[END_REF] and also to [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF]:

Example 1. If X is a random variable such that g X is a concave function, then (29) is satisfied. For instance, assume log X is an infinite divisible random variable such that its Lévy exponent g X = log M X has the form

g X (λ) = dλ + σ 2 2 λ 2 + (0,∞) (e -λx -1 + λx1l x≤1 ) π(dx), λ ≥ 0, (44)
with d ∈ R, σ ≥ 0 and the Lévy measure π satisfy the (0,∞) (x 2 ∧ 1)π(dx) < ∞. It is easy to check that g X is concave. Furthermore, we have

∆ 1 ∆ s g(t) = σ 2 s + (0,∞)
e -tx (1 -e -x )(1 -e -sx ) π(dx), t, s > 0,

and by (43), X satisfies (29) with c = σ 2 .

=

  b a,b g a+b and b a,b+c d = b a,b b a+b,c , for all a, b, c > 0, ,

  where in the first (respectively second) identity b a,b and g a+b (respectively b a,b and b a+b,c ) are assumed to be independent. Biasing on Beta and Gamma variables is nicely expressed by (b a,b ) (t) d = b a+t,b , and (g a ) (t) d = g a+t , for all t > 0.

  (i) convergence (26, case T = N) holds and D Z∞ contains some value λ 0 ∈ (0, ∞);(ii) convergence (26, case T = [0, ∞)) holds and D Z∞ contains some value λ 0 ∈ (0, ∞);(iii) convergence (27, case T = N) holds and D X∞ contains some value λ 0 ∈ (0, ∞);(iv) convergence (27, case T = [0, ∞)) holds and D X∞ contains some value λ 0 ∈ (0, ∞).

  3, it is enough to show that + [s] + 2)/r([t]). It is then immediate that

	lim sup t∈[0,∞), t→∞	ρ t+s ρ t	=	lim sup t∈[0,∞), t→∞	r(t + s) r(t)	≤ lim sup t∈[0,∞), t→∞	r([t] + [s] + 2) r([t]	= lim sup n∈N, n→∞	r(n + [s] + 2) r(n)
			≤ lim sup n∈N, n→∞	r(n + 1) r(n)		[s]+2)	= lim sup n∈N, n→∞	ρ n+1 ρ n	[s]+2)	< ∞.
	On the other hand,		lim sup t∈[0,∞), t→∞	ρ t+1 ρ t	≥ lim sup n∈N, n→∞
	lim sup n∈N, n→∞	ρ n+1 ρ n	< ∞ is equivalent to		lim sup t∈[0,∞), t→∞	ρ t+s ρ t	< ∞, for all s > 0.	(42)
	By Theorem 3, we also know that
						ρ t ∼

+∞ r(t) = E[X ∞ ] E[X (t) ]

and Corollary 1 says that the function t → r(t) is nondecreasing. Let [x] the integer of the real number x. We have [t] ≤ t ≤ t + s ≤ [t] + [s] + 2 for every t, s > 0 and then r(t + s)/r(t) ≤ r

([t]


Pushpendra Singh
email: pushpendrasingh@iitkalumni.org

Breaking the Limits -Redefining the Instantaneous Frequency

Keywords: Analytic signal, Hilbert transform, an increasing (or a nondecreasing) function, Instantaneous frequency, Linearly independent nonorthogonal yet energy preserving (LINOEP) vectors, zero-phase filtering

 introduced the concept of variable frequency as a generalization of the constant frequency. The instantaneous frequency (IF) is the time derivative of the instantaneous phase and it is well-defined only when this derivative is positive. If this derivative is negative, the IF creates problem because it does not provide any physical significance. This study proposes a mathematical solution and eliminate this problem by redefining the IF such that it is valid for all monocomponent and multicomponent signals which can be nonlinear and nonstationary in nature. This is achieved by using the property of the multivalued inverse tangent function which provides basis to ensure that instantaneous phase is an increasing function. The efforts and understanding of all the methods based on the IF would improve significantly by using this proposed definition of the IF. We also demonstrate that the decomposition of a signal, using zero-phase filtering based on the well established Fourier and filter theory, into a set of desired frequency bands with proposed IF produces accurate time-frequency-energy (TFE) distribution that reveals true nature of signal. Simulation results demonstrate the efficacy of the proposed IF that makes zero-phase filter based decomposition most powerful, for the TFE analysis of a signal, as compared to other existing methods in the literature.

INTRODUCTION

The time-domain representation and the frequency-domain representation are two classical representations of a signal. In both domains, the time (t) and frequency (f ) variables are mutually exclusive [START_REF] Boashash | Time Frequency Signal Analysis and Processing: A Comprehensive Reference[END_REF]. The Time-Frequency Distributions (TFD) representation on the other hand, provides localized signal information in time and frequency domain. The TFD provides insight into the complex structure of a signal consisting of several components. There exist many types of timefrequency analysis methods such as short-time Fourier transform (STFT), Gabor transform, wavelet transforms, Wigner-Ville distribution.

The Carson and Fry [START_REF] Carson | Variable frequency electric circuit theory with application to the theory of frequency modulation[END_REF] introduced the concept of variable frequency, required to the theory of frequency modulation (FM), as a generalization of the definition of constant frequency. Moreover, the nonstationary nature of the signals and nonlinear systems require the idea of instantaneous frequency (IF). The IF is the basis of the TFD or time-frequency-energy (TFE) representation and analysis of a signal. The IF is a practically important parameter of a signal which can reveal the underlying process and provides explanations for physical phenomenon in many applications such as vibration, acoustic, speech signal analysis [START_REF] Singh | The Fourier Decomposition Method for nonlinear and non-stationary time series analysis[END_REF], meteorological and atmospheric applications [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and nonstationary time series analysis[END_REF], seismic [START_REF] Singh | The Fourier Decomposition Method for nonlinear and non-stationary time series analysis[END_REF], radar, sonar, solar physics, structural engineering, communications, health monitoring, biomedical and medical applications [START_REF] Cummings | Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand[END_REF], cosmological gravity wave and financial market data analysis.

The IF is the time derivative of the instantaneous phase (IP) and it is well-defined only when this derivative is positive. If this derivative is negative, it creates problem because it does not provide any physical significance. In order to avoid this problem, recently many nonlinear and nonstationary signal representation, decomposition and analysis methods, e.g. empirical mode decomposition (EMD) algorithms [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and nonstationary time series analysis[END_REF][START_REF] Wu | Ensemble Empirical Mode Decomposition: a noise-assisted data analysis method[END_REF][START_REF] Rehman | Multivariate empirical mode decomposition[END_REF][START_REF] Singh | The Hilbert spectrum and the Energy Preserving Empirical Mode Decomposition[END_REF][START_REF] Singh | Some studies on nonpolynomial interpolation and error analysis[END_REF][START_REF] Singh | Nonpolynomial spline based empirical mode decomposition[END_REF][START_REF] Singh | The Linearly Independent Non Orthogonal yet Energy Preserving (LINOEP) vectors[END_REF][START_REF] Meignen | A new formulation for empirical mode decomposition based on constrained optimization[END_REF], synchrosqueezed wavelet transforms (SSWT) [START_REF] Daubechies | Synchrosqueezed Wavelet Transforms: an Empirical Mode Decomposition-like Tool[END_REF], variational mode decomposition (VMD) [START_REF] Dragomiretskiy | Variational Mode Decomposition[END_REF], eigenvalue decomposition (EVD) [START_REF] Jain | An iterative approach for decomposition of multi-componentnonstationary signals based on eigenvalue decomposition of the Hankel matrix[END_REF], empirical wavelet transform (EWT) [START_REF] Gilles | Empirical Wavelet Transform[END_REF], sparse time-frequency representation [START_REF] Hou | Adaptive data analysis via sparse time-frequency representation[END_REF], time-varying vibration decomposition [START_REF] Feldman | Time-varying vibration decomposition and analysis based on the Hilbert transform[END_REF], resonance-based signal decomposition [START_REF] Selesnick | Resonance-based signal decomposition: A new sparsity-enabled signal analysis method[END_REF] and Fourier decomposition methods (FDM) [START_REF] Singh | Some studies on a generalized fourier expansion for nonlinear and nonstationary time series analysis[END_REF][START_REF] Singh | The Fourier Decomposition Method for nonlinear and non-stationary time series analysis[END_REF][START_REF] Singh | Time-Frequency analysis via the Fourier Representation[END_REF][START_REF] Singh | Some studies on multidimensional Fourier theory for Hilbert transform, analytic signal and space-time series analysis[END_REF][START_REF] Singh | LINOEP vectors, spiral of Theodorus, and nonlinear time-invariant system models of mode decomposition[END_REF][START_REF] Singh | Fourier-based Feature Extraction for Classification of EEG Signals Using EEG Rhythms[END_REF] based on the Fourier theory, are proposed. The Fourier theory is the only tool for spectrum analysis of a signal and the FDM has established that it is a superior tool for nonlinear and nonstationary time series analysis. The main objective of all these methods is to obtain the signal representation such that the IF of a signal understudy is always positive.

Unlike these decomposition methods, the IF proposed in this study does not necessitate to decompose a signal into a set of narrow band components, which satisfy certain properties, to generate the TFE distribution of a signal. That is, without any decomposition the TFE distribution of a signal can be obtained. It also provide freedom and potential to decompose a signal into a set of desired (preferably orthogonal or LINOEP [START_REF] Singh | The Hilbert spectrum and the Energy Preserving Empirical Mode Decomposition[END_REF]) frequency bands by a zero-phase filtering approach to obtain TFE distribution of a signal. All these features are obtained by redefining the IF when it is negative and thus defining the IF for both monocomponent and multicomponent signals. In order to redefine the IF, we use the fact that inverse tangent is a multivalued (i.e. one-to-many mapping) function. So defined IF of a signal is always positive and valid for any signal.

The main contributions as well as some important observations of this study are as follows:

1. We use conventional definition of the IF when it is positive, if it is negative then redefine the IF to make it positive. Thus, the proposed IF is always positive and valid for all monocomponent as well as multicomponent signals, which can be nonstationary and nonlinear in nature.

2. Since many decades, there has been a general understanding in the literature, e.g. [1-4, 11, 30, 31, 34], that the Fourier theory (due to linearity, periodicity and stationarity) is not suitable for nonstationary signal analysis. This proposed IF provides an elegant way to use the Fourier and filter theory based zerophase filtering for the decomposition of a signal into a set of suitable bands with desired cutoff frequencies. In order to validate this, in the study, we have used only the Fourier and finite impulse response (FIR) filter theory based decomposition (except when comparing with EMD, FDM, wavelet transform and conventional non zero-phase FIR filtering) to obtain TFE analysis of a signal.

3. We demonstrate that the zero-phase discrete Fourier transform (DFT) filter-bank based decomposition of a signal produces orthogonal components, however it is more natural to obtain LINOEP components with zero-phase FIR filter based decomposition. The both set of orthogonal and LINOEP vectors preserve the energy in decomposition, and present similar TFE distribution of a signal.

4. The proposed method, using the Hilbert spec-trum, produces average frequencies in the TFE distribution with good time resolution when envelope of signal is smooth. However, if envelope of a signal is fluctuating randomly and rapidly, e.g. the Gaussian white noise and Earthquake time series, the TFE plot presents good time and frequency resolution.

5. We demonstrate that the different decomposition methods are producing the different TFE distributions of a signal. Using the proposed method, when a signal is decomposed into more number of orthogonal or LINOEP narrow bands, true frequencies present in the signal are revealed, frequency resolution also increases while the time resolution reduces marginally.

Thus, this study presents a new paradigm for nonlinear and nonstationary data analysis. This paper is organized as follows: The proposed methodology is presented in Section 2. Simulation results and discussion are presented in Section 3. Section 4 presents conclusion of the work.

METHODS

The concept of variable frequency with application to the theory of frequency modulation (FM) is proposed in [START_REF] Carson | Variable frequency electric circuit theory with application to the theory of frequency modulation[END_REF], and it is postulated that the notion of IF is a generalization of the definition of constant frequency. A definition of the IF by analyzing an expression for simple harmonic motion (SHM) is considered in [START_REF] Van Der Pol | The fundamental principles of frequency modulation[END_REF] as

x SHM (t) = a cos t 0 2πf (t) dt + θ = a cos(ψ(t)), (1)
where the argument of the cosine function is the phase ψ(t) = [ t 0 2πf (t) dt + θ]. This leads to the definition of instantaneous frequency [START_REF] Van Der Pol | The fundamental principles of frequency modulation[END_REF]

f (t) = 1 2π dψ(t) dt . (2)
The concept of instantaneous frequency was enhanced in [START_REF] Gabor | Theory of communication[END_REF] where a method for generating a unique complex signal z(t) from a real signal x(t) and its Hilbert transform x(t) was proposed. This method obtains

z(t) = x(t) + j x(t) = a(t)e jφ(t) , (3)
where

a(t) = [x 2 (t) + x2 (t)] 1/2 ≥ 0, φ(t) = tan -1 [x(t)/x(t)] (4)
and z(t) is the Gabor's complex signal (well-known as the analytic signal) and x(t) is the Hilbert Transform (HT) of x(t), defined as

x(t) = p.v. ∞ -∞ x(τ ) π(t -τ ) dτ = p.v. x(t) * 1 π(t) , (5)
where p.v. denotes the Cauchy principal value of the integral [START_REF] Hildebrand | Advanced Calculus for Engineers[END_REF] and * denotes convolution operation. The work done in [START_REF] Carson | Variable frequency electric circuit theory with application to the theory of frequency modulation[END_REF] and [START_REF] Gabor | Theory of communication[END_REF] was unified in [START_REF] Ville | Theorie et application de la notion de signal analytic[END_REF] to define the IF of a signal x(t) = a(t) cos(φ(t)) as

ω(t) = 2πf (t) = d dt (arg[z(t)]) , (6)
where z(t) is the analytic signal given by (3) and arg[z(t)] = φ(t) as defined in (4). The IF, as defined in [START_REF] Singh | Some studies on nonpolynomial interpolation and error analysis[END_REF], provides physical meaning only when it is positive and it becomes meaningless when it is negative [START_REF] Huang | The empirical mode decomposition and Hilbert spectrum for non-linear and nonstationary time series analysis[END_REF][START_REF] Wu | Ensemble Empirical Mode Decomposition: a noise-assisted data analysis method[END_REF][START_REF] Rehman | Multivariate empirical mode decomposition[END_REF][START_REF] Singh | The Hilbert spectrum and the Energy Preserving Empirical Mode Decomposition[END_REF][START_REF] Singh | Nonpolynomial spline based empirical mode decomposition[END_REF][START_REF] Daubechies | Synchrosqueezed Wavelet Transforms: an Empirical Mode Decomposition-like Tool[END_REF][START_REF] Dragomiretskiy | Variational Mode Decomposition[END_REF][START_REF] Jain | An iterative approach for decomposition of multi-componentnonstationary signals based on eigenvalue decomposition of the Hankel matrix[END_REF][START_REF] Singh | Some studies on a generalized fourier expansion for nonlinear and nonstationary time series analysis[END_REF][START_REF] Singh | The Fourier Decomposition Method for nonlinear and non-stationary time series analysis[END_REF][START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 1: Fundamentals[END_REF][START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 2: Algorithms and Applications[END_REF]. This is where we improve and provide the solution to make the IF positive for all time t. From ( 4) and ( 6), one can obtain the TFE distribution by 3-D plot of {t, f (t), a 2 (t)}. However, it is to be noted that the frequencies, of signal x(t) = a(t) cos(φ(t)), obtained by the Fourier transform and Hilbert spectrum [START_REF] Singh | Some studies on nonpolynomial interpolation and error analysis[END_REF] are same only if a(t) is a constant, otherwise Hilbert spectrum gives frequencies of signal cos(φ(t)) and not of signal x(t).

Before evaluating the time derivative of phase in [START_REF] Singh | Some studies on nonpolynomial interpolation and error analysis[END_REF], phase unwrapping is necessary to ensure that all appropriate multiples of 2π have been included in phase φ(t). Phase unwrap operation corrects the radian phase angles by adding multiples of ±2π when absolute jumps between consecutive elements of a phase vector are greater than or equal to the default jump tolerance of π radians [32]. This is being done for phase delay and IF determination by all the methods available in literature.

As is well-known that the tangent is a surjective (many-to-one mapping) function. The domain, range and period of tan(x) are {x|x = π 2 +nπ, ∀n ∈ Z}, all real numbers R = (-∞, ∞) and π, respectively. The inverse tangent is the multivalued function. The domain of tan -1 (x) is all real numbers, R, and range is (-π 2 , π 2 ). If z = x + jy, then the range of tan -1 (y/x) is (-π, π], sign of x and y is used to determine the specific quadrant.

In order to obtain the IF positive for all the time, we consider the discrete signal processing which is the only practical way to process data by a processor. The discrete time version of the equations ( 3), ( 4), ( 5) and ( 6) are [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 2: Algorithms and Applications[END_REF] z

[n] = x[n] + j x[n] = a[n]e jφ[n] , (7a)
a[n] = x 2 [n] + x2 [n] 1/2 ≥ 0, (7b) φ[n] = tan -1 (x[n]/x[n]) , (7c)
x

[n] = x[n] * 1 -cos(πn) πn , (7d)
and

ω[n] = φ d [n], (7e)
respectively, where the differentiation in discretetime can be approximated by [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal-Part 2: Algorithms and Applications[END_REF] forward finite difference (FFD),

φ d [n] = φ[n + 1] -φ[n] , or backward finite difference (BFD), φ d [n] = φ[n] - φ[n-1] or central finite difference (CFD), φ d [n] = φ[n + 1] -φ[n -1] /2.
It is to be noted that the phase in (7c) is computed by the function, atan2(x[n], x[n]), which produces the result in the range (-π, π] and also avoids the problems of division by zero.

By considering the phase unwrapping fact and multivalued nature of the inverse tangent function, we redefine the IF ω[n], defined in (7e) as a discrete version of ( 6), as

ω[n] = φ d [n], if φ d [n] ≥ 0, φ d [n] + π, otherwise, (8)
which makes the IF positive for all time n. This small and trivial but extremely important fact has been illusive for many decades. This IF would not only significantly improve the computational efforts (e.g. sifting process in EMD algorithms) and understanding of any method which uses the IF, but also provides an elegant solution in mathematical terms to use the Fourier and filter theory based zero-phase filtering for nonstationary signal decomposition and TFE analysis. The mathematical validity of this solution can easily be seen by the fact that φ

[n] = tan -1 (x[n]/x[n]) and φ[n] + knπ = tan -1 (x[n]/x[n]) , ∀k, n ∈ Z (due to periodicity of tangent function, i.e. tan(φ[n]) = tan(φ[n] + knπ)). In solution (8), we have taken k = 0 if φ d [n] ≥ 0, otherwise k = 1, thus we can write ω[n] = φ d [n] + kπ. (9)
The equation ( 9) is also valid when φ[n] ∈ (-π, π] is a wrapped phase and in this case we select the value of 9) is estimated by using the multivalued property of the inverse tangent function which provides basis to ensure that the instantaneous phase function φ[n] defined in (7c) is an increasing (or a nondecreasing) function, i.e.

k ∈ Z such that ω[n] ∈ [0, π]. Thus, the proposed IF ω[n] in (8) or (
φ[n + 1] ≥ φ[n], ∀n.
A dual to IF (8), the group delay (GD) is defined as the negative frequency derivative of the phase in the Fourier domain. It measures the relative delay of different frequencies from the input to the output in a system. Thus, similar to (8), we can modify the GD definition to make it always positive and valid for all signals. The proposed definition of IF can easily be extended for multidimensional signals such as spatial data (image) and space-time data (3D wave equation).

A MATLAB implementation code of the proposed method is outlined in Algorithm 2. In order to avoid unnecessary variations in the IF, before evaluating the IF, the mean and dominating low frequency component like trend present in the signal can be removed. These can be easily removed by any zero-phase low pass filtering operation [START_REF] Singh | The Fourier Decomposition Method for nonlinear and non-stationary time series analysis[END_REF][START_REF] Singh | LINOEP vectors, spiral of Theodorus, and nonlinear time-invariant system models of mode decomposition[END_REF]. Moreover, depending upon the requirements, signal can be decomposed into a set of desired frequency bands.

In the examples presented in the following section, to decompose a signal x[n] into a set of desired frequency bands, we have used the signal model

x[n] = c 0 + M i=1 y i [n] = c 0 + M i=1 c i [n], (10)
where c 0 is the mean-value of signal

x[n], {y i [n]} M i=1
and {c i [n]} M i=1 are the M orthogonal and LINOEP components, respectively. A simple block diagram of the zero-phase filter-bank (i.e.

H i [k] ∈ R, ∀i, k) to decompose a signal x[n] into a set {y 1 [n], y 2 [n], • • • , y M [n]} is shown in Figure 1.
The frequency response of i-th band of the DFT based zero-phase filter-bank can be defined by setting H i [k] = 1 at desired frequency band and zero otherwise, i.e.

H i [k] = 1, (K i-1 + 1) ≤ k ≤ K i & (N -K i ) ≤ k ≤ (N -K i-1 -1), = 0, otherwise    (11)
where

i = 1, 2, • • • , M ; K 0 = 0 and K M = N/2 (or K M = (N -1)/2 if N is odd). Using the inverse DFT (IDFT), component y i [n] can be computed as y i [n] = N -1 k=0 H i [k]X[k] exp(j2πkn/N ) , (12)
where

X[k] = 1 N N -1 n=0 x[n] exp(-j2πkn/N ) is the DFT of signal x[n] of length N samples.
In this work, we use non-causal finite impulse response (FIR) and infinite impulse response (IIR) filter to decompose a signal into a set of LINOEP vectors by the filter mode decomposition (FMD) algorithm proposed in [START_REF] Singh | Some studies on a generalized fourier expansion for nonlinear and nonstationary time series analysis[END_REF][START_REF] Singh | The Fourier Decomposition Method for nonlinear and non-stationary time series analysis[END_REF][START_REF] Singh | LINOEP vectors, spiral of Theodorus, and nonlinear time-invariant system models of mode decomposition[END_REF], as summarized We advocate to use zero-phase filtering because it preserves salient features such as minima and maxima in the filtered waveform exactly at the position where those features occur in the unfiltered waveform. It is pertinent to note that the conventional (i.e. non zero-phase) filtering shifts these features in the filtered waveform and therefore cannot be used to obtain a meaningful TFE distribution, which is clearly demonstrated in simulation results. The zero-phase filtering of a signal can be obtained by the DFT, non-causal FIR and IIR filters, or via other decomposition methods like FDM, VMD, EVD, EWT and EMD algorithms.

x[n] DFT . . . H 1 [k] H 2 [k] H M [k] IDFT IDFT . . . IDFT y 1 [n] y 2 [n] . . . y M [n]

SIMULATION RESULTS

AND DISCUSSION

In this section, we consider number of examples that are mostly discussed in literature to validate

i = ZP HP F i (x i , f ci ); r i = x i -y i ; α i = y i ,r i r i ,r i ; c i = y i -α i r i ; ci+1 = (1 + α i )r i ; x i+1 = ci+1 ; c M = cM ; %PART B x 1 = x; for i = 1 to M -1 do y i = ZP LP F i (x i , f ci ); r i = x i -y i ; α i = r i ,y i y i ,y i ; c i = (1 + α i )y i ; ci+1 = r i -α i y i ; x i+1 = ci+1 ; c M = cM ;
the efficacy of method understudy. A complete MATLAB code of the proposed method is available on request and would be made publicly available soon. Discussion: From Examples (1, 2 and 3), it is clear that the zero-phase DFT or zero-phase FIR filter based decomposition is able to track TFE distribution present in a signal. However, conventional filtering cannot be used to obtain meaningful TFE distribution of a signal, which is clearly demonstrated in bottom figures of Figure 5 This representation demonstrate that it is a superposition of equal amplitude sinusoidal functions of all frequencies [0-π). The Nyquist frequency (F s /2) is the highest frequency that can be present at a given sampling rate, F s , in a discrete-time signal. The analytic representation of this signal is given by [START_REF] Singh | The Fourier Decomposition Method for nonlinear and non-stationary time series analysis[END_REF]

z[n] = sin(π(n-n 0 ))+j[1-cos(π(n-n 0 ))] π(n-n 0 ) = a[n] exp(jφ[n]), where a[n] = sin( π 2 (n-n 0 )) π 2 (n-n 0 ))
, and

φ[n] = π 2 (n-n 0 ). Thus, frequency ω[n] = φ[n+1]- φ[n] = π
2 , which corresponds to half of the Nyquist frequency, i.e. F s /4. Figure 10 shows the TFE estimates of this signal (with n 0 = 1999, F s = 1000 Hz and length N = 4000) using the EMD (top figure) and proposed method without decomposition (bottom figure). We again observe that the frequency present in this TFE plot is a average frequency (because delta function contains equal amplitude sinusoids of all frequencies 0 to F s /2). This example also demonstrate that the TFE plot, obtained by the Hilbert spectrum, is not limited by uncertainty principle and signal can be highly concentrated in time and frequency plane. However, it is to be noted that this TFE plot is not providing the true frequencies of delta function.

Discussion: In order to explain a average frequency effect in Figure 2, Figure 6, Figure 8 and Figure 10, let us consider a sum of sinusoids of equal amplitudes x(t) = N k=1 A cos(ω 0 kt). Its analytic representation is given by z

(t) = N k=1 A exp(jω 0 kt) = A sin(ω 0 N 2 t) sin(ω 0 t/2) exp(jω 0 N +1 2 t) = a(t) exp(jφ(t)) which implies phase φ(t) = (ω 0 N +1 2 t) and hence IF f (t) = 1 2π ω 0 N +1 2 .
This is what we observe in these figures (especially Figure 10 bottom one). Here, it is to be noted that if amplitudes of constituent sinusoids are not equal, then resultant IF would not be a constant (average) frequency but it would be a variable one.

Example 5: Figure 11 shows the TFE analysis of a white Gaussian noise (with zero mean, unit variance, 10240 samples and sampling frequency F s = 100 Hz) obtained from the EMD (top one) and this proposed method without decomposition (bottom one), which shows that the energy is randomly distributed across all the frequency range.

Example 6: An Earthquake time series signal is a nonlinear and nonstationary data. The Elcentro Earthquake data (sampled at F s = 50Hz) has been taken from [33] and is shown in Figure 12 (top one). The critical frequency range that matter in the structural design is less than 10Hz, and the Fourier based power spectral density (PSD), Figure 12 (bottom one), show that almost all the energy in this data is within 10Hz. The TFE dis- were true, then we could transmit delta function in almost zero time with zero bandwidth through any system or channel. Thus, we conclude that the TFE representation depends on the number of bands in which data has been divided. In order to further illustrate this point we divided the same data, which is used for Figure 10, by DFT based zero-phase filtering into five and ten bands to obtain top and bottom figures in Figure 15, respectively, which reveal many those frequencies of data which are not present in Figure 10.

From the Figures (2, 6, 8, 10 and 15), it is clear that the proposed method using the Hilbert spectrum produces average frequencies and good time resolution when envelope of signal is smooth. However, if envelope of a signal is fluctuating randomly or rapidly, e.g. consider the case of Gaussian white noise and Earthquake time series, the TFE plot has good time and frequency resolution as shown in Figure 11 and Figure 14 (top one). It is also clear, form Figures (4, 6, 8, 10 and 15), that as the signal is decomposed into more number of narrow bands, true frequencies present in the signal under analysis are revealed, frequency resolution is also increasing while the time resolution is reducing marginally.

CONCLUSION

The instantaneous frequency (IF) is an important parameter for the analysis of nonstationary signals and nonlinear systems. It is the basis of the time-frequency-energy (TFE) analysis of a signal. The IF is the time derivative of the instantaneous phase and, originally, it is well-defined only when this derivative is positive. That is, the IF is valid only for monocomponent signals. If time derivative of instantaneous phase is negative, i.e. the IF is negative, then it does not provide any physical significance. This study proposed a mathematical solution and eliminate this problem by modifying the present definition of IF. This is achieved by using the property of the multivalued inverse tangent function that provides base to ensure that the instantaneous phase is an increasing function.

There are two fundamental and important conceptual innovation of this work. First, the extension of the conventional definition of IF by redefining it such that it is always positive. This proposed IF is valid for all types of signals such as monocomponent and multicomponent, narrowband and wideband, stationary and nonstationary, linear and nonlinear signals. The understanding of the time-frequency-energy representation, by all the methods which are using the IF, would improve significantly by using this definition. Second, we have also demonstrated that the zerophase filtering based decomposition of a signal into a set of desired frequency bands with proposed IF accurately reveals the TFE distribution. Whereas, conventional (non zero-phase) filtering based decomposition cannot be used to obtain correct and meaningful TFE distribution. The Fourier and filter theory are well established, fully matured and developed, thus the zero-phase filter-ing based decomposition of a signal is most powerful which presents full control over the number of bands with desired cutoff frequencies. This kind of control and features are difficult to achieve or may not be possible by the decomposition methods such as empirical mode decomposition (EMD) algorithms, synchrosqueezed wavelet transforms (SSWT), variational mode decomposition (VMD), eigenvalue decomposition (EVD), time-varying vibration decomposition, resonance-based signal decomposition, EMD based on constrained optimization and empirical wavelet transform (EWT) available in the literature. Simulations and numerical results demonstrated the superiority, validity and efficacy of the proposed IF for the TFE analysis of a signal as compared to other existing methods available in the literature.

Figure 1 :

 1 Figure 1: Block diagram of the zero-phase filter-bank to decompose a signal x[n] into a set {y 1 [n], y 2 [n], • • • , y M [n]} of orthogonal desired frequency bands.

Algorithm 2 :

 2 An FMD algorithm to obtain LINOEP vectors c i from decomposition of a signal x such that x = c 0 + M i=1 c i and c i ⊥ M l=i+1 c l . Use PART A (PART B) to obtain {c 1 , • • • , c M } in order of highest to lowest (lowest to highest) frequency components. %PART A x 1 = x; for i = 1 to M -1 do y

Example 1 :

 1 In this example, we consider three nonstationary signals (a) linear chirp (unit amplitude and [1000-2000] Hz), (b) frequency modulated (FM) sinusoid (unit amplitude, carrier f c = 780 Hz and frequency deviation 200 Hz) and (c) mixture of a linear chirp and frequency modulated (FM) signals, i.e sum of signals of cases (a) and (b).

Figure 2 (

 2 top to bottom) shows timefrequency-energy (TFE) estimates of these three nonstationary signals [(a), (b) and (c)] obtained by using the proposed IF without decomposition. We observe that the frequencies in case (c), Figure 2 (bottom one), are average frequencies of the first two cases (a) and (b).

Figure 3

 3 shows the IF estimates of nonstationary signal, which is sum of linear chirp and FM signals i.e. case (c), without decomposition: (1) top figure with conventional IF (7e) where notice the both positive and negative estimates of frequencies, (2) bottom figure with proposed IF (8) that produces only positive and correct values of frequencies. This, clearly, demonstrate that the proposed definition of the IF is able to obtain correct values of frequencies at all times.

Figure 2 :

 2 Figure 2: The TFE analysis of nonstationary signals by this proposed method without decomposition: (top figure) linear chirp, (middle figure) frequency modulated signal and (bottom figure) sum of linear chirp and FM signals.

Figure 4

 4 Figure 4 shows the TFE estimates of nonstationary signal, case (c), by this proposed method with decomposition: (top figure) into 2 bands of [0-1000, 1000-4000] Hz and (bottom figure) into 100 bands of equal frequencies. The proposed method is, clearly, able to track TFE distribution present in all these signals.Figure 5 shows the TFE estimates of nonstationary signal, case (c), by this proposed IF: (top figure) with zero-phase FIR filter based decomposition into 100 bands; and (bottom figure) with conventional (non zero-phase) FIR filter based decomposition into 100 bands of equal frequencies, which is not able to detect and track true frequencies present in the signal. Example 2: We obtain a nonstationary signal by adding five unit amplitude linear chirps of

Figure 5 2 :

 52 Figure 4 shows the TFE estimates of nonstationary signal, case (c), by this proposed method with decomposition: (top figure) into 2 bands of [0-1000, 1000-4000] Hz and (bottom figure) into 100 bands of equal frequencies. The proposed method is, clearly, able to track TFE distribution present in all these signals.Figure 5 shows the TFE estimates of nonstationary signal, case (c), by this proposed IF: (top figure) with zero-phase FIR filter based decomposition into 100 bands; and (bottom figure) with conventional (non zero-phase) FIR filter based decomposition into 100 bands of equal frequencies, which is not able to detect and track true frequencies present in the signal. Example 2: We obtain a nonstationary signal by adding five unit amplitude linear chirps of

Figure 3 :

 3 Figure 3: The instantaneous frequency (IF) estimates of nonstationary signal, which is sum of linear chirp and FM signals, without decomposition: (top figure) with conventional IF (7e) and (bottom figure) with proposed IF (8). Notice the both positive and negative frequencies with conventional IF (top figure); only positive and correct values of frequencies by the proposed IF (bottom figure).

Figure 4 :

 4 Figure 4: The TFE analysis of nonstationary signal, which is sum of linear chirp and FM signals, by this proposed method with DFT based decomposition: (top figure) into 2 bands of [0-1000, 1000-4000] Hz and (bottom figure) into 100 bands of equal frequencies.

Figure 5 :

 5 Figure 5: The TFE analysis of nonstationary signal, which is sum of linear chirp and FM signals, by this proposed method: (top figure) with zero-phase FIR filter based decomposition into 100 bands and (bottom figure) with conventional (non zero-phase) FIR filter based decomposition into 100 bands of equal frequencies.

Figure 7

 7 shows the TFE distribution of a nonstationary signal (sum of five linear chirp signals) (1) by this proposed method with zero-phase FIR filter based decomposition into 10 bands (upper figure), and with conventional (non zero-phase) FIR filter based decomposition into 10 bands of equal frequencies (middle figure); and (2) by the EMD algorithm (lower figure). Clearly, both the conventional and EMD algorithm not able to detect true frequencies present in the signal. Example 3: In Figure 8, we present the TFE analysis of a nonstationary signal, which is sum of a unit amplitude linear chirp signal [500-1500] Hz ([0-1] s chirp signal and [1-1.5] s no signal) and its delayed version ([0-0.5] s no signal and [0.5-1.5] s same chirp signal), by this proposed method.

Figure 6 :

 6 Figure 6: The TFE analysis of a nonstationary signal, which is sum of five linear chirp signals, by this proposed method: (top figure) without decomposition, (middle figure) with DFT based decomposition into 10 bands and (bottom figure) with DFT based decomposition into 20 bands of equal frequencies.

Figure 9

 9 shows the TFE distribution of a nonstationary signal (sum of a linear chirp signal and its delayed version) by this proposed IF: (top figure) with zero-phase FIR filter based decomposition into 10 bands; and (bottom figure) with conventional FIR filter based decomposition into 10 bands of equal frequencies, which clearly not able to reveal true frequencies present in the signal.

Figure 7 :

 7 Figure 7: The TFE analysis of a nonstationary signal, which is sum of five linear chirp signals (1) by this proposed method with zero-phase FIR filter based decomposition into 10 bands (upper figure), and with conventional (non zero-phase) FIR filter based decomposition into 10 bands of equal frequencies (middle figure); (2) by the EMD algorithm (lower figure).

, Fig- ure 7 and Figure 9 . 4 :Figure 8 :Figure 9 :

 9489 Figure 8: The TFE analysis of a nonstationary signal, which is sum of a linear chirp signal and its delayed version, by this proposed method: (top figure) without decomposition, and (bottom figure) with DFT based decomposition into 10 bands of equal frequencies.

Figure 10 :

 10 Figure 10: The TFE analysis of unit sample sequence δ[n -n 0 ] (with, n 0 = 1999, sampling frequency Fs = 1000 Hz, length N = 4000) by ensemble EMD (EEMD) (top figure) and this proposed method without decomposition (bottom figure).

Figure 11 :

 11 Figure 11: The TFE analysis of the Gaussian white noise with zero mean and unit variance (with sampling frequency Fs = 1000 Hz, length N = 10240) by the EMD (top figure) and this proposed method without decomposition (bottom figure).

Figure 12 :Figure 13 :

 1213 Figure 12: The Elcentro Earthquake May 18, 1940 North-South Component data (top figure), Fourier based power spectral density (PSD) (bottom figure).

Figure 14 :

 14 Figure 14: The TFE plot of the Elcentro Earthquake data (top to bottom) using the: (a) proposed method without decomposition and (b) proposed method with DFT based decomposition into four bands [0-5, 5-10, 10-20, 20-25] Hz and (c) DFT based decomposition into 25 bands of 1 Hz each.

Figure 15 :

 15 Figure 15: The TFE analysis of unit sample sequence δ[n -n 0 ] (with, n 0 = 1999, sampling frequency Fs = 1000 Hz, length N = 4000) by proposed method with DFT based decomposition into five (top figure) and ten (bottom figure) bands of equal frequencies.

ACKNOWLEDGMENTS

Author would like to show his gratitude to the Prof. SD Joshi (IITD), Prof. RK Pateny (IITD) and Dr. Kaushik Saha (Director, Samsung R&D Institute India-Delhi) for sharing their wisdom and expertise during the course of this research.