G Mora
email: gaspar.mora@ua.es

Accumulation points of the sets of real parts of zeros of the partial sums of the Riemann zeta function

Keywords: AMS Subject Classification: 30Axx, 30D05 Zeros of partial sums of the Riemann zeta function, Almostperiodic functions, Level curves

Let ζ n (z) := n k=1 1 k z be the nth partial sum of the Riemann zeta function. In this paper it is shown that for every integer n > 2 there exists

Then, for every n > 2, ζ n (z) possesses infinitely many zeros, with negative and positive real part, arbitrarily close to any line contained in the strip n) contains a non-degenerated interval containing 0, so 0 is an accumulation point common to all the sets P ζn := {ℜz : ζ n (z) = 0}, n > 2, and in particular 0 ∈ R (n) for all n ≥ 2.

Introduction

The distribution of the zeros of exponential polynomials was mainly studied in the first third of XX th century linked to the development of the theory of differential equations, as we can see for instance in [START_REF] Langer | On the zeros of exponential sums and integrals[END_REF][START_REF] Tamarkin | The zeros of certain integral functions[END_REF][START_REF] Tamarkin | Some general problems of the theory of ordinary linear differential equations and expansions of an arbitrary function in series of fundamental functions[END_REF][START_REF] Wilder | Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points[END_REF]. On the concrete case of the partial sums of the Riemann zeta function, ζ n (z) := n k=1 1 k z , the interest for knowing the distribution of their zeros had its starting point in 1948 as consequence of a paper of Turán [START_REF] Turán | On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann[END_REF] where he pointed out an intriguing connexion betweeen a particular distribution of such zeros near the line x = 1 and the Riemann Hypothesis (briefly R.H.). Namely, Turán [START_REF] Turán | On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann[END_REF]Th. I] proved that if

ζ n (z) = 0 for ℜz ≥ 1 + n -1 2 +ǫ , n ≥ N 0 (ǫ), then the Riemann Hypothesis is true. The first step to avoid Turán's dream of proving R.H. via the previous theorem was given by Haselgrove [START_REF] Haselgrove | A disproof of a conjecture of Pólya[END_REF] few years later. Indeed, by means of an added note at the end of [START_REF] Haselgrove | A disproof of a conjecture of Pólya[END_REF], Haselgrove showed that the suggestion of Turán [26, (25.1)] on the non-negativity for all x ≥ 1 of the function T (x) := k≤x λ(k) k , was false, where λ(k) := (-1) p(k) and p(k) is the number of prime factors of k counting multiplicities. Hence, for some n, one has and then the corresponding partial sum

ζ n (z) = n k=1 1 k z ,
because the equivalence theorem of Bohr [START_REF] Bohr | Zur Theorie der allgemeinen Dirichletschen Reihen[END_REF], attains the same set of values in any half-plane ℜz > a. That is, for arbitrary real a,

{D n (z) : ℜz > a} = {ζ n (z) : ℜz > a} .

Thus, since for sufficiently large real z, D n (z) is near 1 and, by (1.1), D n (1) < 0, there exists a real zero x > 1 of D n (z). Therefore, there is a zero of ζ n (z) in the half-plane ℜz > 1.

In 1968 Spira [START_REF] Spira | Zeros of sections of the zeta function, I, II[END_REF] demonstrated the same for n = 19. That is, the partial sum ζ 19 (z) has a zero in the half-plane ℜz > 1. Levinson [9, Theorem 1] in 1973 found an asymptotic formula, for large n, for the location of the zeros of ζ n (z) near the point z = 1. In particular, he proved that theses zeros have real part less than 1. Voronin [START_REF] Voronin | On the zeros of partial sums of the Dirichlet series for the Riemann zeta-function[END_REF] in 1974 showed that ζ n (z) has zeros in ℜz > 1 for infinitely many n.

A sharp result on the upper bound b (n) := sup {ℜz :

ζ n (z) = 0}
was demonstrated in 2001 by Montgomery and Vaughan in [START_REF] Montgomery | Mean values of multiplicative functions[END_REF]. Namely, they proved that there exists N 0 such that if n > N 0 then ζ n (z) = 0 whenever

ℜz ≥ 1 + 4 π -1 log log n log n . ( 2
)
But in the opposite direction, in 1983, Montgomery [START_REF] Montgomery | Zeros of approximations to the zeta function[END_REF] had already shown that for every 0 < c < 4 π -1 there exists N 0 (c) such that ζ n (z) has zeros in the half-plane

ℜz > 1 + c log log n log n , n > N 0 (c). (3)
Since for any positive c, n sufficiently large and any 0 < ǫ < 1 2 , one has

c log log n log n > n -1 2 +ǫ ,
Montgomery's result implies the existence of zeros of ζ n (z) having real part much larger than the bound 1 + n - The results of Montgomery and Vaughan, (1.2) and (1.3), imply that, fixed 0 < c < 4 π -1, for all n > N 1 := max {N 0 , N 0 (c)}, the upper bound b (n) satisfies the inequalities

1 + c log log n log n < b (n) ≤ 1 + 4 π -1 log log n log n . ( 4
)
Then, since lim n→∞ log log n log n = 0, it follows that

lim n→∞ b (n) = 1.
On the other hand, Balazard and Velásquez-Castañón [START_REF] Balazard | Sur l'infimum des parties réelles des zéros des sommes partielles de la fonction zêta de Riemann[END_REF] demonstrated that the lower bound

a (n) := inf {ℜz : ζ n (z) = 0} is so that lim n→∞ a (n) n = -log 2. ( 5
)
Therefore lim

n→∞ a (n) = -∞.
Once the estimation of the bounds a (n) and b (n) has been settled, our interest is focused on the distribution of the real parts of the zeros of ζ n (z) in the interior of its critical interval a (n) , b (n) and, in this sense, we suggest to see [START_REF] Dubon | A note on the real projection of the zeros of partial sums of Riemann zeta function[END_REF][START_REF] Mora | A Note on the Functional Equation F (z)+F (2z)+...+F (nz) = 0[END_REF][START_REF] Mora | On the Distribution of Zeros of a Sequence of Entire Functions Approaching the Riemann Zeta Function[END_REF][START_REF] Mora | On the asymptotically uniform distribution of the zeros of the partials sums of the Riemann zeta function[END_REF][START_REF] Mora | On the existence of exponential polynomials with prefixed gaps[END_REF]. A way of knowing the distribution of the zeros of ζ n (z) is by means of the study of the set of their real projections

P ζn := {ℜz : ζ n (z) = 0}
and its closure set, denoted by R (n) . Indeed, the difference between R (n) and the critical interval a (n) , b (n) gives us a certain measure of the uniformity of the distribution of the real part of the zeros of ζ n (z) and, consequently, the position of those zeros with respect to vertical lines contained in the critical strip a (n) , b (n) × R where are located all the zeros of ζ n (z).

The zeros of

ζ 2 (z) = 1 + 1 2 z , or those of the function G 2 (z) := 1 + 2 z , are of the form π(2k + 1)i log 2 , k ∈ Z. ( 6
)
Then n) for all n ≥ 2, the best possible relation between both sets one reaches when they are equal. Thus, whenever R (n) = a (n) , b (n) , we will say that the distribution of the real parts of the zeros of ζ n (z) is completely uniform on its critical interval. It means that, given any vertical line contained in a (n) , b (n) × R, ζ n (z) possesses infinitely many zeros arbitrarily close to that line. We have proved [START_REF] Mora | On the asymptotically uniform distribution of the zeros of the partials sums of the Riemann zeta function[END_REF]Th.12] that this is exactly the case of the partial sums ζ n (z) for sufficiently large n. That is, we have demonstrated in [START_REF] Mora | On the asymptotically uniform distribution of the zeros of the partials sums of the Riemann zeta function[END_REF]Th.12] the existence of a positive integer N such that

a (2) = b (2) = 0, so R (2) = {0} and trivially R (n) = a (n) , b (n) for n = 2. Since R (n) ⊂ a (n) , b (
R (n) = a (n) , b (n) for all n > N . ( 7
)
This result points out that the distribution of the real part of the zeros of ζ n (z), for all n > N , is completely uniform on its corresponding critical interval. Consequently, the partial sums ζ n (z), n > N , in spite of the set of their frequencies {-log k : k = 2, ..., n} is not linearly independent (briefly l.i.) over the rationals for any n > 3, share with the Dirichlet polynomials, with l.i. frequencies and coefficients of modulus 1, the property of having the real parts of their zeros uniformly distributed on their critical intervals (see [START_REF] Mora | On the existence of exponential polynomials with prefixed gaps[END_REF]Th. 10]).

In the present paper, by means of our Theorem 2, we have given a positive answer to an old question, on the negativity of the real part of the zeros of the partial sums of the Riemann zeta function, posed by Bellman and Hooke in 1963 [START_REF] Bellman | Differential-Difference Equations[END_REF]p. 439,Question 25]. It has been also proved in our Theorem 17 that for every integer n > 2 there exists a number

δ n > 0 such that -δ n , b (n) ⊂ R (n) .
Then, whenever n > N , by (1.7), the Theorem 17 is improved of an optimal form because the δ n 's can be taken equal to -a (n) = n log 2 + o(n), noting (1.5). Some relevant consequences are derived from Theorem 17: a) By (1.4) and Corollary 3 of this paper, the set

∩ ∞ n=3 R (n) contains an interval of the form [-δ, b] for some δ ≥ 0 and b := min b (n) : n > 2 , which is a positive number. Then the strip [-δ, b] × R is common to all the nth partial sums ζ n (z), n > 2.
That is, given any vertical line contained in [-δ, b]× R, there are infinitely many zeros of every ζ n (z), for all n > 2, arbitrarily close to that line.

b) In particular, we deduce that 0 is an accumulation point common to all the sets P ζn = {ℜz : ζ n (z) = 0}, n > 2. Moreover, because of Lemma 1, for every n > 2, there exist infinitely many zeros of each ζ n (z), with negative and positive real parts, which are arbitrarily close to the imaginay axis. Hence, the Theorem 17 and its consequences have applications to the stability of the solutions of neutral functional differential equations with delays (see for instance [START_REF] Michiels | Sensitivity to Infinitesimal Delays in Neutral Equations[END_REF]). c) By regarding the partial sum of Riemann zeta function Proof. We firstly note that, since the almost-periodicity implies analyticity [5, p.101], f (z) is analytic on S a,b . Secondly, by [5, p.102], f (z) is bounded on any closed sub-strip S α,β : = {z : a < α ≤ ℜz ≤ β < b} of S a,b . Now, let z 1 be a point on the line x = ℜz 0 such that f (z 1 ) = 0. Thus we claim that there exist positive numbers δ and l such that on any segment of length l of the line x = ℜz 0 there is a point z 1 + iT verifying

ζ 2 (z) := 1 + 1 2 z , noting (1.6), one has ∩ ∞ n=2 R (n) = {0} ,
|f (z 1 + iT )| ≥ δ. (8)
Indeed, by taking δ = |f (z1)| 2 > 0, the almost-periodicity of f (z) involves the existence of a real number l = l(δ) such that every interval of length l on the imaginary axis contains at least a number

T 1 = T 1 (δ), called translation number, satisfying |f (z + iT 1 ) -f (z)| ≤ δ for all z ∈ S a,b . (9)
Then, by setting z = z 1 in (2.2) and according to the choice of δ, the inequality (2.1) follows, as claimed. By reiterating this process for every δ n = δ 2 n-1 , n ≥ 1, the almost-periodicity of f (z) assures the existence of a real number l n = l n (δ n ) such that every interval of length l n contains at least a translation number T n = T n (δ n ) which, without loss of generality, can be chosen in such a way that (T n ) n=1,2,.. be an increasing sequence of positive numbers with T n+1 -T n > 1, for all n, satisfying

|f (z + iT n ) -f (z)| ≤ δ n for all z ∈ S a,b . (10)
Then, by just making z = z 0 in (2.3), we have

lim n→∞ f (z 0 + iT n ) = 0. ( 11
)
We now consider the sequence of analytic functions on S a,b

g n (z) := f (z + iT n ), z ∈ S a,b , n = 1, 2, ..., (12)
which are uniformly bounded on the compacts of S a,b by virtue of the boundedness of f (z) on any substrip of S a,b . Hence, by Montel's theorem [1, p. 165], there exists a uniformly convergent subsequence {g n k (z) : k = 1, 2, } to an analytic function, say g(z), on S a,b . The function g(z), by (2.1), is not identically 0 on S a,b and, because of (2.5) and (2.4), has a zero at the point z 0 which, by analyticity, is not isolated. Then we can determine a positive number This means, by Rouché's theorem [1, p. 88], that every function g n k (z), for all k > k 0 , has in the disk |zz 0 | < r as many zeros as g(z) does, so at least one, say z n k . Then, from (2.5), one has

r < min x 0 -a, b -x 0 , ǫ, 1
f (z n k + iT n k ) = 0 for all k > k 0 .
Finally, since |z n kz 0 | < r < 1 2 for all k > k 0 and T n k+1 -T n k > 1 for all k, the points z n k + iT n k are distinct for all k > k 0 . Hence, the function f (z) has a zero at each point z n k + iT n k for any k > k 0 . Consequently, f (z) has infinitely many zeros in the strip S ǫ . This completes the proof.

Since the partial sums of the Riemann zeta function are almost-periodic functions, from the previous lemma, we deduce a relevant property on the distribution of their zeros with respect to the imaginary axis. Proof. For any n ≥ 2, by regarding the functions

G n (z) := ζ n (-z), n ≥ 2,
the sets of zeros of G n (z) and ζ n (z), denoted by Z Gn(z) and Z ζn(z) respectively, are related by Z Gn(z) = -Z ζn(z) . Then the bounds

a n := inf {ℜz : G n (z) = 0} , b n := sup {ℜz : G n (z) = 0} and a (n) := inf {ℜz : ζ n (z) = 0} , b (n) := sup {ℜz : ζ n (z) = 0} , satisfy a (n) = -b n , b (n) = -a n for all n ≥ 2. ( 13
)
Concerning the sets Z Gn(z) it is needed to remind that in [13, Propositions 1,2,3] was proved: a) For any n ≥ 2, Z Gn(z) is infinite. b) For every n ≥ 2, there exist real numbers r n , s n , such that Z Gn(z) is contained in the vertical strip {z ∈ C : r n ≤ ℜz ≤ s n }. c) For any n ≥ 2, Z Gn(z) is not contained in the imaginary axis, except for n = 2.

As we have seen in (1.6) the zeros of ζ 2 (z) are all imaginary, then the last part of the theorem follows. Hence, from now on, we assume that n > 2. Let suppose that there exists an integer m > 2 such that the corresponding function G m (z) has its zeros, say (α m,l ) l≥1 , of the form ℜα m,l = 0 for l = 1, ..., p, ℜα m,l = 0 if l > p, for some integer p ≥ 1. [START_REF] Mora | On the Distribution of Zeros of a Sequence of Entire Functions Approaching the Riemann Zeta Function[END_REF] Noticing G n (z) = G n (z) for all z ∈ C, (α m,l ) l≥1 are conjugate. Then p is necessarily even, so p = 2q ≥ 2, and the zeros α m,l with l > p are of the form ±iy j with y j > 0. Let denote by P (z) the polynomial with leader coefficient 1 and zeros (α m,l ) l=1,2,...p . Then

P (z) = (z 2 -2a 1 z + |α m,1 | 2 ) • • • (z 2 -2a q z + |α m,q | 2 ), ( 15
)
where a l := ℜα m,l , l = 1, ..., q. Therefore the function

H m (z) := G m (z) P (z)
is entire of order 1 with zeros (±iy j ) j=1,2,... . By Hadamard's factorization theorem [START_REF] Ash | Complex Variables[END_REF]Th. 4.4.3] we can write

H m (z) = e Az+B ∞ j=1 1 + z 2 y 2 j , ( 16
)
where the constants B and A, because of (2.8), satisfy

e B = H m (0) = m |α m,1 ...α m,q | 2 , A = H ′ m (0) e B = log(m!) m + 2 a 1 |α m,1 | 2 + ... + a q |α m,q | 2 .
By (2.9) the function H m (z)e -Az is even, then H m (z)e -Az = H m (-z)e Az for all z ∈ C. Now, substituting H m (z) by Gm(z) P (z) , we get

P (-z) P (z) = G m (-z)e 2Az G m (z) for all z ∈ C. (17)
If we assume that any zero of P (z) is a zero of P (-z), the polynomials P (z) and P (-z) are equal and their zeros (α m,l ) l=1,2,...p are conjugate and opposite. Hence q = p 2 is even,

a 1 |α m,1 | 2 + ... + a q |α m,q | 2 = 0 and then A = log(m!) m
. But this leads us to the following contradiction: noticing (n!) 2 > n n for all n > 2, the right side of (2.10) tends to +∞ as z ∈ R does, whereas its left side is identically 1. Consequently, at least one zero of P (z) cannot be a zero of P (-z). Thus the left side of (2.10) is a meromorphic function having at least a pole. On the other hand, the right side of (2.10) is a quotient of exponential polynomials with a finite number of poles (the zeros of P (z) that are not cancelled with the zeros of P (-z)). Then Shields's theorem [START_REF] Shields | On Quotients of Exponential Polynomials[END_REF] asserts that Gm(-z)e 2Az Gm(z) is an exponential polynomial, which represents a new contradiction because P (-z) P (z) has at least a pole. As consequence, there is no function G m (z), m > 2, having its zeros of the form (2.7). Therefore, taking into account c), every function G n (z), n > 2, has infinitely many zeros in at least one of the half-planes {z : ℜz < 0} and {z : ℜz > 0}. However G n (z), n > 2, cannot have all its zeros, say (z n,l ) l=1,2,... , with real part of the same sign. Indeed, by expressing G n (z) = 1 + 2 z + ... + n z of the form 1 + e z log 2 + ... + e z log n , by Ritt's formula [19, formula (9)], we have

∞ l=1 ℜz n,l = O(1). (18)
Now, by assuming, for instance, ℜz n,l ≥ 0 for all l ≥ 1, from c), there exists at least a zero of G n (z), say z n,l0 , such that ℜz n,l0 > 0. By taking ǫ = ℜz n,l0 and noting that G n (z) is an almost-periodic function, because of Lemma 1, G n (z) has infinitely many zeros in the strip

S ǫ 2 := z : ǫ 2 < ℜz < 3ǫ 2 .
This implies that

∞ l=1
ℜz n,l = +∞, contradicting (2.11). Hence, G n (z) has at least a zero in {z : ℜz < 0} and again by using Lemma 1, G n (z) has infinitely many zeros with negative real part.

Then the theorem follows for the functions G n (z) and, taking into account (2.6), it is also true for the partial sums ζ n (z). The next result is an easy consequence of the above theorem.

Corollary 3

The bounds a n , b n and a (n) , b (n) corresponding to the functions G n (z) and ζ n (z), respectively, satisfy

a n = b n = a (n) = b (n) = 0 if n = 2 and a n < 0, b n > 0, a (n) < 0, b (n) > 0 for all n > 2.
3 The level curves

|G * n (z)| = p x 0 k n , n > 2, x 0 ∈ R
The curves |f (z)| =constant, where f (z) is an analytic function on a given region of the complex plane, are defined as level curves (see for instance Titchmarsh's book [25, p.121]). This concept will be crucial for attaining the main objective of our paper, namely, to prove the existence of non-degenerate intervals contained in the sets

R n := {ℜz : G n (z) = 0} (19)
and

R (n) := {ℜz : ζ n (z) = 0}, (20)
for all integer n > 2.

From the definition of the bounds corresponding to the functions G n (z), ζ n (z), and (3.1), (3.2), it follows

a n , b n ∈ R n ⊂ [a n , b n ] and a (n) , b (n) ∈ R (n) ⊂ a (n) , b (n) ,
for all n ≥ 2. By Corollary 3, the set

R n = R (n) = {0} if and only if n = 2.
Therefore n = 2 is the trivial case and whether n > 2, the sets R n , R (n) have at least two distinct points, namely, the bounds a n , b n and a (n) , b (n) , respectively. For every n > 2 we define the function

G * n (z) := G n (z) -p z kn ,
where G n (z) := 1 + 2 z + ... + n z and p kn is the last prime not exceeding n. The goal is to study properties of the level curves

|G * n (z)| = p x0 kn , (21)
for each integer n > 2 and a given real number x 0 .

In order to emphasize the importance of the level curves, we recall that the sets R n , defined in (3.1), were characterized in [START_REF] Dubon | A note on the real projection of the zeros of partial sums of Riemann zeta function[END_REF] in terms of such curves as follows. By defining the real functions

A n (x, y) := |G * n (x + iy)| -p x kn ; x, y ∈ R, n > 2,
the above result was completed in [START_REF] Mora | On the asymptotically uniform distribution of the zeros of the partials sums of the Riemann zeta function[END_REF] under the form:

Theorem 5 (Mora [15, Th. 2]) For every integer n > 2, a real number x belongs to R n if and only if there exists some y ∈ R such that

A n (x, y) = 0. Furthermore, A n (x, 0) ≥ 0 for all x ∈ [a n , b n ].
By squaring (3.3) we obtain the cartesian equation of the level curve for a given real number x 0 ,

1 + 2 2x + ... + n 2x + 2.1 x n m=2 m x cos y log( m 1 ) + 2.2 x n m=3 m x cos y log( m 2 ) + +... + 2(n -1) x n m=n m x cos y log( m n -1 ) = p 2x0 kn , (22)
where in the left side of (3.4) the integer variables do not take the value p kn . Since, for any value of y, the limit of the left side of (3.4) is +∞ when x → +∞ and the rigth side of (3.4) is fixed, the range of x is always upper bounded for every x 0 . However, as the limit of the left side of (3.4) is 1 when x → -∞, if x 0 = 0, the range of the variable x must be lower bounded. Finally, from equation (3.4), it is immediate that the level curve neither contains a vertical segment nor has a vertical asymptote. Thus, fixed n > 2 and x 0 ∈ R, the variable x in the equation ( For n = 3 one has

p k3 = 3, then G * 3 (z) := G 3 (z) -p z k3 = 1 + 2 z . Hence the level curve is |1 + 2 z | = 3 x0
. By squaring we obtain its cartesian equation,

1 + 2 2x + 2 x+1 cos(y log 2) = 3 2x0 . ( 23
)
The lower extreme associated to Proof. Because of the function of the left side of (3.4) is even with respect to the variable y, we will assume that y ≥ 0. For a given a real number x 0 ≤ 0 we write ( where z j,w is the jth zero of F * n,w (z) with respect to the order defined, transforms the circle |1 + w| = r x0 on a closed curve, whose equation is given by (3.6), and it is the jth arc-connected component of the level curve, and so on. Finally, as the function of the left side of equation (3.6) is even with respect to the variable y, we obtain by simmetry with respect to the real axis the closed arc-connected components of the level curve in the half-plane {z : ℑz < 0}.

x 0 = 0 is b - 3,x0 = log(1-3 x 0 ) log 2 if x 0 < 0 log(3 x 0 -1) log
If x 0 = 0, the right side of (3.6) is 0. Then dividing (3.6) by 2 x+1 , the equation of the level curve is

2 x-1 + ... + n 2 2x 2 x-1 + cos(y log 2) + 1 x n m=3 m 2 x cos y log( m 1 ) + +2 x n m=3 m 2 x cos y log( m 2 ) + ...+ +(n -1) x n m=n m 2 x cos y log( m n -1 ) = 0. ( 27
)
By taking the limit as x → -∞, all the terms of the left side of (3.9), except cos (y log 2), tend to 0. Then the equation cos (y log 2) = 0 gives us the horizontal asymptotes of the level curve. Consequently, the level curve has infinitely many arc-connected components which are open curves (closed on the right) having asymptotes of equations y = (2k + 1) π 2 log 2 , k ∈ Z, oriented to -∞.

For studying the case x 0 > 0, let y 0 be a fixed real number. By substituting it in (3.4), we have an expression of the form 1 + a 2 2 x + a 3 3 x + ... + a n(n-1) (n(n -1))

x + a n 2 n 2x = p 2x0 kn , [START_REF] Wilder | Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points[END_REF] where the a's are real numbers (the left side of (3.10) does not contain the term a p kn p x kn ). Because the left side of (3.10) has been obtained from (3.3) as the sum of the squares of the real and imaginary parts of G * n (z), for z = x + iy 0 , the number of changes of the sign of its coefficients is a nonnegative even integer. Hence, since p 2x0 kn > 1, the number, say W , of changes of the sign of the coefficients of the equation, equivalent to (3.10), (the left side of this equation does not contain the term p x kn ) and we then claim that the above equation has real solution if and only if x 0 > 0. Indeed, clearly the equation has no real solution if x 0 ≤ 0. Then if the equation has solution, x 0 must be positive. Conversely, by assuming x 0 > 0, the number p x0 kn is greater than 1 and W , the number of changes of the sign of the equation 1p x0 kn + 2 x + ... + n x = 0, is 1. Thus, by Pólya and Szëgo's formula [START_REF] Pólya | Problems and theorems in analysis[END_REF], the number, N , of real solutions of the above equation is necessarily

1 -p 2x0 kn + a 2 2 x + a 3 3 x + ... + a n(n-1) (n(n -1)) x + a n 2 n 2x =
p x0 kn = G * n (b - n,x0 + iy) ≤ G * n (b - n,x0 ) < |G * n (c n,x0 )| = p x0 kn , a contradiction. Therefore, b - n,x0 = c n,x0
. Finally, since the variable y takes all real numbers, the unique point that the level curve meets to the real axis is (b - n,x0 , 0) and there is no vertical asymptote, the level curve |G * n (z)| = p x0 kn has only one arc-connected component. The proof is now completed.

From Example 7 and Proposition 8 we obtain the following consequence. kn , defined by the interior and the exterior points of its arc-connected components, respectively. Thus, if

z ′ = (x ′ , y ′ ) is a point satisfy- ing |G * n (z ′ )| < p x0 kn , z ′ is
′ = (x ′ , y ′ ) satisfies |G * n (z ′ )| < p x0
kn , there exists a point z = (x, y) of the level curve on the right of z ′ and, consequently, x > x ′ . The proof is now completed.

The next result shows that the set R n := {ℜz : G n (z) = 0} contains intervals.

Theorem 10 Let [a n, b n ] be the critical interval of G n (z), n > 2, x 0 ∈ R n and b n,x0 its upper extreme associated with respect to the level curve

|G * n (z)| = p x0 kn . Then [x 0 , b n,x0 ] ∩ [a n, b n ] ⊂ R n .
Proof. Since x 0 ∈ R n , because of Theorem 4, there exists a value y 0 such that the point (x 0 , y 0 ) lies on the level curve |G * n (z)| = p x0 kn . From Definition 6, if

x 0 = 0, then x 0 ∈ b - n,x0 , b n,x0 and x 0 ∈ (-∞, b n,x0 ], if x 0 = 0. Therefore, for any x 0 ∈ R n , one has x 0 ≤ b n,x0 . Thus, if x 0 = b n,x0 , since R n ⊂ [a n , b n ], the theorem trivially follows. Let us assume that x 0 < b n,x0 and let x 1 ∈ [a n , b n ] be a point such that x 0 < x 1 < b n,x0 . Then x 1 ∈ b - n,x0 , b n,x0 , or x 1 ∈ (-∞, b n,x0 ], so the line of equation x = x 1 intersects to the level curve |G * n (z)| = p x0 kn at least at a point, say z 1 = x 1 + iy 1 . Consequently |G * n (z 1 )| = p x0
kn . Now, by the definition of A n (x, y), we obtain

A n (x 1 , y 1 ) = |G * n (z 1 )| -p x1 kn = p x0 kn -p x1 kn < 0. On the other hand, since x 1 ∈ [a n , b n ], by Theorem 5, one has A n (x 1 , 0) ≥ 0.
Then, the continuity of A n (x 1 , y) as function of y implies the existence of a value, say y 2 , such that A n (x 1 , y 2 ) = 0. Now, because of Theorem 5, it follows that x 1 ∈ R n and then ] ⊂ R n and hence, in this case, we also get

[x 0 , b n,x0 ) ∩ [a n, b n ] ⊂ R n . ( 29
[x 0 , b n,x0 ] ∩ [a n, b n ] = [x 0 , b n,x0 ] ⊂ R n ,
which proves completely the theorem. An important conclusion on the sets R 3 and R (3) is deduced from Theorem 10.

Corollary 11

The sets R 3 and R (3) are equal to [a 3, b 3 ] and a (3) , b (3) , respectively.

Proof. The bound a 3 := inf {ℜz : G 3 (z) = 0} belongs to R 3 and, by Corollary 3, a 3 is negative. Its upper extreme associated, with respect to the level curve

|G * 3 (z)| = p x0 k3 , determined in Example 7, is b 3,a3 = log(1 + 3 a3 ) log 2 > 0.
Then, by taking x 0 = a 3 in Theorem 10, we have > 0, from (3.12), it deduces in particular that 0 ∈ R 3 . as we have just seen in Example 7, the upper extreme associated to 0 is 1, so, again from Theorem 10, we get

[0, 1] ∩ [a 3, b 3 ] ⊂ R 3 . ( 31
)
From Corollary 3 and [6, Lemma 1], we know that

0 < b 3 ≤ sup {x : 1 + 2 x ≥ 3 x } = 1.
Then, by (3.13), one has

[0, 1] ∩ [a 3, b 3 ] = [0, b 3 ] ⊂ R 3 ,
which, jointly with (3.12), involves that

R 3 = [a 3 , b 3 ] .
Now, taking into account (2.6), one has R (3) = [-b 3 , -a 3 ] = a (3) , b (3) . Then the proof is completed.

From the above corollary it follows that the functions G 3 (z) and ζ 3 (z) have infinitely many zeros arbitrarily near any vertical line contained in [a 3 , b 3 ] × R and a (3) , b (3) × R, respectively. Therefore, the distribution of the real parts of the zeros of G 3 (z) and ζ 3 (z) is completely uniform on their critical intervals.

f n (x 1 ) = b n,x1 < x 2 ≤ b n,x0 = f n (x 0 ), so f n is strictely increasing.
In the next result we find, for each integer n > 2, a lower bound of the set of upper extremes b n,x0 when x 0 ∈ R.

) provided that -1+3 x 0 -1 2 2 1+2.3 x 0 -1 2 1 2 ≤ 1. But 0 < 1+3 x 0 -1 2 2 1+2.3 x 0 -1 2 1 2 ≤ 1 if and only if x 0 ≤ log(6+3 √ 3) log 3 32
. Then, we claim that any x 0 ∈ [0, b 4 ), where b 4 := sup {ℜz : G 4 (z) = 0}, satisfies the above condition. Indeed, because of [6, Lemma 1], one has b 4 ≤ sup {x : 1 + 2 x + 3 x ≥ 4 x } and, on the other hand, it is immediate to check that

sup {x : 1 + 2 x + 3 x ≥ 4 x } < 2 < log 6 + 3 √ 3 log 3 , so x 0 < b 4 < 2 < log(6+3 √ 3) log 3
, and the claim follows. Consequently, for each x 0 ∈ [0, b 4 ), by making x = x 1 in (3.17), there exists a value of y, say y 1 , given by (3.19)

Indeed, if z = x + iy a zero of G 4 (z), one has 1 + 2 z + 4 z = -3 z . Then the real and imaginary parts are equal, 1 + 2 x cos(y log 2) + 4 x cos(y log 4) = -3 x cos(y log 3), 2 x sin(y log 2) + 4 x sin(y log 4) = -3 x sin(y log 3).

By squaring and adding, we obtain the same expression that in (3.9), but the right side term which is equal to 3 2x . That means that x = ℜz, y = ℑz satisfy equation (3.16), with x 0 replaced by x, so necessarily 4 x ≤ 1 + 2.3 x-1 2 . Therefore (3.21) follows, as claimed. In consequence, if x 0 ∈ [0, b 4 ), by (3.21) one has x 0 < β 4 which implies that 4 x0 < 1 + 2.3 x0-1 2 . That is, x 0 satisfies (3.20) and then we have

x 0 < x 1 ≤ b 4,x0 , for all x 0 ∈ [0, b 4 ) . ( 40
Now, if we assume that any zero of P (z) is a zero of P (-z), then (α m,l ) l=1,2,...p are conjugate and opposite, so q is even, and P (z) coincides with P (-z). Thus Corollary 20 For all n > 2, there exist infinitely many zeros, having positive and negative real part, of G n (z) and ζ n (z) arbitrarily close to the imaginary axis. In particular, the point 0 is an accumulation point common to the sets P Gn = {ℜz : G n (z) = 0} and P ζn = {ℜz : ζ n (z) = 0} for all n > 2.

Proof. From Theorem 17, it is enough to take into account that 0 is an interior point to each set R n := P Gn , R (n) := P ζn , for all n > 2.

In [START_REF] Sepulcre | A new approach to obtain points of the closure of the real parts of the zeros of the partial sums 1 + 2 z + ... + n z , n ≥ 2[END_REF]Conjecture 15] Sepulcre and Vidal conjectured that the point 0 ∈ R n for all n ≥ 2. Now, as an easy consequence of Theorem 17, we can give a positive answer to such question.

Corollary 21

The conjecture of Sepulcre and Vidal is true.

Proof. By Corollary 19, ∩ ∞ n=3 R n ⊃ [a, δ], where a < 0 and δ ≥ 0. Then, in particular, 0 ∈ R n for all n > 2. Noting (1.6), the set R 2 = {0}, therefore it follows that 0 ∈ R n for all n ≥ 2. Consequently Sepulcre and Vidal's conjecture is true.

1 2

 1 +ǫ considered by Turán in [26, Th. I]. Hence, this old theorem of Turán of 1948 became vacuous and consequently the hope of proving R.H. via such hypothetical localization of the zeros of the partial sums of the Riemann zeta function died.

Lemma 1

 1 which proves Sepulcre and Vidal's conjecture [20, Conjecture 15]. 2 Distribution of the zeros of ζ n (z) on the halfplanes ℜz > 0 and ℜz < 0 Let f (z) be an almost-periodic function not identically null on a strip S a,b := {z : a < ℜz < b}. If f (z) has a zero at a point z 0 ∈ S a,b then it has infinitely many zeros in S ǫ := {z : ℜz 0ǫ < ℜz < ℜz 0 + ǫ} for arbitrary ǫ > 0.

2 such

 2 that |g(z)| = 0 for all z on the circle |zz 0 | = r, so the number m := min {|g(z)| : |zz 0 | = r} > 0. Noticing |zz 0 | = r is a compact set in the strip S a,b and lim k→∞ g n k (z) = g(z) is uniform on the compacts of S a,b , there exists a positive integer k 0 such that |g(z)g n k (z)| < m ≤ |g(z)| for all |zz 0 | = r and for all k > k 0 .

Theorem 2

 2 Every partial sum of the Riemann zeta function, ζ n (z) := n k=1 1 k z , n ≥ 2, possesses infinitely many zeros in each half-plane {z : ℜz < 0} and {z : ℜz > 0} except for n = 2 whose zeros are all on the imaginary axis.

Theorem 4 (

 4 Dubon, Mora, Sepulcre, Ubeda, Vidal [6, Th. 2]) A real number x 0 belongs to R n , n > 2, if and only if the vertical line x = x 0 intersects the level curve |G * n (z)| = p x0 kn .

Definition 6 Example 7

 67 3.4) either runs on a finite interval b - n,x0 , b n,x0 , if x 0 = 0, or on the infinite interval (-∞, b n,x0 ] , if x 0 = 0, where the extremes b - n,x0 , b n,x0 are defined as follows. Fixed an integer n > 2 and a real number x 0 , we define b n,x0 , called upper extreme associated to x 0 , as the maximum value that we can assign to the variable x to have, for at least a value of y, a point (b n,x0 , y) of the level curve |G * n (z)| = p x0 kn . When x 0 = 0, we define b - n,x0 , called lower extreme associated to x 0 , as the minimum value that we can assign to the variable x to have, for at least a value of y, a point b - n,x0 , y of the level curve |G * n (z)| = p x0 kn . To illustrate the above definition we determine in the next example the points extremes corresponding to the level curve |G * n (z)| = p x0 kn for n = 3. The level curve |G * 3 (z)| = p x0 k3 , x 0 ∈ R.

2 if x 0 > 0 and the upper extreme associated to any x 0 is b 3 2 ,Proposition 8

 20328 x 0 = 0, by dividing (3.5) by 2 x+1 , the equation of the level curve is 2 x-1 + cos(y log 2) = 0 and then x runs on (-∞, 1].Thus, for x 0 < 0, (3.5) represents an infinite family of closed curves containing each of them one zero of G * 3 (z), so the level curve has infinitely many arc-connected components. For x 0 = 0, the above equation represents an infinite family of open curves having horizontal asymptotes oriented to -∞ of equations y = (2k + 1) π 2 log 2 , k ∈ Z, so the level curve has infinitely many arcconnected components. Finally, for x 0 > 0,(3.5) is the equation of a unique open curve, that is the level curve has only one arc-connected component. The curve meets to the real axis at the point b - 3,x0 = log(3 x 0 -1) log and the variable y takes all real numbers. The next result proves that the level curve |G * n (z)| = p x0 kn for n > 3 is essentially the same as the case n = 3. Fixed an integer n > 2 and a real number x 0 , let us consider the level curve |G * n (z)| = p x0 kn . Then, if x 0 < 0, the level curve has infinitely many arc-connected components which are closed curves; if x 0 = 0, the level curve has infinitely many arc-connected components which are open curves with horizontal asymptotes, as x → -∞, of equations y = (2k + 1) π 2 log 2 , k ∈ Z. Moreover, the level curve only meets to the real axis at a point, say c n,x0 , if and only if x 0 > 0; in this case the lower extreme b - n,x0 = c n,x0 and the level curve has only one arc-connected component which is an open curve where the variable y takes all real values.

π 2 log

 2 (n-1) , if n is prime. Now assume that x 0 < 0. By writing |G * n (z)| = p x0 kn as |1 + w| = r x0 , where w := 2 z + ...+ n z (this expression does not contain the term p z kn ) and r x0 := p x0 kn , the equation (3.6) describes the continuous motion of each zero z = (x, y) of the function F * n,w (z) := G * n (z) -1w when w runs on the circle |1 + w| = r x0 . Indeed, fixed n > 2, for each w of the circle |1 + w| = r x0 the function F * n,w (z) has infinitely many zeros on the halfplane {z : ℑz ≥ 0} situated on a vertical strip. These zeros will be supposed lexicografically ordered. That is, given a point w of the circle |1 + w| = r x0 , a zero z w = (x, y) of F * n,w (z) precedes to other zero z ′ w = (x ′ , y ′ ) if and only if either y < y ′ or x < x ′ , if y = y ′ . Then, by taking the interval (0, 2π] as a determination of the argument (the domain corresponding to the log function is Ω := C \ [0, +∞)), let us define the map ϕ 1 (w) := z 1,w , that assigns to each w of the circle |1 + w| = r x0 the first zero, z 1,w , of F * n,w (z) with respect to the order defined. Because of 0 < r x0 < 1, the circle |1 + w| = r x0 is in Ω and the argument of w varies on a certain interval [πα x0 , π + α x0 ] for some α x0 such that 0 < α x0 < π 4 , so [πα x0 , π + α x0 ] ⊂ (0, 2π]. Thus, the function ϕ 1 transforms the circle |1 + w| = r x0 on a closed curve whose equation is given by (3.6) and it constitutes the first arc-connected component of the level curve. By analogy, the function ϕ j (w) := z j,w , j > 1,

1 .

 1 Hence the above equation has only a real solution, say c n,x0 , and the claim follows. Furthermore, since (c n,x0 , 0) is a point of the level curve, from Definition 6, one has b - n,x0 ≤ c n,x0 . But, if we suppose that b - n,x0 < c n,x0 , again from Definition 6, for some y, the point b - n,x0 , y belongs to the level curve and then G * n (b - n,x0 + iy) = p x0 kn . Now, because G * n (z) is strictely increasing on the real axis, we obtain

Corollary 9

 9 For a given integer n > 2 and a real number x 0 , let |G * n (z)| = p x0 kn be the corresponding level curve. If a point z ′ = (x ′ , y ′ ) satisfies |G * n (z ′ )| < p x0 kn , then there exists a point z = (x, y) of the level curve such that x > x ′ . Proof. By Example 7 and Proposition 8, if x 0 < 0, the level curve |G * n (z)| = p x0 kn has infinitely many arc-connected components which are closed curves. Thus, the level curve divides to the complex plane in two regions w : |G * n (w)| < p x0 kn and w : |G * n (w)| > p x0

  in the interior of a closed arc-connected component and consequently it must have a point z = (x, y) of the level curve situated on the right of z ′ , so x > x ′ . The case x 0 = 0 is analogous to the previous one because the arc-connected components (open curves) are closed on the right side and open on the left side with asymptotes oriented to -∞. Finally, if x 0 > 0, the variable y take all real values and x runs on the interval b - n,x0 , b n,x0 . Then, since for any n > 2, max |G * n (w)| : ℜw ≤ b - n,x0 = G * n (b - n,x0 ), the set w : |G * n (w)| < p x0 kn coincides exactly with the region situated on the left of the level curve. Hence, if z

)

  By supposing b n,x0 > b n , from(3.11), we get[x 0 , b n,x0 ] ∩ [a n, b n ] = [x 0 , b n,x0 ) ∩ [a n, b n ] ⊂ R nand then the theorem follows. If b n,x0 ≤ b n , since x 0 ∈ R n ⊂ [a n, b n ], again by (3.11), one has [x 0 , b n,x0 ) ∩ [a n, b n ] = [x 0 , b n,x0 ) ⊂ R n , but, as R n is closed, the point b n,x0 also belongs to R n . Then, [x 0 , b n,x0

a 3 , 2 ≥ b 3 2 < b 3 .

 32323 log(1 + 3 a3 ) log 2 ∩ [a 3, b 3 ] ⊂ R 3 . (30) Hence, the corollary follows, if log(1+3 a 3 ) log , because R 3 ⊂ [a 3, b 3 ]. Let us assume log(1+3 a 3 )log Then, since a 3 < 0 and log(1+3 a 3 ) log 2

Lemma 13

 13 For every n > 2, let us denote b * n = sup {ℜz : G * n (z) = 0}. Then b * n < b n,x0 for all x 0 ∈ R. Proof. Let us assume that G * n (z) has a zero, say w, on the vertical line x = b * n . Then ℜw = b * n . Now, let x 0 be a real number. Then, since G * n (w) = 0, one has |G * n (w)| < p x0 kn . From Corollary 9, there exists a point z 1 = (x 1 , y 1 ) of the level curve |G * n (z)| = p x0 kn such that b * n < x 1 . By Definition 6 it follows that x 1 ≤ b n,x0 and then b * n < x 1 ≤ b n,x0 , which proves the lemma under the above assumption. If G * n (z) has no zero on the line x = b * n , according to the definition of b * n , there exists a sequence of zeros (z m ) m of G * n (z) with ℜz m < b * n such that lim m ℜz m = b * n . Let us consider a real number x 1 < x 0 . Then, as G * n (z m ) = 0, we get |G * n (z m )| < p x1 kn for all m. This implies, from Corollary 9, the existence of a sequence of points of the level curve |G * n (z)| = p x1 kn , say (w m ) m , such that ℜz m < ℜw m for all m. On the other hand, since every w m is on the level curve |G * n (z)| = p x1 kn , from Definition 6, one has ℜw m ≤ b n,x1 for all m. Hence, we have ℜz m < ℜw m ≤ b n,x1 for all m and, taking the limit when m → ∞, we obtain b * n ≤ b n,x1 . (

) Let x 0

 0 be an arbitrary point such that 0 < x 0 < b 4 . The point b - 4,x0 , by Proposition 8, is obtained by making y = 0 in equation(3.16), so b - 4,x0 is the real solution of the equation1 + 2 x + 4 x = 3 x0 . (41)Then we claim that b - 4,x0 < x 0 . Indeed, if b - 4,x0 ≥ x 0 , from (3.23), we would have3 x0 = 1 + 2 b - 4,x 0 + 4 b - n,x 0 ≥ 1 + 2 x0 + 4 x0 > 3 x0 ,where a l := ℜα m,l , l = 1, ..., q. By considering the functionH * m (z) := G * m (z) P(z) , entire of order 1 with zeros α m,l for l > p, by Hadamard's factorization theorem [1, Th. 4.4.3], we have H * m (z) = e Amz+Bm constants B m and A m are given by the relationse Bm = H * m (0) = m -1 |α m,1 ...α m,q | 2 , A m = (H * m ) ′ (0) e B = log( m! p km ) m -1 + 2 a 1 |α m,1 | 2 + ... + a q |α m,q | 2 ,where p km denotes, as usual, the last prime not exceeding m. Noticing (4.3), H * m (z)e -Amz is an even function, thenH * m (z)e -Amz =H * m (-z)e Amz for all z ∈ C. Hence, substituting H * m (z) by G * m (z) P (z) , we get P (-z) P (z)G * m (-z) = e 2Amz G * m (z) , for all z ∈ C.

a1 |αm,1| 2 +

 2 ... + aq |αm,q| 2 = 0, soA m = log( m! p km ) m -1 . (46)By writing G * m (z) = 1 + 2 z + ... + (m * ) z (it does not have the term p z km ), wherem * := m -1, if m is prime m, otherwise. ,we claim that 2A m -log m * > 0 for all m > 4. Indeed, by assuming that m is prime, m = p km , so m * = m -1 and, by (4.5), one has2A m -log m * = 2 log((m -1)!) m -1 -log(m -1). (47)Remark 18 Observe that the proof of Theorem 17, valid for all n > 2, is essentially based on the old notion of level curve. Nevertheless (1.7) follows for all n > N , for some positive integer N whose existence depends of the Prime Number Theorem (see[START_REF] Mora | On the asymptotically uniform distribution of the zeros of the partials sums of the Riemann zeta function[END_REF] Lemma 7]). Then [15, Th.12] improves the Theorem 17 of an optimal form, provided that n > N , because the δ n 's are equal to b n = -a (n) = n log 2 + o(n), n → ∞, by virtue of (1.5). The same occurs when n = 3, 4, by virtue of Corollary 11 and Theorem 14, respectively.An important conclusion is deduced from Theorem 17, namely, each set∩ ∞ n=3 R n , ∩ ∞ n=3 R (n) contains a non-degenerated closed interval.Corollary 19 There exist δ ≥ 0 and a < 0 such that∩ ∞ n=3 R n ⊃ [a, δ] and ∩ ∞ n=3 R (n) ⊃ [-δ, -a].Proof. By virtue of Theorem 17, we define δ := inf {δ n : n > 2} and then, since δ n > 0 for all n > 2, necessarily δ ≥ 0. By (1.4), all the terms of the sequence b (n) n>2 , except at most a finite quantity of them, are arbitrarily close to 1. By Corollary 3, b (n) > 0 for all n > 2, then b := min b (n) : n > 2 is a positive number. On the other hand, by (2.6), one has -a n = b (n) for all n > 2, so a := max {a n : n > 2} = -b < 0. Now the proof is completed.

  0, is odd. Then, by Pólya and Szëgo's formula[18, p. 46], W -N is an even nonnegative integer, where N denotes the number of real zeros (counting multiplicities) of the preceding equation. Therefore, N must be positive, which implies that equation (3.10) has solution. Consequently, as y 0 is arbitrary, the variable y takes all the real values. On the other hand, by making y = 0 in (3.4), one has 1 + 2 x + ... + n x = p x0 kn

  Lemma 12 For every n > 2, let us denote by f n the function that assigns to each real number x 0 its upper extreme associated b n,x0 with respect to the level curve |G * n (z)| = p x0 kn . Then f n is a strictely increasing function. Proof. Let us consider two real numbers x 1 < x 0 . Then, since p kn ≥ 2 for all n ≥ 2, one has p x1 kn < p x0 kn . From Definition 6, given f n (x 1 ) := b n,x1 , there exists some y 1 such that the point z 1 = (b n,x1 , y 1 ) belongs to the level curve |G * n (z)| = p x1 kn . Then, since p x1 kn < p x0 kn , the point z 1 satisfies |G * n (z 1 )| < p x0 kn . By applying Corollary 9, there exists a point z 2 = (x 2 , y 2 ) of the level curve |G * n (z)| = p x0 kn such that b n,x1 < x 2 . Now, as z 2 belongs to the level curve |G * n (z)| = p x0 kn , again from Definition 6, we have x 2 ≤ b n,x0 = f n (x 0 ). Thus

  , determining a point (x 1 , y 1 ) of the level curve |G * 4 (z)| = 3 x0 . Thus, by Definition 6, b - 4,x0 ≤ x 1 ≤ b 4,x0 . On the other hand, from the definition of x 1 , one has x 1 > x 0 if and only if 4 x0 < 1 + 2.3 x0-

	1 2 .	(38)
	We define β

4 := sup x : 4 x ≤ 1 + 2.3 x-1 2

. Then, from Pólya and Szëgo's formula

[18, p. 46

], the equation 4 x = 1 + 2.3 x-1 2 has a unique real solution, so equal to β 4 . Now we claim that b 4 ≤ β 4 .

Now, by Lemma 12, since x 1 < x 0 , it follows that b n,x1 < b n,x0 and, noting inequality (3.14), we get b * n ≤ b n,x1 < b n,x0 , which completes the proof.

The above lemma, jointly with Theorem 10, allows us to determine the sets R n and R (n) in the case n = 4.

Theorem 14

The sets R 4 and R (4) are equal to [a 4, b 4 ] and a (4) , b (4) , respectively.

Proof. The last prime not exceeding 4 is 3, so p k4 = 3. Then G * 4 (z) = 1+2 z +4 z , and its zeros are 2π(3k + 1)i 3 log 2 , 2π(3k + 2)i 3 log 2 , k ∈ Z.

Hence, b * 4 := sup {ℜz : G * 4 (z) = 0} = 0. By applying Theorem 10 to x 0 = a 4 , which is negative by Corollary 3, one has

Since cos (y log 4)) = 2 cos 2 (y log 2) -1, from (3.16), we obtain

and then the variable x must satisfy (

Now, for x 0 ≥ 0, is 1 -2.3 x0-1 2 < 0. Thus, noting that 4 x > 0 for any x, from (3.18) it follows that the variable x satisfies

Then, by taking x = x 1 :=

)) 2 = 0), there exists a value for y, given by cos(y log 2) = - ), the set R (4) corresponding to the partial sum ζ 4 (z) also satisfies 4) , b (4) .

The proof is now completed.

4 Distribution of the zeros of G * n (z) on the halfplanes ℜz > 0 and ℜz < 0 Now it is needed to prove a similar result to that of Theorem 2 for the functions G * n (z).

Theorem 15 Every function G * n (z), n > 2, possesses infinitely many zeros in each half-plane {z : ℜz < 0} and {z : ℜz > 0} except for n = 3, 4, whose zeros are all imaginary.

Proof. For n = 3 one has p k3 = 3. Then G * 3 (z) = 1 + 2 z , whose zeros are

For n = 4, p k4 = 3, so G * 4 (z) = 1 + 2 z + 4 z and then its zeros are

Hence, for n = 3, 4, the zeros of G * n (z) are all imaginary and, consequently, the theorem follows in both cases.

Assume that, for some m > 4, there exists a function G * m (z) whose zeros (α m,l ) l=1,2,... are of the form

for some positive integer p. By noting that G * m (z) = G * m (z) for all z ∈ C, the zeros of G * m (z) are conjugate and then necessarily p is even. Hence, for some integer q, the number p = 2q ≥ 2 and the zeros α m,l , if l > p, are of the form ±iy j with y j > 0 for all j ≥ 1. Let us denote by P (z) the polynomial with leader coefficient 1 and zeros (α m,l ) l=1,2,...p . Then

But, 2 log((m-1)!) m-1

-log(m -1) is positive by virtue of the inequality

which can be easily proved by induction and by the mean value theorem applied on the log function. Hence the claim follows provided that m be a prime number. Let us suppose that m is a composite number greater than 5. From (4.5) and the definition of m * , we get

Then 2A m -log m * is positive if and only if (m!) 2 > m m-1 (p km ) 2 . But this last inequality, taking into account that p km ≤ m -1, can be immediately deduced from the inequality, sharpest than (4.7),

which can be also demonstrated by using induction and by the mean value theorem applied on the log function. Hence,

as we claimed. Now, by making z = x > 0 in (4.4) and by taking the limit when x → +∞, because of (4.8), the left side of (4.4) tends to 1 whereas the right side tends to +∞, a contradiction. Therefore, there is at least one zero of P (z) which is not a zero of P (-z). Then, by writing (4.4) of the form

the left side of (4.9) is a meromorphic function with at least a pole, whereas the right side is a quotient of exponential polynomials with a finite number of poles. It means, by Shields's theorem [START_REF] Shields | On Quotients of Exponential Polynomials[END_REF], that the right side of (4.9) is an exponential polynomial and then we are led to a new contradiction. Therefore, there is no integer m > 4 for which the function G * m (z) have its zeros of the form (4.1) and, in particular, are not all of them situated on the imaginary axis. Indeed, if this were so, by defining the polynomial P (z) as identically 1, we would be led to the following contradiction: by taking the limit when z = x → +∞ in (4.4), the left side of (4.4) tends to 1 whereas the right side one tends to +∞. Consequently, all the functions G * n (z), n > 4, have infinitely many zeros in at least one of the half-planes {z : ℜz < 0}, {z : ℜz > 0}. Let (z * n,l ) l=1,2,... be all the zeros of G * n (z) for a fixed n > 4. Suppose, for example, that ℜz * n,l ≥ 0 for all l ≥ 1 and let z * n,j be a zero of G * n (z) such that ℜz * n,j > 0. Then if we take ǫ = ℜz * n,j , noticing G * n (z) is an almost-periodic function, because Lemma 1, G * n (z) has infinitely many zeros in the strip

On the other hand, since G * n (z) is an exponential polynomial with all its coefficients equal to 1, by Ritt's formula [19, formula (9)], we have The next result is obtained directly from the above theorem.

Corollary 16

The bounds corresponding to the functions


Zhiqiang Gao

Quansheng Liu
email: quansheng.liu@univ-ubs.fr

Hesong Wang
email: hesongwang@tom.com

Central limit theorems for a branching random walk in a random environment

Keywords: 2000 Mathematics Subject Classification. 60J10, 60F05, 60J80 Branching random walk, random environment in time, central limit theorems

HAL is

Introduction

The model of branching random walks has been well studied in the literature: see e.g. [START_REF] Asmussen | Branching random walks. I[END_REF][START_REF] Biggins | Martingale convergence in the branching random walk[END_REF][START_REF] Biggins | The central limit theorem for the supercritical branching random walk, and related results[END_REF][START_REF] Kaplan | Branching random walks. II[END_REF][START_REF] Révész | Random walks of infinitely many particles[END_REF] and the reference therein. In the classical branching random walk, the point processes indexed by the particles u, formulated by the number of its children and their displacements, have a fixed constant distribution for all particles u. In reality these distributions may vary from generation to generation according to a random environment, just as in the case of a branching process in a random environment introduced in [START_REF] Athreya | On branching processes with random environments. I. Extinction probabilities[END_REF][START_REF] Athreya | On branching processes with random environments. II. Limit theorems[END_REF][START_REF] Smith | On branching processes in random environments[END_REF]. In other words, the distributions themselves may be realizations of a stochastic process, rather than being fixed. Here, we study such a model which we call a branching random walk with a random environment in time. In this model, the offspring distributions of a particle of generation n and the distributions of the displacements of its children depend on an environment indexed by n. This model was first introduced by Biggins and Kyprianou (2004, [START_REF] Biggins | Measure change in multitype branching[END_REF]). It is different to the usual branching random walks in a random environment (see e.g. [START_REF] Baillon | A variational approach to branching random walk in random environment[END_REF][START_REF] Greven | Branching random walk in random environment: phase transitions for local and global growth rates[END_REF]), in which the authors considered the case where the offspring distribution of a particle situated at z ∈ R depends on a random environment indexed by z, while the moving mechanism is controlled by a fixed deterministic law. Some limit theorems on this model were announced in Liu (2007, [15]). Here we are interested in central limit theorems for the counting measure Z n (•) which counts the number of particles of generation n situated in a given set. For the classical branching random walk, such central limit theorems were proved by Kaplan and Assmussen (1976, [12]), which confirms a conjecture of Harris in 1963 ([10]). For more results on this subject, see e.g. Kaplan (1982, [11]), Klebaner (1982, [13]) and Biggins (1990, [6]).

The main objective of the present paper is to extend the basic results of Kaplan and Assmussen (1976, [12]) to the random environment case. The main results will be stated in Section 2, while their proofs will be given in Sections 3 ∼ 5.

2 Description of the model and the main results

Description of the model

A random environment in time ξ = (ξ n ) is formulated as a sequence of random variables independent and identically distributed with values in some measurable space (Θ, F). Each realization of ξ n corresponds to two probability distributions p(ξ n ) and G ξn , where p(ξ n ) = {p k (ξ n ) : k ∈ N} is a probability law on N and G ξn a probability law on R. Without loss of generality, we can take ξ n as coordinate functions defined on the product space (Θ N , F ⊗N ), equipped with a probability law τ which is invariant and ergodic under the usual shift transformation θ on Θ N :

θ(ξ 0 , ξ 1 , • • • ) = (ξ 1 , ξ 2 , • • • ) .
When the environment ξ = (ξ n ) is given, the process can be described as follows. The process begins at time 0 with one initial particle ∅ of generation 0 located at

S ∅ = 0 ∈ R; at time 1, it is replaced by N = N ∅ new particles of generation 1, located at L i = L ∅i , 1 ≤ i ≤ N,
where N is of distribution p(ξ 0 ) and each L i is of distribution G ξ0 , given the environment ξ. In general, each particle u = u 1 ...u n of generation n is replaced at time n + 1 by N u new particles of generation n + 1, with displacements L u1 , L u2 , • • • , L uNu , so that the i-th child is located at

S ui = S u + L ui ,
where N u is of distribution p(ξ n ) and each L ui is of distribution G ξn , given the environment ξ. All the random variables N u and L u , indexed by all finite sequences u of positive integers, are independent of each other, given the environment ξ. We abbreviate G ξn as G n in the rest of this chapter.

Let (Γ, P ξ ) be the probability space under which the process is defined when the environment ξ is fixed. As usual, P ξ is called quenched law. The total probability space can be formulated as the product space (Γ × Θ N , P), where P = P ξ ⊗ τ in the sense that for all measurable and positive function g, we have gdP = g(y, ξ)dP ξ (y)dτ (ξ), (recall that τ is the law of the environment ξ). The probability P is called annealed law. The quenched law P ξ may be viewed as the conditional probability of the annealed law P given ξ. We will use E ξ to denote the expectation with respect to P ξ . Other expectations will be denoted simply E (there will be no confusion according to the context).

Let T be the genealogical tree with {N u } as defining elements. By definition, we have: (a)

∅ ∈ T; (b) ui ∈ T implies u ∈ T; (c) if u ∈ T, then ui ∈ T if and only if 1 ≤ i ≤ N u . Let
T n = {u ∈ T : |u| = n} be the set of particles of generation n, where |u| denotes the length of the sequence u and represents the number of generation to which u belongs.

The main results

Let Z n (•) be the counting measure of particles of generation n: for B ⊂ R,

Z n (B) = u∈Tn 1 B (S u ).
Then {Z n (R)} constitute a branching process in a random environment (see e.g. [START_REF] Athreya | On branching processes with random environments. I. Extinction probabilities[END_REF][START_REF] Athreya | On branching processes with random environments. II. Limit theorems[END_REF][START_REF] Smith | On branching processes in random environments[END_REF]).

For n ≥ 0, define

m n = m n (ξ) = ∞ k=1 kp k (ξ n ), l n = l n (ξ) = xdG n (x), σ 2 n = σ 2 n (ξ) = x 2 dG n (x) -l 2 n .
Throughout the paper, we consider the supercritical case where

E ln m 0 > 0 and E 1 m 0 ∞ k=2 (k ln k)p k (ξ 0 ) < ∞ (2.1)
For simplicity, we assume that p 0 (ξ 0 ) = 0 a.s.

(2.2)

We also need the following assumptions:

El 0 < +∞ and 0 < Eσ 2 0 < +∞. (2.3)
The following the σ-fields will be used: for n ≥ 1,

F n = σ(ξ, N u : |u| < n), D n = σ(ξ, N u , L ui : i ≥ 1, |u| < n), I n = σ(ξ k , N u , L ui : k < n, i ≥ 1, |u| < n).
We introduce the notations:

ℓ n = ℓ n (ξ) = l 0 + l 1 + • • • + l n-1 , s 2 n = s 2 n (ξ) = σ 2 0 + σ 2 0 + • • • + σ 2 n-1 , m n (2) = m n (2, ξ) = ∞ k=1 k 2 p k (ξ n ), γ n = γ n (ξ) = (m n (2) -m 2 n ) 1/2 , π 0 = 1 and π n = π n (ξ) = m 0 • • • m n-1 for n ≥ 1.
It is well known that

W n = (π n ) -1 Z n (R), n ≥ 1
constitute a martingale with respect to F n . Under (2.1), the limit

W = lim n W n
exists a.s. with EW = 1 (see e.g. [START_REF] Athreya | On branching processes with random environments. II. Limit theorems[END_REF]); by (2.2), W > 0 a.s.. Now our main results can be formulated as follows.

Theorem 2.1. Assume that (2.1)-(2.3) hold. Then for all t ∈ R, as n → ∞,

(π n ) -1 Z n (ℓ n + s n t) → Φ(t)W in probability, (2.4
)

where Z n (x) = Z n ((-∞, x]) for x ∈ R and Φ(t) = t -∞ (2π) -1/2 e -x 2 /2 dx.
Under some additional moment conditions, the convergence in probability can be improved to the a.s. convergence. Theorem 2.2. Assume that (2.1)-(2.3) hold and that for some η > 0,

E 1 m 0 ∞ k=2 k(ln k) 2+η p k (ξ 0 ) < +∞, (2.5)
then it is almost sure that for all t ∈ R, as n → ∞,

(π n ) -1 Z n (ℓ n + s n t) → Φ(t)W. (2.6)
Also by a theorem of Klebaner (1982, [13]), we can prove the following a.s. convergence under another additional moment condition. We should note the condition (2.7) here and the above one (2.5) do not imply each other.

Theorem 2.3. Assume that (2.1)-(2.
3) hold and that

E ln + γ 2 0 m 2 0 < +∞, (2.7)
then it is almost sure that for all t ∈ R, as n → ∞,

(π n ) -1 Z n (ℓ n + s n t) → Φ(t)W. (2.8)
Remark 2.4. By the monotonicity of Z n (x) and the continuity of Φ(x), it can be easily seen that (2.4) implies that for all t ∈ R, as n → ∞,

(π n ) -1 Z n (ℓ n + s n t + o( √ n)) → Φ(t)W in probability. (2.9)
Here we have use the fact that s n ∼ √ nσ (where σ = Eσ 2 0 ). By the same reason, (2.4) (or (2.9)) implies that for all t ∈ R, as n → ∞,

(π n ) -1 Z n (ℓ n + √ nσt + o( √ n)) → Φ(t)W in probability. (2.10)
Similarly, (2.6) (or (2.8)) implies that a.s. for all t ∈ R, as n → ∞,

(π n ) -1 Z n (ℓ n + √ nσt + o( √ n)) → Φ(t)W. (2.11)
For the classical branching random walk, (2.10) reduces to the result of Kaplan and Asmussen [START_REF] Kaplan | Branching random walks. II[END_REF]. We believe that it is possible to relax the moment condition (2.7) for the a.s. result (2.8) and extend the result to the more general case where the displacements of the children of the same parent can have different distributions. This is true for the deterministic case: see e.g. Kaplan (1982, [11]) and Biggins (1990, [6]).

3 Proof of Theorem 2.1

Notations and a key decomposition

We first introduce some notations. As usual, we write N * = {1, 2, 3, • • • } and denote by

U = ∞ n=0 (N * ) n
the set of all finite sequences, where (N * ) 0 = {∅} contains the null sequence ∅.

For all u ∈ U , let T(u) be the shifted tree of T at u with defining elements {N uv }: we have 1)

∅ ∈ T(u), 2) vi ∈ T(u) ⇒ v ∈ T(u) and 3) if v ∈ T(u), then vi ∈ T(u) if and only if 1 ≤ i ≤ N uv . Define T n (u) = {v ∈ T(u) : |v| = n}. Recall that T = T(∅) and T n = T n (∅).
For u ∈ (N * ) k (k ≥ 0) and n ≥ 1, let S u be the position of u and write

Z n (u, B) = v1•••vn∈Tn(u) 1 B (L uv1 + • • • + L uv1•••vn ),
then the law of Z n (u, B) under P ξ is that of Z n (B) under P θ k ξ . Define

W n (u, B) = (π n (θ k ξ)) -1 Z n (u, B), W n (u, t) = W n (u, (-∞, t]), W n (B) = (π n ) -1 Z n (B), W n (t) = W n ((-∞, t]). By definition, π n (θ k ξ) = m k • • • m k+n-1 , Z n (B) = Z n (∅, B) (∅ is the initial particle), W n (B) = W n (∅, B), W n = W n (R) and fixed the environment ξ, for k ≤ n, Z n (B) = u∈T k Z n-k (u, B -S u ). (3.1)
Notice that for u ∈ U ,

Z n-k (u, B -S u ) = v1•••v n-k ∈T n-k (u) 1 B (S uv1•••v n-k )
represents the descendants of u at time n situated in B.

Hereafter, we will use the following notations for the convention:

P ξ,n (•) = P ξ (•|I n ), P n (•) = P(•|I n ); E ξ,n (•) = E ξ (•|I n ), E n (•) = E(•|I n ).
For each n, we choose an integer k n < n as follows. Let β with 1/2 < β < 1 and α > 2/(β -1 -1). For j ∈ N and j α/β ≤ n < (j + 1) α/β , set k n = a j = [j α ]. Let t n = ℓ n + s n t for t ∈ R and n ≥ 1. Then by (3.1),

W n (t n ) = A n + B n + W kn Φ(t), (3.2)
where

A n = 1 π kn u∈T kn {W n-kn (u, t n -S u ) -E ξ,kn (W n-kn (u, t n -S u ))} , B n = 1 π kn u∈T kn [E ξ,kn (W n-kn (u, t n -S u )) -Φ(t)].
Here we remind that the random variables W n-kn (u, t n -S u ) are independent of each other under the conditional probability P ξ,kn .

For the proof of Theorem 2.1, as in Asmussen and Kaplan [START_REF] Kaplan | Branching random walks. II[END_REF], the main idea is to use the decomposition formula (3.2), proving that A n → 0 and B n → 0 in probability. However, new ideas will be needed for the proof of the later due to the appearance of the random environment. For the proof of Lemma 3.1, we shall use some method which is different from that of Kaplan and Asmussen [START_REF] Kaplan | Branching random walks. II[END_REF].

Two lemmas

Proof of Lemma 3.1. For convenience, we introduce the following notations: for n ≥ 1, u ∈ T kn ,

X n,u = W n-kn (u, t n -S u ) -E ξ,kn (W n-kn (u, t n -S u ), Y n = 1 Z kn u∈T kn X n,u ; X n,u = X n,u 1 {|Xn,u|<Z kn } , Y n = 1 Z kn u∈T kn X n,u .
We shall prove that ∀ε > 0,

P kn (|Y n | > ε) n→∞ ----→ 0. (3.5)
From this and that

Z kn (R) π kn
→ W > 0 a.s., we shall have that ∀ε > 0

P kn (|A n | > ε) n→∞ ----→ 0.
Now by the dominated convergence theorem, (3.3) follows.

We proceed the proof of (3.5) in 4 steps.

Observe that ∀ε > 0,

P kn (|Y n | > 2ε) ≤ P kn (|Y n -E ξ Y n | > ε) + P kn (|E ξ Y n | > ε) ≤ P kn (Y n = Y n ) + P kn (|Y n -E ξ Y n | > ε) + P kn (|E ξ Y n | > ε). (3.6)
Step 1. We first prove that

P kn (Y n = Y n ) n→∞ ----→ 0. (3.7)
By the stationarity of the environment, we have

P kn (Y n = Y n ) ≤ u∈T kn P kn (X n,u = X n,u ) = u∈T kn P kn (|X n,u | ≥ Z kn ) ≤ u∈T kn P kn (W n-kn (u) + 1 ≥ Z kn ) = r n P(W n-kn + 1 ≥ r n ) rn=Z kn ≤ E (W n-kn + 1)1 {W n-kn +1≥rn} rn=Z kn n→∞ ----→ 0,
where in the last step, we have used the facts that {W n-kn +1} is uniformly integrable and Z kn → ∞ a.s.. (The alert reader might have noticed that:

r n P(W n-kn + 1 ≥ r n ) rn=Z kn = Z kn P(W n-kn + 1 ≥ Z kn )).
Step 2. Next we prove that

P kn (|Y n -E ξ,kn Y n | > ε) n→∞ ----→ 0. (3.8)
Notice that

P kn (|Y n -E ξ,kn Y n | > ε) = E kn P ξ,kn (|Y n -E ξ,kn Y n | > ε) ≤ ε -2 E kn   Z -2 kn u∈T kn E ξ,kn X 2 n,u   = ε -2 Z -2 kn u∈T kn E kn X 2 n,u = ε -2 Z -2 kn u∈T kn E kn X 2 n,u 1 {|Xn,u|≤Z kn } ≤ ε -2 Z -2 kn u∈T kn E kn (W n-kn (u) + 1) ∧ Z kn 2 = ε -2 Er -1 n (W n-kn + 1) ∧ r n 2 rn=Z kn n→∞ ----→ 0,
where in the last step, we have used the facts that {W n-kn + 1} is uniformly integrable, that the martingale {W n } converges to W and that Z kn n→∞ ----→ ∞ a.s.. Step 3. We then prove that

P kn (|E ξ,kn Y n | > ε) n→∞ ----→ 0. (3.9)
By Chebyshev's inequality, we see that

P kn (|E ξ,kn Y n | > ε) ≤ ε -1 E kn |E ξ,kn Y n | ≤ ε -1 Z -1 kn u∈T kn E kn |E ξ,kn X n,u | = ε -1 Z -1 kn u∈T kn E kn -E ξ,kn X n,u 1 {|Xn,u|≥Z kn } ≤ ε -1 Z -1 kn u∈T kn E kn |W n-kn (u) + 1|1 {|W n-kn (u)+1|≥Z kn } = ε -1 E |W n-kn + 1|1 {|W n-kn +1|≥rn} rn=Z kn → 0,
where in the last step, we have used the facts that {W n-kn + 1} is uniformly integrable and Z kn n→∞ ----→ ∞ a.s.. For the proof of Lemma 3.2, we first prove the following result. Lemma 3.3. Let u ∈ T kn . For a.e. ξ,

E ξ W n-kn (u, B) = G kn * • • • * G n-1 (B), u ∈ T kn ; E ξ W n-kn (u, t n ) → Φ(t) unif ormly in t as n → ∞, where t n = ℓ n-kn + s n-kn t, ℓ n-kn = l kn + • • • + l n-1 and s 2 n-kn = σ 2 kn + • • • + σ 2 n-1 .
Proof of Lemma 3.3. It is easy to check the first equality, which implies that the second assertion is equivalent to the following : for almost every ξ,

G kn * • • • * G n-1 ( t n ) → Φ(t) uniformly in t as n → ∞. (3.10)
To prove this, we will use the Lindeberg-Feller theorem for an array of random variables, so it suffices to verify the following Lindeberg condition (see e.g. [START_REF] Petrov | Limit theorems of probability theory[END_REF]) : for every ε > 0,

lim n→∞ 1 s 2 n-kn n j=kn {x:|x-lj |>εsn} (x -l j ) 2 dG j (x) = 0 a.s. ( 3.11)
By the Birkhoff ergodic theorem,

lim n→∞ s 2 n-kn n -k n = lim n→∞ s 2 n -s 2 kn n • n n -k n = lim n→∞ s 2 n n -lim n→∞ k n n s 2 kn k n = Eσ 2 0 a.s.
and hence the above condition is equivalent to

lim n→∞ 1 n n j=kn {x:|x-lj |>ε √ n} (x -l j ) 2 dG j (x) = 0 a.s. (3.12)
To see that (3.12) holds, it suffices to notice that for all K > 0 and n > K,

1 n n j=kn {x:|x-lj |>ε √ n} (x -l j ) 2 dG j (x) ≤ 1 n n j=1 {x:|x-lj |>ε √ K} (x -l j ) 2 dG j (x);
by the Birkhoff ergodic theorem, the right-hand side converges a.s. to

E {x:|x-l0|>ε √ K} (x -l 0 ) 2 dG 0 (x) ,
which can be as small as we want when K is large enough . Thus (3.12) holds and (3.10) follows.

So the lemma has been proved.

Proof of Lemma 3.2. We begin with the fact that for almost every environment ξ and u ∈ T kn ,

E ξ,kn (W n-kn (u, t n -S u )) = G kn * • • • * G n-1 (t n -S u ). (3.13)
By Lemma 3.3, we see that

sup x∈R Φ(x) -G kn * • • • * G n-1 x s n-kn + ℓ n-kn → 0 as n → ∞.
Combining this with the fact that t n -S u = s n-kn t + ℓ n-kn + o( √ n), we see that a.s.,

D n-kn (ξ) = sup t∈R |Φ(t) -G kn * • • • * G n-1 (t n -S u )| → 0 as n → ∞. (3.14)
Notice that

|B n | = (π kn ) -1 u∈T kn {E ξ,kn (W n-kn (u, t n -S u )) -Φ(t)} = (π kn ) -1 u∈T kn G kn * G kn+1 * • • • * G n-1 (t n -S u ) -Φ(t) ≤ W kn D n-kn (ξ) + (π kn ) -1 u∈T kn Φ   t n -S u -(ℓ n -ℓ kn ) s 2 n -s 2 kn   -Φ(t) ≤ W kn D n-kn (ξ) + W kn Φ   s n t s 2 n -s 2 kn   -Φ(t) + C π kn u∈T kn |S u -ℓ kn | s 2 n -s 2 kn ,
where C is a suitable constant independent of ξ. The last inequality follows from elementary calculus. In the last expression, the first term tends to 0 a.s. by (3.14) and the a.s. convergence of W n ; it is clear that the second term tends to 0 a.s.. So the lemma will be proved once we prove that the third term tends to 0 a.s.. In view of the definition of k n , it suffices to show that Proof of Lemma 4.1. We shall follow the notations defined in the proof of Lemma 3.1. By the extended Borel-Cantelli Lemma, we only need to prove that ∀ε > 0,

M j = (π aj ) -1 u∈Ta j j -α/2β |S u -ℓ aj | → 0 as j → +∞. (3.15) Notice that ∞ j=1 E ξ,Fa j {M j } = ∞ j=1 (π aj ) -1 Z aj j -α/2β E ξ,Fa j {|S u -ℓ aj |} ≤ ∞ j=1 W aj j -α/2β E ξ,Fa j {|S u -ℓ aj | 2 } = ∞ j=1 W aj j -α/2β a 1/2 j s 2 aj /a j < ∞ a.s. where E ξ,Fi (•) := E ξ (•|F i ). Therefore ∞ j=1 M j < ∞ a.
∞ n=1 P kn (|A n | > ε) < ∞. (4.4) As Z kn (R) π kn
→ W > 0 a.s., then we shall prove that ∀ε > 0,

∞ n=1 P kn (|Y n | > ε) < ∞. (4.5)
We will get (4.5) if we can prove that the three terms in (3.6) are summable. We deal with them in three steps. We remind that X n,u ≤ W n-kn (u) + 1 = W n-kn • (θ |u| ξ) + 1 for u ∈ T kn .

Step 1. We first prove that Then (4.6) follows.

Step 2. We next prove that ∀ε > 0, Step 3. We then prove that ∀ε > 0,

3 ) 3 . 2 .

 332 Using the decomposition formula (3.2), we proceed the proof of Theorem 2.1 by establishing two lemmas:Lemma 3.1. If (2.1) holds, thenA n → 0 in probability under P.(3.Lemma If (2.3) holds, thenB n → 0 P-a.s. .(3.4)

Step 4 .

 4 By (3.6), (3.7), (3.8) and (3.9), we see that ∀ε > 0, P kn (|Y n | > 2ε) n→∞ ----→ 0. Therefore (3.5) has been proved.

∞ n=1 P 1 n

 n=11 kn (Y n = Y n ) < ∞. (4.6)We observe thatP kn (Y n = Y n ) ≤ E (W n-kn + 1)1 {W n-kn +1≥rn} rn=Z kn ≤ E (W * + 1)1 {W * +1≥rn} rn=Z kn . Notice that r n = Z kn satisfies the condition lim inf n→∞ ln rn n β > 0, because Z kn /π kn → W > 0 a.s., (π n ) → exp{E ln m 0 } a.s. and k n ∼ n β . From this and Lemma 4.2, we obtain that ∞ n=1 E (W * + 1)1 {W * +1≥rn} rn=Z kn < +∞. (4.7)

∞ n=1 P 1 1 c 3 + c 4 (

 n=11134 kn (|Y n -E ξ Y n | > ε) < ∞. (4.8)Observe that ∀u ∈ T kn , n ≥ 1,E kn X 2 n,u = E kn X 2 n,u 1 {|Xn,u|<Z kn } = ∞ 0 2yP kn (|X n,u |1 {|Xn,u|<Z kn } > y)dy ≤ Z kn 0 2yP kn (|X n,u | > y)dy ≤ Z kn 0 2yP kn (|W n-kn (u) + 1| > y)dy = Z kn 0 2yP(|W n-kn + 1| > y)dy ≤ Z kn 0 2yP(|W * + 1| > y)dy.Then we have∞ n=1 P kn (|Y n -E ξ Y n | > ε) = ∞ n=1 E kn P ξ,kn (|Y n -E ξ,kn Y n | > ε) {y<Z kn } • 2yP(|W * + 1| > y)dy ≤ ε -2 c 2 + ∞ ln y) 1/β-1 P(|W * + 1| > y)dy ≤ ε -2 (c 2 + c 5 E(W * + 1) ln η (W * + 1)) < ∞,where in the second line we have used Chebyshev's inequality, in the last two inequalities we use the results on {Z kn } and the constant c 2 depends only on {Z kn }. Hence (4.8) is proved.

∞ n=1 P

 n=1 kn (|E ξ Y n | > ε) < ∞. (4.9)

  s. by the extended Borel-Cantelli Lemma. Then(3.15) follows.Proof of Theorem 2.1. Theorem 2.1 is a direct consequence of Lemmas 3.1 and 3.2.As in Section 2, we will need the decomposition (3.2) but here we choose a new β such that 1 > β > 1 1+η and then define k n . The proof of this theorem relies on Lemma 3.2 and the following lemma:For the proof of Lemma 4.1, we shall use the following result on the the weighted moments for W * defined by W * = sup n {W n }.

	4 Proof of Theorem 2.2
	Lemma 4.1. If (2.1), (2.2) and (2.5) hold, then
	A n → 0 P-a.s. .	(4.1)
	Lemma 4.2. ([14]) Assume that (2.1), (2.2) and (2.5) hold, then
	E(W * + 1)(ln(W * + 1)) 1+η < ∞.	(4.2)
	Furthermore, for β > 1 1+η and {r n } with lim inf n→∞	ln rn n β > 0,

∞ n=1 E (W * + 1)1 {W * +1≥rn} < +∞. (

4.3)

Acknowledgements

The project is partially supported by National Nature Science Foundation of China (Grant No. 11101039) and the Research Fund for the Doctoral Program of Higher Education (Grant No. 20100003110004).

With the same procedure as in step 3 of the proof of Lemma 3.1, we can show that

Combining this with (4.7), we prove (4.9).

Then the lemma has been proved.

Proof of Theorem 2.2. Theorem 2.2 is a direct consequence of Lemmas 4.1 and 3.2.

5 Proof of Theorem 2.3

In this section, we give a proof of Theorem 2.3, using a result of Klebaner (1982, [13]) on a branching random walk in a varying environment.

Proof of Theorem 2.3. We only need to prove that for almost every ξ,

By Theorem 2 of Klebaner (1982, [START_REF] Klebaner | Branching random walk in varying environments[END_REF]), it suffices to verify the following two conditions for a.e. ξ:

The first condition in (I) can be verified using the method in the proof of Lemma 3.3 and the second in (I) follows from the Birkhoff ergodic theorem. We now check (II). Note that by calculus and the Birkhoff ergodic theorem,

where the last inequality follows from (2.1). Thus the Cauchy's root test ensures the convergence of the first series in (II). The left thing is to prove the convergence of the second series in (II). By (2.7), we see that

Now the Cauchy's root test shows the convergence of the second series in (II). Therefore by Theorem 2 in [START_REF] Klebaner | Branching random walk in varying environments[END_REF], (5.1) holds; using the dominated convergence theorem again, we see that (2.8) holds.


Hesong Wang
email: hesongwang@tom.com

Zhiqiang Gao

Quansheng Liu
email: quansheng.liu@univ-ubs.fr

Gao Zhiqiang

Central limit theorems for a supercritical branching process in a random environment

Keywords: Branching processes, random environment, central limit theorem, martingale, rate of convergence. 2000 MSC: 60J80, 60F05

For a supercritical branching process (Z n ) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population W n = Z n /E[Z n |ξ] to its limit W ∞ : we show a central limit theorem for W ∞ -W n with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for W n+k -W n for each fixed k ∈ N * .

Introduction

Galton-Watson processes have been studied by many authors, due to a wide range of applications. See for example the books by [START_REF] Harris | The theory of branching processes[END_REF] and [START_REF] Athreya | Branching processes[END_REF]. In a Galton-Watson process {Z n , n = 0, 1, ...}, particles behave independently, each gives birth to a random number of particles of the next generation with a fixed distribution {p k : k = 0, 1, ...}.

A branching process in a random environment is a natural and important extension of the Galton-Watson process. It is a class of non-homogeneous Galton-Watson processes indexed by a time-environment ξ = (ξ 0 , ξ 1 , ξ 2 , ...), which is supposed to be stationary and ergodic; given the environment ξ, the particles of n-th generation have offspring distribution {p k (ξ n ) : k ∈ N} depending on ξ n . For first important works on the subject, see [START_REF] Smith | On branching processes in random environments[END_REF] and Athreya and Karlin (1971a,b).

For a Galton-Waston process with Z 0 = 1 and m = EZ 1 ∈ (0, ∞), it is well known that {W n = Z n /m n : n = 0, 1, ...} forms a non-negative martingale, and converges almost surely to a random variable W ∞ . For the convergence rate of the martingale, [START_REF] Heyde | Some central limit analogues for supercritical Galton-Watson processes[END_REF] and [START_REF] Bühler | Ein zentraler Grenzwertsatz für Verzweigungsprozesse[END_REF] obtained respectively that if Var(Z 1 ) = σ 2 < ∞, then conditioned on Z n > 0, the conditional laws of (m 2m)

1 2 σ -1 Z n -1 2 m n (W ∞ -W n ) and m k /(m k -1) 1 2 (m 2 -m) 1 2 σ -1 Z n -1 2 m n (W n+k -W n ) k ∈ N *
converge to the normal law N (0, 1); [START_REF] Heyde | An invariance principle and some convergence rate results for branching processes[END_REF] gave an estimation of its convergence rate under a third moment condition. The object of this paper is to extend the theorems of [START_REF] Bühler | Ein zentraler Grenzwertsatz für Verzweigungsprozesse[END_REF], [START_REF] Heyde | Some central limit analogues for supercritical Galton-Watson processes[END_REF] and [START_REF] Heyde | An invariance principle and some convergence rate results for branching processes[END_REF] to a branching process in a random environment. The main results are Theorems 2.1 and 2.2.

Main Results

As usual, we write

N = {0, 1, 2, • • • }, N * = {1, 2, • • • } and R for the set of real numbers.
Let us first recall the definition of a branching process in a random environment. For reference on the subject, see for example Athreya and Karlin (1971a,b), and [START_REF] Athreya | Branching processes[END_REF].

A random environment ξ = (ξ n ) is formulated as a stationary and ergodic sequence of random variables taking values in some measurable space (Θ, F). Each realization of ξ n corresponds to a probability distribution p(ξ

n ) = {p i (ξ n ) : i ∈ N} where p i (ξ n ) ≥ 0, ∞ i=0 p i (ξ n ) = 1, 0 < ∞ i=0 ip i (ξ n ) < ∞.
(1)

Without loss of generality, we can take ξ n as coordinate functions defined on the product space (Θ N , F ⊗N ), equipped with a probability law τ , which is invariant and ergodic under the usual shift transformation θ on

Θ N : θ(ξ 0 , ξ 1 , • • • ) = (ξ 1 , ξ 2 , • • • ) . A branching process (Z n ) n≥0
in the random environment ξ is a class of non-homogeneous branching processes indexed by ξ. By definition,

Z 0 = 1, Z n+1 = Zn i=1 X n,i n ≥ 0, (2)
where given ξ, {X n,i : n ≥ 0, i ≥ 1} is a family of (conditionally) independent random variables, each X n,i has the common law p(ξ n ). Notice that when all ξ n are the same constant, (Z n ) reduces to the classical Galton-Waston process.

Let (Γ, P ξ ) be the probability space under which the process is defined when the environment ξ is given. As usual, P ξ is called quenched law. The total probability space can be formulated as the product space (Γ × Θ N , P ), where P = P ξ ⊗ τ in the sense that for all measurable and positive function g, we have

gdP = g(ξ, y)dP ξ (y)dτ (ξ), (3)
(recall that τ is the law of the environment ξ). The total probability P is usually called annealed law. The quenched law P ξ may be considered to be the conditional probability of the annealed law P given ξ. The expectation with respect to P ξ (resp. P ) will be denoted E ξ (resp. E).

For n ≥ 0, define

m n (a) = m(ξ n , a) = ∞ i=1 i a p i (ξ n ), a ∈ R, (4)
m n = m n (1), σ 2 n = m n (2) -m 2 n , (5)
π 0 = 1 and π n = π n (ξ) = m 0 • • • m n-1 for n ≥ 1. (6)
Then

π n = E ξ Z n for n ≥ 0. It is well known that W n = Z n /π n (7)
is a martingale with respect to the filtration

F 0 = {∅, Ω}, F n = σ{ξ, X j,i : j ≤ n -1, i ≥ 1} (n ≥ 1), (8)
so that the limit

W ∞ = lim n→∞ W n (9)
exists almost surely (a.s.) with EW ≤ 1 by Fatou's lemma.

Throughout the paper, we always assume that

E ln m 0 > 0 and E Z 1 m 0 ln + Z 1 < ∞. ( 10
)
The first assumption ensures that the process is supercritical (cf. Athreya and Karlin (1971a)); the second one together with the first implies that EW ∞ = 1; moreover, s., (11) where q(ξ) = lim n→∞ P ξ (Z n = 0) is the extinct probability.

P ξ (W ∞ > 0) = P ξ (Z n → ∞) = lim n→∞ P ξ (Z n > 0) = 1 -q(ξ) > 0 a.
In this paper, we search for central limit theorems on W ∞ -W n and W n+k -W n for fixed k ≥ 1 with an appropriate normalization. Assum that m 0 (2) < ∞ a.s., and let

∆ 2 k = ∆ 2 k (ξ) = 0≤i<k 1 π i σ 2 i m 2 i for k ∈ N * ∪ {∞}. Then for k ∈ N * , ∆ 2 k (ξ) is the variance of W k under P ξ ; ∆ 2 ∞ (ξ) is the variance of W ∞ if the series converges (i.e. ∆ 2 ∞ (ξ) < ∞): see Lemma 3.2.
We can now formulate our first main result.

Theorem 2.1. Suppose that ( 10) holds and that m 0 (2) < ∞ a.s.. In the case where k = ∞, assume additionally that

E ln + (σ 2 0 /m 2 0 ) < ∞. Write U n,k = π n (W n+k -W n ) √ Z n ∆ k (θ n ξ) f or k ∈ N * ∪ {∞},
where by convention

W n+k = W ∞ if k = ∞. Then for each k ∈ N * ∪ {∞}, as n → ∞, sup x∈R |P ξ (U n,k ≤ x|Z n > 0) -Φ(x)| → 0 in L 1 , (12)
and

sup x∈R |P (U n,k ≤ x|Z n > 0) -Φ(x)| → 0. (13)
We believe that for each k ∈ N ∪ {∞}

lim n→∞ sup x∈R |P ξ (U n,k ≤ x|Z n > 0) -Φ(x)| = 0 a.s..
We notice that in the classical Galton-Waston process, (13) reduces to the results of [START_REF] Bühler | Ein zentraler Grenzwertsatz für Verzweigungsprozesse[END_REF] and [START_REF] Heyde | Some central limit analogues for supercritical Galton-Watson processes[END_REF]. Our second main result concerns the rate of convergence in the above central limit theorem for a branching process with an independent and identically distributed environment.

Theorem 2.2. Let the environment {ξ n } be independent and identically distributed. Assume that (10) holds and that m 0 (2) < ∞ a.s.. In the case where k = ∞, assume additionally that

E ln + (σ 2 0 /m 2 0 ) < ∞. For each k ∈ N * ∪ {∞}, if E| W k -1 ∆ k | 2+δ < ∞ for some δ ∈ (0, 1], then sup x∈R |P (U n,k ≤ x|Z n > 0) -Φ(x)| ≤ C δ Em 0 (-δ 2 ) n E W k -1 ∆ k 2+δ P (Z n > 0) , ( 14
)
where U n,k is defined in Theorem 2.1 and C δ is the Berry-Esseen constant.

Remark 2.3. It maybe useful to notice that if

E(Z 1 /m 0 ) 2+δ < ∞, Em 0 -(1+δ) < 1 and m 0 (2)/m 2 0 ≥ A for some constant A > 1, then E W∞-1 ∆∞ 2+δ
< +∞. In fact by Theorem 3 of [START_REF] Guivarc'h | Propriétés asymptotiques des processus de branchement en environnement aléatoire[END_REF], the first two conditions imply that E|W ∞ -1| 2+δ < ∞, while the last one implies that ∆ 2 ∞ ≥ A -1 > 0. For the classical Galton-Watson process with δ = 1, Theorem 2.2 reduces to Theorem 2 of (Heyde and Brown, 1971, p.272).

Proof of Theorem 2.1

In this section, we consider a central limit theorem under a second moment condition in proving Theorem 2.1. We first give some lemmas.

Lemma 3.1 (Grincevičjus (1974)). Let {(α n , β n ), n = 0, 1, 2, • • • } be a station- ary and ergodic sequence of random variables with values in R 2 . If E ln |α 0 | < 0 and E ln + |β 0 | < ∞, then ∞ n=0 |α 0 α 1 • • • α n-1 β n | < ∞ a.s.
In fact, the result is a direct consequence of the ergodic theorem and Cauchy's criterion for the convergence of series.

Using the above lemma, we can easily obtain the following result.

Lemma 3.2. Under the assumptions in Theorem 2.1, for each k ∈ N * ∪ {∞},

Var ξ (W k ) = ∆ 2 k (ξ) = 0≤i<k 1 π i σ 2 i m 2 i . ( 15
)
This has been known for branching processes in varying environment, see e.g. (Jagers, 1974, p.175) in a slightly different form. For reader's convenience, we present a proof in the following.

Proof of Lemma 3.2. By (2) and the definition of W n , we have

W n+1 -W n = 1 π n Zn j=1 ( X n,j m n -1).
Recall that under P ξ , the random variables {X n,j } are independent of each other and have the common distribution p(ξ n ) with expectation m n . Hence a direct calculation shows that

E ξ ((W n+1 -W n ) 2 ) = E ξ E ξ ((W n+1 -W n ) 2 |F n ) = E ξ Z n π n 2 σ 2 n m 2 n = 1 π n σ 2 n m 2 n .
As {W n } is a martingale, it follows that

E ξ W 2 k = E ξ W 2 0 + k-1 i=0 E((W i+1 -W i ) 2 ) = 1 + k-1 i=0 1 π i σ 2 i m 2 i .
Therefore for each fixed integer k,

Var ξ (W k ) = E ξ (W 2 k ) -1 = k-1 i=0 1 π i σ 2 i m 2 i .
Now we turn to the calculation of Var ξ (W ∞ ). By Lemma 3.1, when E ln m 0 > 0

and E ln + σ 2 i m 2 i < ∞, sup n E ξ (W 2 n ) = 1 + ∞ i=0 1 π i σ 2 i m 2 i < ∞ a.s. So W n converges to W ∞ in L 2 under P ξ and E ξ (W 2 ∞ ) = lim k→∞ E ξ (W 2 k ) = 1 + ∞ i=0 1 π i σ 2 i m 2 i . It follows that Var ξ (W ∞ ) = E ξ (W 2 ∞ ) -1 = ∆ 2 ∞ (ξ) = ∞ i=0 1 π i σ 2 i m 2 i < ∞ a.s.
To give our next lemma, we will need some notations, which will also be used in the proof of the main theorems. By definition,

Z n+k = Zn j=1 Z k (n, j), (16)
where Z k (n, j) denotes the number of descendants in the (n + k)-th generation of the j-th particle among the Z n particles in n-th generation.

Writing W k (n, j) = Z k (n,j) π k (θ n ξ) and using ( 16), we obtain the following decomposition:

π n (W n+k -W n ) = Zn j=1 (W k (n, j) -1). (17) Letting k → ∞, it follows that π n (W ∞ -W n ) = Zn j=1 (W ∞ (n, j) -1), (18)
where under P ξ , the random variables {W ∞ (n, j)} j are independent of each other and have the common conditional distribution

P ξ (W ∞ (n, j) ∈ •) = P θ n ξ (W ∞ ∈ •).
Lemma 3.3. Suppose that the assumptions of Theorem 2.1 hold. Let

r n ∈ N with r n → ∞. For k ∈ N * ∪ {∞}, define Y k,n = 1 √ r n rn j=1 W k (n, j) -1 ∆ k (θ n ξ) .
Fix k ∈ N * ∪ {∞}. Then for each subsequence {n ′ } of N with n ′ → ∞, there is a subsequence {n ′′ } of {n ′ } with n ′′ → ∞ such that for a.e. ξ and all x ∈ R, as

n ′′ → ∞, P ξ (Y k,n ′′ ≤ x) → Φ(x).
Proof. Fix k ∈ N * ∪ {∞}. In order to use Lindeberg's theorem, for n ∈ N and ǫ > 0, we consider the quantity

L k (ξ, ǫ, n) = 1 r n rn j=1 E ξ W k (n, j) -1 ∆ k (θ n ξ) 2 ; W k (n, j) -1 ∆ k (θ n ξ) √ r n > ǫ ,
where for a set A, we write E ξ (x; A) for E ξ (X1 A ), 1 A denoting the indicator function of A. By the stationarity and ergodicity of the environment, for all ǫ > 0, as n → ∞,

EL k (ξ, ǫ, n) = E W k -1 ∆ k 2 ; W k -1 ∆ k > √ r n ǫ → 0. ( 19
)
Let {n ′ } be a subsequence of N. Notice that from (19), we can choose a subsequence {n ′′ } for which L k (ξ, ε, n ′′ ) → 0 a.s., but this sequence may depend of ǫ. We will use a diagonal argument to select a subsequence {n ′′ } of {n ′ } such that a.s. L k (ξ, ε, n ′′ )

n ′′ →∞ ----→ 0 for all ǫ > 0. Set ǫ m = 1/m for m ≥ 1.
Let {n 0,i } = {n ′ }. Because of ( 19), there is a subsequence {n 1,i } of {n 0,i } and a set Λ 1 with τ (Λ 1 ) = 1 such that ∀ξ ∈ Λ 1 ,

lim i→∞ L k (ξ, ǫ 1 , n 1,i ) = 0.
Inductively for m ≥ 1, when Λ m and {n m,i } are defined such that τ (Λ m ) = 1 and ∀ξ ∈ Λ m , L k (ξ, ǫ m , n m,i ) → 0, there is a subsequence {n m+1,i } ⊂ {n m,i } and a set Λ m+1 with τ (Λ m+1 ) = 1 such that ∀ξ ∈ Λ m+1 ,

lim i→∞ L k (ξ, ǫ m+1 , n m+1,i ) = 0.
We now consider the diagonal sequence {n i.i } i≥1 and Λ = ∞ j=1 Λ j . For each fixed ǫ > 0, let m ≥ 1 ǫ . Then ǫ m ≤ ǫ and by the monoticity of L k (ξ, ǫ, n) in ǫ, we see that ∀ξ ∈ Λ,

L k (ξ, ǫ, n m,i ) ≤ L k (ξ, ǫ m , n m,i ) → 0 as i → ∞.
As {n i,i } is a subsequence of {n m,i } whenever i > m, this implies that

lim i→∞ L k (ξ, ǫ, n i,i ) = 0. ( 20
)
Since τ (Λ) = 1, we have shown that for all ǫ > 0, (20) holds a.s.. It follows that a.s. ( 20) holds for all rational ǫ > 0, and therefore for all real ǫ > 0 by the monoticity of L k (ξ, ǫ, n ii ) in ǫ. So by Lindeberg's theorem, it is a.s. that for all

x ∈ R, as i → ∞, P ξ (Y k,ni,i ≤ x) → Φ(x).
Thus the lemma has been proved with {n ′′ } = {n i,i }.

Proof of Theorem 2.1. We shall only deal with the case where k = ∞, as the case where k ∈ N * can be treated similarly. We first prove the following assertion: for each sequence {n ′ } of N with n ′ → ∞, there exist a subsequence {n ′′ } of {n ′ } with n ′′ → ∞ such that for a.e. ξ and all x, as n ′′ → ∞,

P ξ (U n ′′ ,∞ ≤ x|Z n ′′ > 0) → Φ(x). ( 21
)
By the definition of U n,∞ and the relation ( 18), we get :

U n,∞ = π n (W ∞ -W n ) √ Z n ∆ ∞ (θ n ξ) = 1 √ Z n Zn j=1 W ∞ (n, j) -1 ∆ ∞ (θ n ξ) ,
where we recall that under P ξ , {W ∞ (n, j), j ≥ 1} is a family of random variables independent of each other and independent of Z n , each has the same law as W ∞ under P θ n ξ . Set

u n (r, x) = P ξ   1 √ r r j=1 W ∞ (n, j) -1 ∆ ∞ (θ n ξ) ≤ x   , r ∈ N * , x ∈ R. Then P ξ (U n,∞ ≤ x|Z n > 0) = [P ξ (Z n > 0)] -1 ∞ r=1 P ξ (U n,∞ ≤ x, Z n = r) = ∞ r=1 u n (r, x) P ξ (Z n = r) P ξ (Z n > 0) . ( 22
)
To show the main idea, let us first consider the special case where q(ξ) = 0 a.s., i.e. for a.e. ξ, Z n → ∞ P * ξ -a.s.. In this case, the relation ( 22) becomes

P ξ (U n,∞ ≤ x) = ∞ r=1 u n (r, x)P ξ (Z n = r) = E ξ u n (Z n , x).
By Lemma 3.3, for each subsequence {n ′ } of N with n ′ → ∞, there exist a subsequence {n ′′ } of {n ′ } with n ′′ → ∞ such that for a.e. ξ and all x, as n ′′ → ∞,

u n ′′ (Z n ′′ , x) → Φ(x).
By the dominated convergence theorem, for a.e. ξ and all x, as n ′′ → ∞,

P ξ (U n ′′ ,∞ ≤ x) = E ξ [u n ′′ (Z n ′′ , x)] → Φ(x).
So we have proved (21) when q(ξ) = 0 a.s.. We now consider the general case where 0 ≤ q(ξ) < 1 a.s.. For each ξ ∈ Θ N , let Z * n be random variables defined on some probability space (Γ * , P * ξ ) with law

P * ξ (Z * n = r) = P ξ (Z n = r) P ξ (Z n > 0) , r ∈ N * . Then P ξ (U n,∞ ≤ x|Z n > 0) = E * ξ u n (Z * n , x)
, where E * ξ denotes the expectation with respect to P * ξ . Let {n ′ } be a sequence of N with n ′ → ∞. If for a.e. ξ,

Z * n ′ → ∞ P * ξ -a.s.,
then as above we can use Lemma 3.3 and the dominated convergence theorem to show that there is a sequence {n ′′ } of {n ′ } with n ′′ → ∞ such that for all x, as n ′′ → ∞,

E * ξ u n ′′ (Z * n ′′ , x) → Φ(x)
. By the fact that Z * n → ∞ in probability under P * ξ , we can choose a subsequence for which Z * n → ∞ P * ξ -a.s.. But to apply Lemma 3.3, we need that the sequence does not depend on ξ. We therefore pass to the probability P * to overcome this difficulty, where P * = P * ξ ⊗ τ is defined on the product space Γ * × Θ N just as P was defined on Γ × Θ N .

Notice that for each r ∈ N * , as n → ∞,

P * ξ (Z * n = r) = P ξ (Z n = r) P ξ (Z n > 0) → 0,
where the last step holds as Z n → ∞ a.s. on the survival event S = {Z n > 0, ∀n ≥ 1} (see (11) or [START_REF] Tanny | Limit theorems for branching processes in a random environment[END_REF] for this fact). Then Z * n → +∞ in probability under P * ξ . By the dominated convergence theorem, this implies that Z * n → +∞ in probability under P * . Therefore for each subsequence {n ′ } of N with n ′ → ∞, there is a subsequence {ñ} ⊂ {n ′ } with ñ → ∞ such that Z * ñ → +∞ a.s. under P * . This implies that for a.e. ξ, as ñ → ∞,

Z * ñ → +∞ P * ξ -a.s.
Now by Lemma 3.3, there exists a subsequence {n ′′ } of {ñ} such that for a.e. ξ and all x, as n ′′ → ∞,

u n ′′ (Z * n ′′ , x) → Φ(x).
By the dominated convergence theorem, for almost every ξ and each x, as

n ′′ → ∞, P ξ (U n ′′ ,∞ ≤ x|Z n ′′ > 0) = E ξ u n ′′ (Z * n ′′ , x) → Φ(x). (23)
So combining the above two cases, we have proved ( 21). Since P ξ (U n ′′ ,∞ ≤ x|Z n ′′ > 0) are distribution functions and Φ(x) is a continuous distribution function, by Dini's Theorem we see that for a.e. ξ, as

n ′′ → ∞, sup x |P ξ (U n ′′ ,∞ ≤ x|Z n ′′ > 0) -Φ(x)| → 0. (24)
By the dominated convergence theorem, (24) implies that as n ′′ → ∞,

E sup x |P ξ (U n ′′ ,∞ ≤ x|Z n ′′ > 0) -Φ(x)| → 0. ( 25
)
Therefore we have proved that for each sequence {n ′ } of N with n ′ → ∞, there is a subsequence {n ′′ } of {n ′ } with n ′′ → ∞ such that (25) holds. Hence

E sup x |P ξ (U n,∞ ≤ x|Z n > 0) -Φ(x)| → 0.
This gives (12) for k = ∞. The proof for k ∈ N * is similar. We now begin to prove (13).

As we have proved that for each subsequence {n ′ } of N, there is a subsequence {n ′′ } so that (24) holds, which implies: for a.e. ξ and all x ∈ R, as

n ′′ → ∞, |P ξ (U n ′′ ,∞ ≤ x|Z n ′′ > 0) -Φ(x)| → 0.
It follows that for a.e. ξ and all x ∈ R,

|P ξ (U n ′′ ,∞ ≤ x, Z n ′′ > 0) -P ξ (Z n ′′ > 0)Φ(x)| → 0.
So by the dominated convergence theorem, we see that for each x ∈ R, as

n ′′ → ∞, |P (U n ′′ ,∞ ≤ x, Z n ′′ > 0) -P (Z n ′′ > 0)Φ(x)| → 0,
and hence

P (U n ′′ ,∞ ≤ x|Z n ′′ > 0) → Φ(x).
By Dini's Theorem, it follows that

sup x |P (U n ′′ ,∞ ≤ x|Z n ′′ > 0) -Φ(x)| → 0. (26)
Therefore we have proved that for each sequence {n ′ } of N, there is a subsequence {n ′′ } of {n ′ } with n ′′ → ∞ such that (26) holds. Hence

sup x |P (U n,∞ ≤ x|Z n > 0) -Φ(x)| → 0.
Thus the proof is completed.

Proof of Theorem 2.2

In this section, we consider the rate of convergence in the central limit theorem under a moment condition of order 2 + δ, in proving Theorem 2.2.

Notice that by the definition (4) of m n (a), we have

m n (a) = E ξ X a n,i if a > 0, m n (a) = E ξ X a n,i 1 {Xn,i>0} if a ≤ 0, (27)
where X n,i is as in (2). For a > 0, define

R n = [m 0 (-a) • • • m n-1 (-a)] -1 Z -a n 1 {Zn>0} , n ≥ 0.
Lemma 4.1. (R n , F n ) n≥0 is a supermartingale, where F n were defined in (8).

Proof. Using the decomposition (2) of Z n+1 , we have

Z -a n+1 1 {Zn+1>0} = Zn i=1 X n,i -a 1 {Zn>0} 1 {Zn+1>0} = Z -a n 1 Z n Zn i=1 X n,i 1 {Xn,i}>0 -a 1 {Zn>0} 1 {Zn+1>0} ≤ Z n -a 1 Z n Zn i=1 (X n,i 1 {Xn,i}>0 ) -a 1 {Zn>0} 1 {Zn+1>0} ,
where the last inequality is due to the convexity property of the function x -a (a > 0).

Taking conditional expectation with respect to F n and P ξ on both sides of the above inequality, we obtain that

E ξ (Z -a n+1 1 {Zn+1>0} |F n ) ≤ Z -a n 1 {Zn>0} m n (-a), (28)
which gives the desired result.

Since Z 0 = 1, by (28), we immediate obtain the following Lemma 4.2. For a > 0, we have

E ξ Z -a n 1 {Zn>0} ≤ m 0 (-a) • • • m n-1 (-a) ( 29
)
If the environment sequence {ξ n } is independent and identically distributed, then

EZ -a n 1 {Zn>0} ≤ (Em 0 (-a)) n . ( 30
)
Now we give the proof of Theorem 2.2.

Proof of Theorem 2.2. We shall only deal with the case k = ∞, as the case where k ∈ N * can be treated similarly.

Consider the probability space (Γ * × Θ N , P * ) and define random variables Z * n as in the proof of Theorem 2.1. By definition,

u n (Z * n , x) = P ξ   1 Z * n Z * n j=1 W ∞ (n, j) -1 ∆ ∞ (θ n ξ) ≤ x   .
By our hypothesis and the Berry-Esseen theorem (see e.g. Theorem 6 of (Petrov, 1995, p.115)), we have

|u n (Z * n , x) -Φ(x)| ≤ C δ (Z * n ) 1+ δ 2 Z * n j=1 E ξ W ∞ (n, j) -1 ∆ ∞ (θ n ξ) 2+δ = C δ (Z * n ) -δ 2 E θ n ξ W ∞ -1 ∆ ∞ 2+δ
, where C δ is the Berry-Esseen constant. Using this evaluation, we can derive that

|P ξ (U n,∞ ≤ x|Z n > 0) -Φ(x)| ≤ E * ξ |u n (Z * n , x) -Φ(x)| ≤ C δ E * ξ (Z * n ) -δ 2 E θ n ξ W ∞ -1 ∆ ∞ 2+δ
.

By the definition of Z * n , this implies that

|P ξ (U n,∞ ≤ x, Z n > 0) -P ξ (Z n > 0)Φ(x)| ≤ C δ E ξ Z -δ 2 n I {Zn>0} E θ n ξ W ∞ -1 ∆ ∞ 2+δ . (31)
Using ( 31) and the fact that the sequence {ξ n } is independent and identically distributed, we get

|P (U n,∞ ≤ x, Z n > 0) -P (Z n > 0)Φ(x)| ≤ E|P ξ (U n,∞ ≤ x, Z n > 0) -P ξ (Z n > 0)Φ(x)| ≤ C δ E Z -δ 2 n I {Zn>0} E W ∞ -1 ∆ ∞ 2+δ
.

Acknowledgements

The authors would like to thank the Editor and an anonymous referee for their comments and remarks. The research was the supported by the National Natural Science Foundation of China (Grant No. 10771021, 10871064 and 50907005), the Key Labor. of Comput. Stoch. Math., Univ. of Hunan (No. 09K026) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education of China (Grant No. [2008] 890).

Together with (30), we obtain that

.

Then the proof is completed.