John Stachurski

Cuong Le Van

Parametric Continuity of Stationary Distributions

The paper gives conditions under which stationary distributions of Markov models depend continuously on the parameters. It extends a well-known parametric continuity theorem for compact state space to the unbounded setting of standard econometrics and time series analysis. Applications to several theoretical and estimation problems are outlined.

Introduction

In macroeconomic dynamics, time series econometrics and other related fields, one frequently considers economies where the sequence of state variables (X t ) ∞ t=0 is stationary. Here X t is a vector of endogenous and exogenous variables, jointly following a Markov process generated by some underlying model. By stationary is meant the existence of a "stationary distribution" µ, such that if X t has law µ, then so does X t+j for all j ∈ N. If such a µ exists, is unique and has some stability properties, then it naturally becomes the focus of equilibrium analysis.

For example, in these settings a law of large numbers result often holds, in which case sample moments from the series (X t ) ∞ t=0 can be identified with integrals of the relevant functions with respect to the stationary distribution µ.

Typically, the underlying laws which drive the process (X t ) ∞ t=0 depend on a vector of parameters, which may for example be policy instruments, or regression coefficients to be estimated from the data. In this case the parameters themselves determine the stationary distribution. The study of how this distribution varies with the parameters is a stochastic analogue of standard comparative dynamics. Our paper investigates conditions under which the functional relationship between parameters and stationary distribution is continuous.

The main results extend directly a well-known and useful argument in Stokey, Lucas and Prescott for Markov processes on a compact state space (1989, Theorem 12.13), which is apparently due to R. Manuelli.

Although compactness of the state space is indeed convenient where it can be assumed, for many empirical studies it fails. An elementary example is the scalar AR(1) models with normal shocks.

Instead of compactness, various uniform stability and monotonicity conditions are adopted. The conditions are stated in terms of the transition laws and distributions of the shocks, with a view to simple verification.

On related literature, another study of parametric continuity of stationary distributions in the compact case is Santos and Peralta-Alva (2003). They obtain detailed error bounds when the transition rule is uniformly contracting on average, as well as addressing the implications for accuracy of numerical simulations. We do not pursue the compact state case here.

The remainder of the paper is structured as follows. In Section 2 three examples illustrating the importance of parametric continuity are given. In Section 3 the general problem is formulated and results are given. Section 4 concludes with proofs.

Examples

2.1. Simulation and Estimation. Parametric continuity of stationary distributions is a component of various problems in estimation, simulation, numerical dynamic programming and economic theory. A well-known example is the Simulated Moments Estimator of Duffie and [START_REF] Singleton | Simulated Moments Estimation of Markov Models of Asset Prices[END_REF], which can be described as follows. Let (X t ) T t=0 be observable data-a vector of state variables that are taken as generated Cahiers de la MSE -2004.59 by some dynamic general equilibrium model. Let ϕ be a function on the state space S which associates to current state X t real number ϕ(X t ). For concreteness, suppose the problem is one of asset pricing, and ϕ(X t ) is the price of a given asset, the value of which depends on the current state. The transition rule for the model which is assumed to generate the data has reduced form [START_REF] Aliprantis | Infinite Dimensional Analysis[END_REF] X t+1 = H α (X t , ξ t ), t = 0, 1, . . . , where α ∈ W , a parameter space, and (ξ t ) ∞ t=0 is a sequence of independent shocks.

The true parameter α is not known. However, the econometrician observes the sequence (X t ) T t=0 , can compute H β for given β ∈ W as the solution to the model, and has access to a sequence of shocks ( ξt ) with the same (joint) distribution as (ξ t ). From these one can simulate for each β ∈ W a sequence (X β t ) N t=0 defined recursively by

X β t+1 = H β (X β t , ξt ), t = 0, 1, . . .
The simulated moments estimator is the value of β which minimizes the distance between the moments of the observed price process ϕ(X t )

and the simulated process ϕ(X β t ).

In order to prove consistency of the estimator, Duffie and The dynamic programming problem is to choose a policy function σ mapping state space S into control space A which maximizes a discounted sum of period utilities, or minimizes discounted loss. The law of motion for the system is given by some rule

(2)

X t+1 = F (X t , U t , ξ t ), t = 0, 1, . . . ,
where for now (ξ t ) is an i.i.d. sequence and U t ∈ A is the current value of the control. Suppose that we obtain numerically a candidate solution σn for the optimal policy σ, where n is some index of dedicated CPU cycles. Suppose further that theoretical bounds are available for the distance σn -σ in terms of n, where • is the norm in some appropriate function space F . Substituting σn into (2) gives the Markov process

(3) X n t+1 = F (X n t , σn (X n t ), ξ t ).

Let T σn (X, ξ) := F (X, σn (X), ξ). The point of this notation is that since in the theory below the parameter space W is required only to be a metric space, we can treat the candidate policy σn as a parameter.

Set W := (F , • ), which includes all feasible policies.

The ultimate objective is to simulate time series at the stationary distribution µ under the optimal policy σ. The method is to substitute σn for σ, as in (3), and use the approximating sequence (X n t ) ∞ t=0 . If the process is ergodic, X n t will be nearly distributed according to the stationary distribution µ n corresponding to (3) when t is large. If σn -σ → 0 as n → ∞ we hope that µ n → µ in some topology, in which case, by taking both t and n large, X n t is approximately distributed according to µ. This problem again reduces to the kind of parametric continuity issue considered in the paper.

Cahiers de la MSE -2004.59 2.3. The Solow-Phelps Stochastic Golden Rule. Next we consider a more specific example, which is of independent interest, and also helps to illustrate how one might verify the conditions we impose.

The example is the Solow-Phelps stochastic golden rule problem for a one-sector economy. The framework is a Solow growth model, to which a random component is added. The problem is to choose a savings rate which maximizes expected utility of consumption at the steady state.

Surprisingly the existence of a solution to this problem has not been treated to our knowledge. 1 Since the savings rate takes values in compact set [0, 1], determining existence of a solution reduces in effect to proving continuity of the map from savings rate to stationary distribution.

In more detail, suppose initially a fixed savings rate s from current income. At each period t production takes place, deriving from capital stock k t random output y t = f (k t ) ξ t ; where f : S → S is the production function, S := (0, ∞), and ξ t is a productivity shock with distribution ν on S. The parameter is the savings rate, and we take the parameter space W to be (0, 1]. Here s = 0 is excluded so as to eliminate trivialities.

For simplicity, depreciation is taken to be total. Given savings rate s,

next period capital is (4) k t+1 = s • f (k t ) ξ t .
The process then repeats, generating Markov chain (k t ) ∞ t=0 starting from k 0 ∈ S given. In order that this process have a stationary distribution, some restrictions on f and ν are necessary: The restrictions on f are presented in the current form because they are so familiar, but a quick reading of the proofs shows they are much stronger than what is actually required. The assumptions E[ξ t ] < ∞ and E[1/ξ t ] < ∞ restrict the right and left-hand tails of the shock respectively. 2 The assumption that ν is positive on sets of positive Lebesgue measure-which is used to prove uniqueness-is also much stronger than required, but simplifies the analysis, and holds for many standard econometric shocks. For example, all lognormal distributions satisfy the three restrictions in Assumption 2.2.

The golden rule problem is to maximize expected utility of consumption at the steady state. Let u : [0, ∞) → R be a utility function.

Assumption 2.3. The function u is continuous, bounded, strictly increasing and, by adding or subtracting a constant if necessary, satisfies

u(0) = 0.
Suppose that for each savings rate s, (4) has a unique stationary distribution µ s (for a precise definition of stationary distribution see Definition 3.2 below). Consumption c t is just (1 -s)f (k t )ξ t . Recalling that k t and ξ t are independent, the distribution of c t in equilibrium is [START_REF] Futia | Invariant Distributions and the Limiting Behavior of Markovian Economic Models[END_REF]

ϕ s (B) := Prob{c t ∈ B} = 1 B [(1 -s)f (k)z]ν(dz)µ s (dk),
2 With unbounded shocks some restrictions of this nature are necessary for the stationary distribution to exist-the first restriction prevents unbounded growth;

the second collapse to the origin.

Cahiers de la MSE -2004.59

k t k t+1 45 0 s n f (k t ) sf (k t ) µ sn d → µ s - 6 Figure 1. Convergence of stationary distributions.
where

1 B [(1 -s)f (k)z] = 1 when (1 -s)f (k)z ∈ B and zero otherwise.
The golden rule problem is to select the "lottery" ϕ s which maximizes expected utility:

(6) max 1. For each savings rate s, the process (4) has a unique stationary distribution, which depends continuously on s.

2.

Moreover, an interior solution s * to the stochastic Phelps golden rule problem (6) exists, at which the expected utility of long run equilibrium consumption is maximized.

The General Model

In what follows, for measurable space (E, E ) let P(E) denote the probabilistic measures on (E, E ), and bM (S) the finite signed measures.

Cahiers de la MSE -2004.59

Functional notation is often used for integration: if h : E → R is Emeasurable and µ ∈ bM (E) then µ(h) := hdµ when the latter is defined. If E has a topology, the σ-algebra E is always the Borel sets.

Throughout the paper we use the convention that if X is a random variable taking values in measurable space (E, E ), then L(X) is the law (distribution) of X, an element of P(E).

a collection of probabilities {µ n } ∞ n=1 ∪ {µ} ⊂ P(E), we write µ n d → µ when the sequence (µ n ) converges to µ in w(P(E), C b (E))-that is when µ n (h) → µ(h) in R for every h ∈ C b (E).
We say that (µ n ) converges to µ in distribution (Stokey, Lucas and Prescott, 1989, Chapter 12).

3.1. The Model. Now let α ∈ W be a parameter. Consider an economy that evolves according to the rule (7)

X t+1 = T α (X t , ξ t ), X 0 ≡ x 0 ∈ S given.
The sequence of shocks (ξ t ) ∞ t=0 is assumed to be uncorrelated and identically distributed. 3 Together, the transition rule T α and the distribution of the shock-call it ν-determine a Markov chain (X t ) ∞ t=0 . The timing is that X t is observed at the start of time t, planning of economic activity takes place on the basis of this information, and is then implemented through the period. Implementation is perturbed by a shock ξ t , giving rise to X t+1 at the start of the next period via [START_REF] Huggett | When are comparative dynamics monotone?[END_REF]. It is clear from the timing that X t and ξ t are independent.

The random variable X t evolves in state space S. The state space has a topology, and B(S) denotes the Borel sets of S. Also, let (Z, Z ) be any measurable space, with the shocks that perturb the economy in each period taking values in Z, so that T α : S ×Z → S. 4 The parameter space W is any metric space.

3.2. Basic Assumptions. Some basic requirements are now given which will at least assure the existence and uniqueness of stationary distributions.

Assumption 3.1. The space S is separable and completely metrizable.

Such spaces are called Polish. For example, in the golden rule problem the state space is S = (0, ∞) with the usual topology. This space is Polish. In fact every open (indeed, every G δ ) subset of a complete metric space is completely metrizable by Alexandrov's Lemma (cf. e.g., Aliprantis and Border, 1999, Theorem 3.22). Assumption 3.2. For each fixed α ∈ W , x → h[T α (x, z)]ν(dz) is continuous and bounded whenever h : S → R is.

While it may not immediately appear so, this assumption is easy to verify in applications. For example, it holds when x → T α (x, z) is continuous for each fixed α and z, as follows from Dominated Convergence.

The continuity provided by Assumption 3.2 is paired with a compactness requirement below to obtain existence of stationary distributions via a fixed point argument. To define the compactness requirement, first we introduce norm-like functions, which play a role somewhat analogous to Liapunov functions in dynamical systems theory.

Definition 3.1. A continuous function V : S → [0, ∞) is called norm- like if there is an increasing sequence of compact sets (K j ) ∞ j=1 with ∪ ∞ j=1 K j = S and inf x / ∈K j V (x) → ∞ as j → ∞. L(X t ) L(X t+1 ) V ? ?
Figure 2. A norm-like function on (0, ∞).

A norm-like function V gets larger and larger at the "edges" of the state space, so we can be sure that L(X t ) stays near the "center" of the space by checking that the expectation of V (X t ) is bounded in t (Figure 2). The next condition will guarantee precisely this. It means that there is on average a drift of the state variable towards parts of the state space on which V is remains small. Assumption 3.3. For each α ∈ W , there exists a norm-like function

V and constants λ, b ∈ [0, ∞) with λ < 1 and (8) V [T α (x, z)]ν(dz) ≤ λV (x) + b, ∀x ∈ S.
Finally, the question of parametric continuity of stationary distributions is most interesting when this distribution is unique (for each parameterization). Fortunately, uniqueness is far more common in stochastic models than in deterministic ones, as a result of the mixing provided by noise. In order to generate uniqueness we assume throughout the paper the following, although any other uniqueness condition will do.

Assumption 3.4. For each α ∈ W , the process ( 7) is ψ-irreducible. In other words, there is a ψ ∈ P(S), possibly depending on the parameter Cahiers de la MSE -2004.59

α, such that ψ(B) > 0 implies

Prob x {X t ∈ B for some t ∈ N} > 0, ∀x ∈ S,
where Prob x is the distribution of (X t ) ∞ t=0 when X 0 ≡ x.

Again, this assumption is often easier to verify in practice than it looks.

See the proof for the golden rule example below. ). The compactness requirement can be weakened to the condition that at least one "trajectory" of the process (7) be precompact.

Intuitively, if this is the case then up to an epsilon, probability mass stays in a compact set at least for one starting point-by Prohorov's theorem, see below-and the compact state proof can be generalized appropriately. The details follow.

The long run equilibrium of the economy ( 7) is identified-when it exists-with the stationary distribution µ α of the Markov chain

(X t ) = (X α t ). Definition 3.2. Distribution µ α ∈ P(S) is called stationary for (7) if (9) µ α (B) = S Z 1 B [T α (x, z)]ν(dz) µ α (dx), ∀B ∈ B(S).
Here states that if L(X t ) = µ α , then so does L(X t+1 ).

1 B [T α (x, z)] = 1 if T α (x, z) ∈ B
Cahiers de la MSE -2004.59

To clarify further, these ideas can be recast in a more general form. 5 Define for each α ∈ W a map P α : P(S) → P(S) by ( 10)

P α µ(B) = S Z 1 B [T α (x, z)]ν(dz) µ(dx), B ∈ B(S).
The operator P α is called the Markov (or Foias) operator associated with [START_REF] Huggett | When are comparative dynamics monotone?[END_REF]. 6 It is well-known that if L(X t ) = µ, then L(X t+1 ) = P α µ (cf.

e.g. We provide a simple direct proof. 5 Most of the following is standard in economics. See for example [START_REF] Futia | Invariant Distributions and the Limiting Behavior of Markovian Economic Models[END_REF], or Stokey, Lucas and Prescott (1989). 6 In some literatures it is the adjoint of P α which is called the Markov operator.

Our terminology and notation closely follows [START_REF] Lasota | Chaos, Fractals and Noise: Stochastic Aspects of Dynamics[END_REF].

Cahiers de la MSE -2004.59

Theorem 3.1. Let P be a Markov operator on (S, B(S)). If P is Feller and there is an x ∈ S such that {P t δ x : t ∈ N} is precompact, then P has a fixed point µ in P(S). 7 Proof. Let A be the closure of the convex hull of {P t δ x : t ∈ N}. From the fact that P is invariant on {P t δ x : t ∈ N}, Feller and linear, it is easy to see that P maps A into A. By the Schauder fixed point theorem, P has a fixed point µ ∈ A ⊂ P(S). 8 We now explore the properties of the Markov operator generated by [START_REF] Huggett | When are comparative dynamics monotone?[END_REF] under Assumptions 3.1-3.4.

Lemma 3.1. Let α ∈ W , and let P α be defined by [START_REF] Phelps | The Golden Rule of Accumulation: A Fable for Growthmen[END_REF]. If Assumption 3.2 holds, then P α has the Feller property.

This result is elementary and well-known (cf. e.g., Stokey, Lucas and Prescott, 1989, p. 376).

Lemma 3.2. Let α ∈ W , and let P α be defined by [START_REF] Phelps | The Golden Rule of Accumulation: A Fable for Growthmen[END_REF]. If Assumptions 3.1 and 3.3 hold, then {P t α δ x : t ∈ N} is precompact for every x ∈ S.

The proof is given in Section 4. The intuition is that from any X 0 ≡

x, Assumption 3.3 bounds the growth of the expectation of V (X t ), which-from the definition of norm-like functions-means that up to an epsilon, L(X t ) is concentrated on a compact set for all t. This property is called tightness, which is equivalent to precompactness in the Polish setting. 7 In the statement, precompact means having compact closure. The topology is w(bM (S), C b (S)). 8 To be more precise, P(S) is a subset of bM (S), the space of finite signed Borel measures on S with the w(bM (S), C b (S)) topology. This space is locally convex

Hausdorff-here we need the Polish assumption-so Schauder's fixed point theorem applies (cf. e.g. Aliprantis and Border, 1999, Theorem 16.52). That the closed convex hull of a precompact set is compact in this setting follows from the same reference, Theorem 5.20 and the remark thereafter.

Cahiers de la MSE -2004.59

We now collect these results and combine them with Assumption 3.4

to obtain existence and uniqueness. Two different parametric continuity results will be presented. The first requires two additional conditions. Specifically, we strengthen Assumptions 3.2 and 3.3 to hold in a uniform way over the parameters.

In the statement of the theorem, is any metric compatible with the topology on S.

Theorem 3.3. Let Assumptions 3.1-3.5 hold, and let {α n } ∞ n=1 ∪{α} ⊂ W . Suppose that for each compact K ⊂ S, there is a constant M < ∞ which can be chosen independent of α n to satisfy [START_REF] Santos | Accuracy of Simulations for Stochastic Dynamic Models[END_REF] [T αn (x, z), T αn (x , z)]ν(dz) ≤ M (x, x ) whenever x, x ∈ K.

Suppose further that V , λ and b in Assumption 3.3 can be chosen independent of α n . Let µ αn and µ α in P(S) be the unique stationary distributions of (7) for each α n and α respectively. In this case, if

α n → α, then µ αn d → µ α .
Cahiers de la MSE -2004.59

The proof is in Section 4. The condition [START_REF] Santos | Accuracy of Simulations for Stochastic Dynamic Models[END_REF] requires that the model ( 7) satisfy a local Lipschitz condition "on average," where the Lipschitz constant can be chosen independent of the points in the sequence α n . For an example of how to check this and the other hypotheses of Theorem 3.3, see the proof of Theorem 2.1 (the golden rule problem) in Section 4.

The second result (Theorem 3.4, below) depends on various order properties, which replace the uniform continuity and uniform boundedness assumptions in Theorem 3.3. In particular, the set of stationary distributions may form a (stochastic dominance) order interval in P(S).

These order intervals are compact in the topology of weak convergence for certain spaces S. The compactness of this set can substitute in some sense for the compactness of the state space assumed in earlier results.

A number of order concepts are introduced. 9 Recall that a relation ≤ E on a space E is called an order if it is reflexive, transitive and antisymmetric. If (X, ≤ X ) and (E, ≤ E ) are two ordered spaces, a map When E has a topology, ≤ E is called separating if, whenever x E y, there is an increasing continuous bounded h : E → R with h(x) > h(y).

T : X → E is called increasing if T x ≤ E T x whenever x ≤ X x , decreasing if T x ≤ E T x whenever x ≤ X x ,
For example, R n with the usual topology and order is separating. 9 For more details we refer to the excellent survey of Torres (1990).

Cahiers de la MSE -2004.59

If E is metrizable with separating order ≤ E , then ≤ E induces on P(E)

the order ≤ P(E) of stochastic dominance. 10 It is defined by

µ ≤ P(E) µ iff µ(h) ≤ µ (h), ∀h ∈ ibE , E the Borel sets.
In what follows let the state space S and the parameter space W be given orders ≤ S and ≤ W respectively. We require some restrictions on the order ≤ S and its interaction with the topology of S:

Assumption 3.6. The order ≤ S on S is separating. In addition, if K is a compact subset of S, then i(K) ∩ d(K) is again compact in S. 11 For example, Assumption 3.6 holds on the space R n with the usual order and topology.

We have the following preliminary result, which extends Hopenhahn In other words, the stationary distribution is (stochastic dominance)increasing in the parameter. The second parametric continuity result is as follows.

10 Without some restrictions on the topology and order ≤ E , the induced relation ≤ P(E) may not be antisymmetric. (See Torres, 1990, Theorem 5.3.) 11 This last condition is used in Torres (1990, p 33) to generate w(P(S), C b (S))compact ≤ P(S) -order intervals in P(S).

Cahiers de la MSE -2004.59

Theorem 3.4. Let the hypotheses of Proposition 3.1 hold, and let

{α n } ∞ n=1 ∪ {α} ⊂ W .
Let µ αn and µ α in P(S) be the unique stationary distributions of (7) for each α n and α respectively. In this case, if

α n → α monotonically in W , then µ αn d → µ α .
The conditions (i) x → T α (x, z) increasing for fixed α ∈ W and z ∈ Z, and (ii) α → T α (x, z) increasing for fixed x ∈ S and z ∈ Z hold for many economic models. For example, the transition rule for the stochastic optimal growth model is well-known to be increasing in the state variable (cf. e.g. [START_REF] Hopenhayn | Stochastic Monotonicity and Stationary Distributions for Dynamic Economies[END_REF]. Also, the transion rule is increasing in the discount factor (a parameter) for each value of the state and shock (cf. e.g. [START_REF] Danthine | Stochastic Properties of Fast vs. Slow Growth Economies[END_REF].

Corollary 3.1. If, in addition, W ⊂ R, then the conclusion holds even when the sequence (α n ) ∞ n=1 is not monotone.

Proof. Write µ n for µ αn and µ for µ α . Since d → is convergence in a topology, it is sufficient to show that every subsequence (µ n(k) ) ∞ k=1 of µ n has a subsubsequence (µ n(k(j)) ) ∞ j=1 converging to µ. So pick any (µ n(k) ) ∞ k=1 . Since every sequence in R has a monotone subsequence, and since every subsequence of (α n ) ∞ n=1 converges to α, there is a subsequence of (α n(k) ) ∞ k=1 -call it (α n(k(j)) ) ∞ j=1 -which converges monotonically to α. In which case µ n(k(j)) d → µ by Theorem 3.4, completing the proof.

h[sf (k n )z]ν(dz) → h[sf (k)z]ν(dz) whenever k n → k ∈ S.
Regarding Assumption 3.3, pick any s ∈ W . On S = (0, ∞) let V (k) := k+k -1 . Clearly V is norm-like on S. By Assumption 2.2, we can choose a real number γ strictly larger than E(1/ξ). By the Inada conditions and differentiability there is a δ > 0 such that sf (k) ≥ γk on (0, δ),

and then a c > 0 such that sf (k) ≥ c on [δ, ∞) by monotonicity, in which case 1 sf (k) ≤ 1 γk + 1 c for all k in (0, ∞).
In addition, by the diminishing returns Inada condition, concavity and the finite mean of the shock, there is an

a 1 < 1 and b 1 < ∞ such that sf (k)Eξ ≤ a 1 k + b 1 , ∀k.
From these bounds we get

V [sf (k)z]ν(dz) = sf (k)zν(dz) + 1 sf (k)z ν(dz) ≤ a 1 k + b 1 + E(1/ξ) γk + E(1/ξ) c . Setting λ := a 1 ∨ [E(1/ξ)/γ] < 1 and b := b 1 + [E(1/ξ)/c] gives (8).
Regarding Assumption 3.4, fix s ∈ (0, 1] =: W and let ψ be any probability on (0, ∞) with a density. Now take any B ⊂ (0, ∞) with positive ψ-measure and any k ∈ (0, ∞) =: S. It is easy to check that the set B/sf (k) has positive Lebesgue measure, so that from Assumption 2.2

we have

Prob k {k t ∈ B for some t ∈ N} ≥ Prob k {k 1 ∈ B} = ν{z : sf (k)z ∈ B} = ν[B/sf (k)] > 0.
Here Prob k is the distribution of (k t ) ∞ t=0 when k 0 ≡ k.

From Theorem 3.1 we now already know that (k t ) ∞ t=0 has a unique stationary distribution. For Theorem 3.3 to hold, however, there are some additional conditions. Let {s n } ∞ n=1 ∪ {s} ⊂ W . First, pick any compact K ⊂ S = (0, ∞). Since f is continuously differentiable, there

Cahiers de la MSE -2004.59 is a number A such that |f (k) -f (k )| ≤ A|k -k | for all k, k ∈ K. But then |s n f (k)z -s n f (k )z|ν(dz) ≤ zν(dz)|f (k) -f (k )| ≤ zν(dz)A|k -k |.
Thus M := zν(dz)A verifies the bound in [START_REF] Santos | Accuracy of Simulations for Stochastic Dynamic Models[END_REF] independent of the value s n .

To finish checking the hypotheses of Theorem 3.3, it remains only to verify that V , λ and b in Assumption 3.3 can be chosen independent of s n . Let V (k) = 1/k + k as before. By restricting attention to the tail of the sequence (s n ) if necessary we can assume that r := inf n s n > 0, because s n → s ∈ W = (0, 1]. As above, we can choose a real number γ strictly larger than E(1/ξ). By the Inada conditions and differentiability there is a δ > 0 such that rf (k) ≥ γk on (0, δ), and

then a c > 0 such that rf (k) ≥ c on [δ, ∞) by monotonicity, in which case 1 s n f (k) ≤ 1 rf (k) ≤ 1 γk + 1 c ∀ n ∈ N, ∀ k ∈ (0, ∞).
In addition, by the diminishing returns Inada condition, concavity and the finite mean of the shock, there is an a 1 < 1 and b 1 < ∞ such that

s n f (k)Eξ ≤ f (k)Eξ ≤ a 1 k + b 1 , ∀k.
From these bounds we get [START_REF] Lasota | Chaos, Fractals and Noise: Stochastic Aspects of Dynamics[END_REF].

V (s n f (k)z)ν(dz) = s n f (k)zν(dz) + 1 s n f (k)z ν(dz) ≤ a 1 k + b 1 + E(1/ξ) γk + E(1/ξ) c . Setting λ := a 1 ∨ [E(1/ξ)/γ] < 1 and b := b 1 + [E(1/ξ)/c] gives
We have now verified all the hypotheses of Theorem 3.

that if L(k(s n )) d → L(k(s)), then L(c(s n )) d → L(c(s)). Further, if L(c(s n )) d → L(c(s)), then Eu[c(s n )] → Eu[c(s)
] by the definition of weak convergence and the assumption that u is continuous and bounded. This will complete the proof of continuity of the objective function. In other words, the probability that in the steady state the capital stock exceeds a converges to zero when savings converges to zero.

That L(k(s n )) d → L(k(s)) when s ∈ W = (0,
So pick any a > 0. From (9),

µ n ([a, ∞)) = 1 [a,∞) [s n f (k)z]ν(dz) µ n (dk) = ν([a/(s n f (k)), ∞))µ n (dk).
Without loss of generality, s n ≤ s 1 for all n. In that case bounded, by the definition of stochastic dominance

(14) µ n ([a, ∞)) ≤ ν([a/(s n f (k)), ∞))µ 1 (dk).
In addition, lim n→∞ ν([a/(s n f (k)), ∞)) = 0 by Dominated Convergence, so applying Dominated Convergence to [START_REF] Stokey | Recursive Methods in Economic Dynamics[END_REF] gives [START_REF] Schenk-Hoppé | Is There a Golden Rule for the Stochastic Solow Growth Model?[END_REF] as required.

We conclude that the objective function ( 6) is continuous for s ∈ [0, 1], a compact set. Therefore a solution to the golden rule problem exists.

That it is interior is immediate from [START_REF] Santos | Analysis of a Numerical Dynamic Programming Algorithm Applied to Economic Models[END_REF], since u is strictly positive on (0, ∞) and integrals of strictly positive functions are strictly positive.

(Therefore s = 0 and s = 1 are not solutions, as in both cases the value of the objective function is zero.) As usual, we say that Definition 4.1. A collection P 0 ⊂ P(S) is tight if, for every ε > 0, there is a compact set K ⊂ S such that sup µ∈P 0 µ(S \ K) < ε. Also, Proof of Lemma 3.2. Fix α ∈ W . By Meyn and Tweedie (1993, Lemma D.5.3), a set P 0 ⊂ P(S) is tight if (and only if) there is a norm-like function V such that sup µ∈P 0 µ(V ) < ∞. This is true for P 0 = {P t α δ x } ∞ t=1 , because if V , λ and b are as in Assumption 3.3, then

P 0 ⊂ P(S)
P t α δ x (V ) = P α P t-1 α δ x (V ) = S Z V [T α (u, z)]ν(dz) P t-1 α δ x (du).
(Substitute P t-1 α δ x for µ and V for 1 B in [START_REF] Phelps | The Golden Rule of Accumulation: A Fable for Growthmen[END_REF].) But

S Z V [T α (u, z)]ν(dz) P t-1 α δ x (du) ≤ S [λV (u) + b]P t-1 α δ x (du) = λP t-1 α δ x (V ) + b.
Repeating this argument t times gives (16)

P t α δ x (V ) ≤ λ t V (x) + b 1 -λ ≤ V (x) + b 1 -λ . ∴ sup t∈N P t α δ x (V ) < ∞.
Thus {P t α δ x } ∞ t=1 is tight, and hence precompact by Prohorov's Theorem.

In order to prove Theorems 3.3-3.4 and Proposition 3.1 we will make use of the following lemma. Proof of Theorem 3.3. For arbitrary β ∈ W , we know by Theorem 3.2 that the process (X t ) ∞ t=0 = (X β t ) ∞ t=0 has a unique stationary distribution. In other words, P β has a unique fixed point µ β ∈ P(S). Let (α n ) Cahiers de la MSE -2004.59 and α be as in the statement of the theorem. In what follows, write µ n for µ αn , µ for µ α , and analogously for T αn , T α and so on.

We first claim that the set of stationary distributions {µ n } ∞ n=1 is precompact in w(P(S), C b (S)). To see this, note that for every x ∈ S, (i) P x := {P t n δ x : t, n ∈ N} is a tight subset of P(S); and (ii) so is

Q x := { 1 t t j=1 P j n δ x : t, n ∈ N}.
The reason for (i) is as follows. By hypothesis, the function V and constants λ and b on the right hand side of ( 16) can be chosen independent of α n (as well as t), so that sup t,n P j n δ x (V ) < ∞. This implies tightness as discussed in the proof of Lemma 3.2. From (i), (ii) is immediate. By Prohorov's theorem, tightness of Q x implies precompactness. Finally, note that {µ n } ∞ n=1 is in the closure of Q x for some fixed x ∈ S by (17). The claim follows.

As a result, every subsequence of (µ n ) has a convergent subsubsequence, written here as (µ j ). Denote the limit of (µ j ) by µ 0 . Suppose we can show for this arbitrary subsequence that the limit point µ 0 is equal to µ. In this case every subsequence of (µ n ) has a subsubsequence converging to µ, and-since convergence in distribution is convergence in a topology-it must be that the sequence (µ n ) itself converges to µ, which is what we wish to prove. So let (µ j ) be the above subsubsequence, µ j

d → µ 0 . Pick any h ∈ BL(S, ), h BL ≤ 1. Fix ε > 0. We have |Pµ 0 (h) -µ 0 (h)| dominated by |Pµ 0 (h) -Pµ j (h)| + |Pµ j (h) -µ j (h)| + |µ j (h) -µ 0 (h)|. (18) ∴ |Pµ 0 (h) -µ 0 (h)| ≤ |Pµ j (h) -µ j (h)| + ε, j large,
where we have used Lemma 3.1. Now consider the remaining term. We have

|Pµ j (h) -µ j (h)| = |Pµ j (h) -P j µ j (h)| ≤ h(T j (x, z))ν(dz) -h(T (x, z))ν(dz) µ j (dx).
Cahiers de la MSE -2004.59

(Substitute h for 1 B in (10).) Since {µ n } and hence {µ j } is precompact it is tight, and there is a compact K ⊂ S with sup j µ j (S \ K) < ε.

Define real functions g j and g on K by

g j (x) := h[T j (x, z)]ν(dz), g(x) := h[T (x, z)]ν(dz).
It follows from Assumption 3.5 and Dominated Convergence that g j converges to g pointwise on K. Also, {g j : j ∈ N} as a collection of real functions is uniformly bounded by 1 and uniformly equicontinuous, as, for any x, x ∈ K,

|g j (x) -g j (x )| ≤ |h(T j (x, z)) -h(T (x , z))|ν(dz) ≤ [T j (x, z), T j (x , z)]ν(dz) (∵ h BL ≤ 1) ≤ M (x, x ).
From the Arzelà-Ascoli Theorem, {g j } is precompact in the supremum norm topology, and therefore has a uniformly convergent subsequence.

Obviously the limit of this subsequence is g, so that, for some j ∈ N,

|g j (x) -g(x)| ≤ ε for all x ∈ K. But |Pµ j (h) -P j µ j (h)| ≤ K |g j (x) -g(x)|µ j (dx) + S\K |h(T j (x, z)) -h(T (x , z))|ν(dz) µ j (dx) ≤ ε + 2ε.
Combining this with (18) gives

|Pµ 0 (h) -µ 0 (h)| < 4ε, ∀ε > 0. ∴ |Pµ 0 (h) -µ 0 (h)| = 0, ∀h ∈ BL(S, ), h BL ≤ 1. ∴ sup{|Pµ 0 (h) -µ 0 (h)| : h BL ≤ 1} = d F M (Pµ 0 , µ 0 ) = 0. ∴ Pµ 0 = µ 0 (∵ d F M a metric).
Since P has only one fixed point in P(S), we conclude that µ 0 = µ.

This completes the proof.

Cahiers de la MSE -2004.59

Proof of Proposition 3.1. Let α and β be as in the statement of the proposition. Pick any x ∈ S. Consider the Markov chains (X α t ) ∞ t=0 and (X β t ) ∞ t=0 when X α 0 = X β 0 ≡ x. From Huggett (2003, Theorem 2) we have L(X α t ) = P t α δ x ≤ P(S) P t β δ x = L(X β t ), ∀t ∈ N whenever the following two conditions hold:

(i) x ≤ S x implies P γ δ x ≤ P(S) P γ δ x , all γ ∈ W .

(ii) α ≤ W β implies P α δ x ≤ P(S) P β δ x , all x ∈ S.

Both of these are easy to verify from the hypotheses of the proposition. Proof of Theorem 3.4. Once again, for each β ∈ W , P β has a unique fixed point and (17) holds. Take (α n ) monotone and converging to α ∈ W . We can and do assume that the sequence is increasing: α n ↑ α. 12 As before we write µ n for µ αn , µ for µ α , and analogously for T αn , T α and so on.

The first important observation is that By Proposition 3.1,

µ 1 ≤ P(S) µ n ≤ P(S) µ, ∀ n ∈ N.
Moreover, in the present setting and order intervals are compact (Assumption 3.6 and Torres ,1990, Theorem 6.6). Indeed, it follows from Assumption 3.6, and Torres (1990, Proposition 6.7) that µ n d → µ 0 for some µ 0 ∈ P(S). If we can show that µ 0 = µ then the proof is done.

12 Otherwise define another order ≤ on W by α ≤ β iff β ≤ W α.

Cahiers de la MSE -2004.59

From Assumption 3.6 and Torres (1990, Proposition 6.2), the increasing This verifies (20), which completes the proof.

1Assumption 2 . 1 .

 21 [START_REF] Schenk-Hoppé | Is There a Golden Rule for the Stochastic Solow Growth Model?[END_REF] considers maximization of expected consumption in a very general stochastic environment. This is a direct generalization of the deterministic problem. However in a stochastic setting maximization of expected utility of consumption is perhaps the more natural criterion.Cahiers de la MSE -2004.59 The function f is continuously differentiable on S, increasing, concave, and satisfies f (k) = 0 ⇐⇒ k = 0 and the Inadaconditions lim k↓0 f (k) = ∞, lim k↑∞ f (k) = 0.Assumption 2.2. Both E[ξ t ] and E[1/ξ t ] are finite. The Lebesgue measure is absolutely continuous with respect to ν (non-Lebesgue-null sets have positive ν-measure).

0≤s≤1E 1 . 2 . 1 .

 121 s u(c), E s u(c) := u dϕ s . We now have the following result. All definitions are made precise in Section 3. The proofs are in Section 4. Part 1 of Theorem 2.1 is illustrated in Figure Theorem Regarding the Solow-Phelps golden rule problem, let Assumptions 2.1, 2.2 and 2.3 hold. The following statements are true.

Let

  E have a topology, and let C b (E) be the continuous bounded functions on E. To each h ∈ C b (E) there corresponds the linear functional on bM (E) defined by µ → µ(h) ∈ R. The weak topology w(bM (E), C b (E)) on bM (E) is the smallest topology which makes all such functionals continuous. Also, w(P(E), C b (E)) is the weak topology on P(E) (i.e., the relative topology inhereted from bM (E)). For

  and zero otherwise, so the inner integral has the interpretation Prob{X t+1 ∈ B | X t = x}. Equation[START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF]

Theorem 3 . 2 .Assumption 3 . 5 .

 3235 If Assumptions 3.1-3.4 hold, then the Markov chain (7) has a unique stationary distribution µ α for each α ∈ W . Proof. Existence follows from Theorem 3.1 and Lemmas 3.1 and 3.2. Uniqueness under Assumption 3.4 is well-known. See, for example, Meyn and Tweedie (1994, Theorem 10.0.1 and Proposition 10.1.1). 3.4. Parametric Continuity. We can now progress to the main results of the paper, concerning parametric continuity of stationary distributions. First, though, in order to have any hope of getting parametric continuity in the stationary distributions, we at least need continuity in the primitives: The transition rule T α is continuous in α. That is, α → T α (x, z) is continuous for each x ∈ S and z ∈ Z.

and

  Prescott (1992, Corollary 3) to a noncompact setting. The proof uses a result of Huggett (2003) on monotone comparative dynamics, a convergence result of Meyn and Tweedie (1994), and the fact that separating orders are closed. Proposition 3.1. Let Assumptions 3.1-3.6 hold. Suppose further that for fixed α ∈ W and z ∈ Z, x → T α (x, z) is increasing, and for fixed x ∈ S and z ∈ Z, α → T α (x, z) is increasing. Let α, β ∈ W , and let µ α and µ β be the corresponding stationary distributions. In this case, if α ≤ W β, then µ α ≤ P(S) µ β .

4 . Proofs 4 . 1 .

 441 The Golden Rule Problem. First we give the proof of Theorem 2.1 based on the parametric continuity results in Section 3. The proof is based on Theorem 3.3. (Alternatively, the same results can be established through Corollary 3.1.) The hypotheses of Theorem 3.3 are now progressively verified. That Assumption 3.1 holds for S = (0, ∞) was verified after the statement of the assumption. Assumption 3.2 is immediate from the Dominated Convergence Theorem and continuity of f : For all s ∈ W := Cahiers de la MSE -2004.59 (0, 1] and h ∈ C b (S),

  3, and hence established Part 1 of Theorem 2.1. Regarding Part 2, we first establish continuity of the objective function (6). Take any s ∈ W . Let µ(s) be the corresponding stationary Cahiers de la MSE -2004.59 distribution for capital. Let k(s) denote a random variable with this distribution. Steady state consumption given s is (12) c(s) = (1 -s)f (k(s))ξ. Now take any s n → s. It follows from (12), the continuity of f and the Slutsky's Theorem (Dudley, 2002, Theorem 11.7.1)

  1] was proved in Part 1. It only remains then to check that L(k(s n )) d → L(k(0)) = δ 0 when s n → 0. So take such a sequence (s n ) ⊂ (0, 1]. Write µ n for µ sn := L(k(s n )), which we now regard as measures on [0, ∞) putting zero mass on {0}. By the Portmanteau Theorem (Aliprantis and Border, 1999, Theorem 14.2) µ n d → δ 0 if and only if lim inf n→∞ µ n (G) ≥ δ 0 (G) for every relatively open set G ⊂ [0, ∞). Evidently this is equivalent to lim n µ n (G) = 1 for all open G containing 0, which in turn is equivalent to (13) lim n→∞ µ n ([a, ∞)) = 0, ∀a > 0.

  µ n ≤ P(S) µ 1 by Proposition 3.1. Since k → ν([a/(s n f (k)), ∞)) is increasing and Cahiers de la MSE -2004.59

4. 2 .

 2 Remaining Proofs. We first give some general discussion.It is known that when S is Polish, P(S) with topology w(P(S), C b (S)) is itself Polish, completely metrized by the Fortet-Mourier metric described below. Let BL(S, ) be the collection of bounded Lipschitz functions on (S, ). This space is given the norm (15) h BL := sup x∈S |h(x)| + sup x =y |h(x) -h(y)| (x, y) . Now set d F M (µ, ν) := sup |µ(h) -ν(h)|, where the supremum is over all h ∈ BL(S, ), h BL ≤ 1. Then d F M metrizes w(P(S), C b (S)) (cf. e.g. Dudley 2002, Theorem 11.3.3).

Lemma 4 . 1 .→

 41 Fix β ∈ W . Under Assumptions 3.1-3.4, not only does P β have a unique fixed point µ β ∈ P(S), but in addition the averages of the marginal distributions converge to it in distribution. That is, µ β , ∀x ∈ S. Proof. Immediate from Lemma 3.2, Theorem 3.2 and Meyn and Tweedie (1994, Theorem 12.1.4).

  δ x ∀t ∈ N 0 .Taking limits with respect to t now gives µ α ≤ P(S) µ β , where we have used (17) and the fact that ≤ P(S) is a w(P(S), C b (S))-closed order on P(S) (Assumption 3.6-separating implies closed-and Torres, 1990, Theorem 6.1).

  functions in C b (S) separate P(S), in the sense that if µ and µ are distinct elements of P(S), then there is an h ∈ ibB(S) ∩ C b (S) with µ (h) = µ (h). Given that µ is the only fixed point of P, then, it suffices to show that (20)Pµ 0 (h) = µ 0 (h), ∀ h ∈ ibB(S) ∩ C b (S).So take h ∈ ibB(S) ∩ C b (S) and ε > 0. As in the proof of Theorem 3.3,|Pµ 0 (h) -µ 0 (h)| is dominated by |Pµ 0 (h) -Pµ n (h)| + |Pµ n (h) -µ n (h)| + |µ n (h) -µ 0 (h)|. (21) ∴ |Pµ 0 (h) -µ 0 (h)| ≤ |Pµ n (h) -µ n (h)| + ε, n large,using Lemma 3.1. Consider the remaining term. We have|Pµ n (h) -µ n (h)| = |Pµ n (h) -P n µ n (h)| ≤ h(T n (x, z))ν(dz) -h(T (x, z))ν(dz) µ n (dx).(Substitute h for 1 B in[START_REF] Phelps | The Golden Rule of Accumulation: A Fable for Growthmen[END_REF].) Since {µ n } is precompact by Prohorov's theorem it is tight, so there is a compact K ⊂ S with sup n µ n (S \ K) < ε. Define real functions g n and g on K byg n (x) := h[T n (x, z)]ν(dz), g(x) := h[T (x, z)]ν(dz).It follows from Assumption 3.5 and Dominated Convergence that g n converges to g pointwise on K. Since h is increasing, the pointwise convergence is monotone. Since h ∈ C b (S) and Assumption 3.2 holds, g n and g are continuous. From Dini's Lemma, then, g n also converges uniformly to g on K, so that, for some n ∈ N,|g n (x) -g(x)| ≤ ε for all x ∈ K. But then |Pµ n (h) -P n µ n (h)| ≤ K |g n (x) -g(x)|µ n (dx) + S\K |h(T n (x, z)) -h(T (x , z))|ν(dz) µ n (dx) ≤ ε + 2Bε,Cahiers de la MSE -2004.59 where B := sup x∈S |h(x)|. Combining this with (21) gives |Pµ 0 (h) -µ 0 (h)| < (2 + 2B)ε.

[ 15 ]

 15 Torres, R. (1990): "Stochastic Dominance." DP n. 905, September 1990, Center for Mathematical Studies in Economics and Management Science, Northwestern University Center for Operations Research and Econometrics, Université Catholique de Louvain, 34 Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium Cahiers de la MSE -2004.59

  Stokey, Lucas and Prescott, 1989, p. 219, or Lasota and Mackey, 1994, p. 414). In particular, We now give the main existence result, which can be thought of as a direct generalization of the well-known compact state result given in Stokey, Lucas and Prescott (1989, Theorem 12.10). The result is not essentially new. It can be deduced in a similar setting from Meyn and Tweedie (1994, Theorem 12.1.2, Proposition 12.1.3 and Lemma D.5.3).

if L(X 0 ) = δ x , then L(X t ) = P t α δ x ,

where δ x ∈ P(S) is the distribution concentrated at x, and the superscript t means t iterations of P α . Comparing

[START_REF] Phelps | The Golden Rule of Accumulation: A Fable for Growthmen[END_REF]

with (

9

), clearly µ α ∈ P(S) is a stationary distribution of

[START_REF] Huggett | When are comparative dynamics monotone?[END_REF]

if and only if it is a fixed point of P α .

The Markov operator

[START_REF] Phelps | The Golden Rule of Accumulation: A Fable for Growthmen[END_REF]

can be thought of as a map not just on P(S), but also from bM (S) to itself. It satisfies P α P(S) ⊂ P(S). More generally, let us define a Markov operator P on (S, B(S)) to be any linear map from bM (S) into itself such that PP(S) ⊂ P(S). Definition 3.3. A Markov operator P on (S, B(S)) is said to have the Feller property if it is w(bM (S), C b (S))-continuous.

  is called precompact if it has compact closure.

	We will make extensive use of Prohorov's Theorem (cf. e.g. Aliprantis
	and Border, 1999, Theorem 14.22): For S Polish and P(S) with the
	w(P(S), C b (S)) topology, a subset of P(S) is tight if and only if it is
	precompact.

Cahiers de la

MSE -2004.59

As is well-known, correlations of finite order can be treated in this framework by redefining state variables and, if necessary, adjusting the dimensions of the state space.Cahiers de laMSE -2004.59

Of course T α is required to be (B(S) ⊗ Z , B(S))-measurable, ∀ α ∈ W . Cahiers de laMSE -2004.59


Nicolas Grollier
email: nicolas.grollier@imt-atlantique.fr

Sébastien Houcke
email: sebastien.houcke@imt-atlantique.fr

Michaël Pelissier
email: michael.pelissier@cea.fr

Performance Analysis of an Enhanced Spectrally Aware RF front end Receiver under Non-linearity

In most of RF receivers (RX) the analog chain aims to improve the overall Signal to Interference plus Noise Ratio (SINR) by filtering out interferences in the Signal Of Interest (SOI) vicinity. However, cognitive radio applications require a high dynamic range RX, pushing a high constraint on the analog section. In particular, nonlinearities may damage the overall SINR and Bit Error Rate (BER) due to blocking and intermodulation processes in presence of strong interference. RX linearization state of the art methods that rely on the feedforward concept have shown the RX linearization improvement on the BER. Nonetheless, benefits of other linearization methods such as the feedback loop concept, are unstudied. In this article, our aim is to decrease the power of unwanted harmonics thanks to a feedback loop. We first explain the variable gain enhancement concept. Then, we present an algorithm that dynamically adapts the RX parameters (gains and ADC margins) to make it work in a linear regime. We analyze the effect of such a method in terms of BER. Important BER improvement is noticed in blocking and saturating situations due to co-channel interferers.

I. INTRODUCTION

In this article we consider the context of Professional Mobile Radios (PMR) radios. The FITNESS project (see [START_REF] Fitness | Fitness scope website[END_REF]) aims to prepare the future of PMR in Europe and in the world. New functionalities requested by missioncritical markets are added, while preserving backwards compatibility with existing narrowband PMR systems. The considered receiver is narrowband, and has stringent specifications to make it compliant with the four European PMR norms in terms of receiver selectivity.

Several PMR communications channels close to the signal of interest (SOI) may be present at ADC level and 60 dB above the SOI. Commercial automatic gain control (AGC) are unable to deal with a dynamic up to 141 dB with channel bandwidth of 12.5 kHz. This specification is particularly severe. So, a specific architecture that allows amplification stage to be reconfigurable was defined. This architecture is meant to be fully integrated, and able to cope with tight linearity and gain requirements of a PMR receiver (RX) front-end. To ensure that ADCs will not clip the received signal, large back-off digitization margins have to be left. One other issue is caused by the nonlinear behavior of RX analog components. Indeed, amplifiers or mixers (see [START_REF] Razavi | RF Microelectronics (2Nd Edition)[END_REF]) used outside of their linear region are responsible for intermodulation products harmonic and even gain blocking. Such terms lay into the SOI bandwidth, as developed in [START_REF] Zou | Digital compensation of cross-modulation distortion in software-defined radios[END_REF]. In the following, unwanted signals are referred as interferers.

The nonlinearity cancellation is a vast topic that is generally addressed in an analog way. This is usually a well-known designer problem, and examples of mechanisms are proposed in [START_REF] Razavi | RF Microelectronics (2Nd Edition)[END_REF], [START_REF] Kenington | High-linearity RF Amplifier Design, ser. Artech House microwave library[END_REF] to limit such effects under specific bounds. Devices specifications could be adjusted by a proper analog design to maximize one or several criteria amongst power efficiency, linearity, peak to average power ratio or adjacent channel power ratio. Three main principles exist to compensate for nonlinearities in the analog domain: feedback, feedforward and predistortion.

Nonetheless, in this paper we consider the whole RX as a series of nonlinear blocks, instead of a single device (e.g. a power amplifier). We assume that there is no knowledge of the signal input, which at first discards the feedback and the predistortion methods. To our best knowledge, it appeared that all state of the art methods of RX linearization rely on the feedforward principle. Two main approaches are to be distinguished. On one hand, there are many fully digital techniques [START_REF] Zou | Digital compensation of cross-modulation distortion in software-defined radios[END_REF], [START_REF] Valkama | Advanced digital signal processing techniques for compensation of nonlinear distortion in wideband multicarrier radio receivers[END_REF]- [START_REF] Myer | Automatic reduction of intermodulation products in high power linear amplifiers[END_REF]. In these papers, various ways allows the cancellation of almost all the power of Inter Modulation Distortion of order 3 (IMD3) harmonics. Their overall ideas are similar: modeling IMD3 harmonics and then subtract it from the received signal with an adaptive mechanism. On the other hand, a mixed analog-digital strategy was proposed in [START_REF] Keehr | Equalization of third-order intermodulation products in wideband direct conversion receivers[END_REF] and is refined in [START_REF]Successive regeneration and adaptive cancellation of higher order intermodulation products in rf receivers[END_REF]. In theses papers IMD terms are generated thanks to an analog mechanism that model the nonlinear mechanism. This signal is digitized and subtracted from the amplified input signal thanks to a least mean square algorithm.

However, none of these methods allow to relax significantly ADCs dynamic range and sampling frequency requirements. Indeed, to be modeled, interferers have to be digitized with respect of Shannon's rule. In [START_REF] Grollier | Enhanced spectrally aware rf front end receiver under non-linearity[END_REF] we have already proposed a novel method to decrease IMD3 products power as a proof of concept. It showed that thanks to a sensing mechanism, the RX could be partially aware of its spectral environment and able to change its parameters (gains and ADC margins) to work in its linear region. The proposed sensing mechanism is able to detect nonlinear harmonics before the compression point and gain clipping.

In this article, we design a feedback loop to adjust the RX gain. We rely on the detection method [START_REF] Grollier | Enhanced spectrally aware rf front end receiver under non-linearity[END_REF] to detect the nonlinear harmonics presence. Our aim is to maintain the RX at the limit of detectable harmful harmonics. It allows to take advantage of the largest gain while decreasing the power of unwanted components. We analyze the effect of such a system on decoding performance, in blocking tests bench. In section II, we develop the nonlinear system model and remind the variable gain control concept. Section III is devoted to practical implementation issues and the simulation counterpart. Main detection mechanism principles are developed as well. To underline possible improvements obtained with our method, we show the BER improvement through simulations in section IV. Section V concludes the paper.

II. SYSTEM MODEL A. Received Signal Model

Let us define useful and interferer continuous baseband signal models in [START_REF] Fitness | Fitness scope website[END_REF].

z i (t) = β i ∞ k=-∞ a k,i h(t -kT i ) (1)
Let T i be the symbol period, a k,i be the complex random symbols with unit variance, h(t) be a continuous shaping filter (normalized in energy) and β i the square root power of z i (t). Here z 1 (t) is meant to be the useful signal and z 2 (t) the unwanted signal. In the following, only the h(t) shape is known as a root raised cosine filter. We consider that the SOI and the interferer are independent from each other, and a k,i are independent MPSK symbols. The signal y i (t) at carrier frequency f ci can be expressed as:

y i (t) = [z i (t)e 2πfcit ] (2)
y 2 (t) parameters are assumed to be unknown except for its carrier frequency f c2 . The composite received signal is the combination of SOI and interferer. This signal is defined in [START_REF] Zou | Digital compensation of cross-modulation distortion in software-defined radios[END_REF].

x(t) = 2 i=1 y i (t) (3)

B. Nonlinear Model

Our whole receiver is modeled as a memoryless polynomial expressed as:

p(t) = α 1 x(t) + α 2 x(t) 2 + α 3 x(t) 3 + w(t) (4)
where α k , k ∈ {1, 2, 3} are characteristics of the RX front-end and w(t) is an additive white Gaussian noise. In the following, α 2 is assumed zero since harmonics in |x(t)| 2 are easily discarded in a heterodyne architecture with proper filtering stages and a front-end differential structure. The nonlinear amplification coefficient α 3 is obtained thanks to (5) (see [START_REF] Razavi | RF Microelectronics (2Nd Edition)[END_REF]), where V IIP 3 is the RX Input Interception Point of third order.

α 3 = - 4 3 α 1 V 2 IIP 3 (5)
The substitution of (3) into (4) leads to the complete formula of harmonic creation for the nonlinear model. All terms are listed in [START_REF] Zou | Digital compensation of cross-modulation distortion in software-defined radios[END_REF], but α 2 is not considered here. We focus on the received signal baseband representation in a given sub-band of interest. The SOI in this sub-band is given by:

p(t) = α 1 + 3 4 α 3 |z 1 (t)| 2 + 3 2 α 3 |z 2 (t)| 2 z 1 (t)e j2πfc1t + w(t) (6)
We clearly see that in [START_REF] Rebeiz | Spectrum sensing under rf non-linearities: Performance analysis and dsp-enhanced receivers[END_REF] the third right hand-side term depends on interferer's power. Hence, when the interference power comes too high, the overall linear gain α 1 is blocked.

C. Variable Gain Enhancement Concept

In a conventional RF receiver, the dynamics of the useful signal is determined by compromising between linear amplification and the background noise (NF) added by the chain components. The presence of unwanted signals is also taken into account in this choice, by limiting amplification and setting additional digitization margins (e.g. PAPR). However, as presented in [START_REF] Grollier | Enhanced spectrally aware rf front end receiver under non-linearity[END_REF], having an RX that is aware of the presence of interferers allows the gain to be dynamically changed to remain in a linear regime despite the presence of unwanted signals. This method also reduces digitization margins by increasing the conversion dynamic. There are two different use scenarios:

1) Sensitivity: which reflects the presence of a lowpowered useful signal (there are no jammers, β 2 =0). If there is no disturbance signal, the gain can be increased to reduce the background noise. 2) Linearity: which corresponds to two situations: a) the useful signal is the only signal present but has a high power; b) the wanted signal and at least one high power interferer are present. In ( 6), these cases correspond to a gain blocking situation due to the third order term of (4). Thus, the gain should be reduced to allow a better linearity, but at the cost of a degradation of the NF.

III. VARIABLE GAIN IMPLEMENTATION ASPECTS

In this section we first develop the gain adaptation issue in a real RX. Then, we go on with main principles of the chosen spectrum sensing method. Finally, we explain the practical method we used in simulation.

A. Gain Adaptation Mechanisms

For several cascaded nonlinear stages, the circuit nonlinearity is described by a relation equivalent to the well known Friis formula. From [START_REF] Razavi | RF Microelectronics (2Nd Edition)[END_REF], we have:

1 V 2 IIP 3,tot ≈ 1 V 2 IIP 3,1 + α 2 1,1 V 2 IIP 3,2 +. . .+ n-1 k=1 α 2 1,k V 2 IIP 3,n (7)
An analysis of this equation is useful to underline mathematically what we propose to do. The ( 7) is the general expression for n cascaded nonlinear stages. Let define α 1,i the linear gain of stage i and V IIP 3,i their corresponding IIP3, with i ∈ {1, . . . , n}. From ( 5), we recall that V IIP 3,i coefficients are linked to (4) by the nonlinear gain α 3 such as:

α 3,i = - 4 3 α 1,i V 2 IIP 3,i (8)
(8) is a relation that can be applied indifferently for each stage i or for the overall cascaded stage. Hence, any modification on a particular analog component impacts all the following RX chain.

In particular, if we consider the linearity scenario: the linear gain has to be decreased. A short analysis of [START_REF] Myer | Automatic reduction of intermodulation products in high power linear amplifiers[END_REF] shows that when coefficients α 1,i decrease, so do the quantity 1/V 2 IIP 3,tot . In [START_REF] Keehr | Equalization of third-order intermodulation products in wideband direct conversion receivers[END_REF], when V IIP 3i increases the nonlinear gain α 3,i decreases. So, increasing V IIP 3,tot makes the circuit more linear, which is the searched effect in the linearity scenario. However, the maximum gain decrease is limited by the noise floor and the Friis formula in [START_REF]Successive regeneration and adaptive cancellation of higher order intermodulation products in rf receivers[END_REF]. For m stages [START_REF] Razavi | RF Microelectronics (2Nd Edition)[END_REF] indicates that:

N F tot = 1 + (N F 1 -1) + N F 2 -1 α 1,1 + . . . + N F m -1 m-1 k=1 α 1,k (9)
Hence, a gain decrease makes the noise floor increases. In practice the gain can be decreased until the noise floor becomes too high. This approach is also valid in the sensing scenario. In the two cases, the minimum affordable gain makes the noise floor too important. On the contrary, the maximum amplification makes the circuit too nonlinear.

In an implementation point of view, the gain of each RX components is not tunable. It means that, one has to adapt specific gains coefficients in order to obtain the desired amplification. Moreover, changing the gain of a unique component makes the RX characteristic different.

In the following we assume that α 1,tot and V IIP 3,tot are continuous and continuously modifiable.

Furthermore, to realize computer simulations a relation between α 1,tot and α 3,tot has to be modeled. In the aim to change the linear gain parameter, one has to change in accordance with the V IIP 3,tot value. Nonetheless, as seen in ( 7) V IIP 3,tot is a function of the linear gain such as:

(V IIP 3,tot ) |dB = f (α 1,tot ) |dB ( 10
)
This relation is useful only in a simulation context. Indeed, in a real RX the gain is adapted to the needs and the nonlinear amplification is a parameter controlled by the chosen linear gain value. So, updating the V IIP 3,tot value is not necessary.

B. Detection of nonlinear harmonics

To make the receiver aware of its spectral environment, we must identify whether the RX is working in its non-linear region, and why. The presence of an unwanted signal and/or an excess of power of the wanted signal creates many harmonics whose expressions are detailed in [START_REF] Zou | Digital compensation of cross-modulation distortion in software-defined radios[END_REF]. We wish to use the presence of these harmonics to detect the non-linear operation of the RX. We based our work on the criterion developed in [START_REF] Grollier | On carrier qpsk signal detector based on second order cyclic-moments[END_REF], which requires few symbols and also allows a quick adaptation of the gain. In the following, we refer to this method as the Cyclostationary Nonlinear Harmonics Detector (CNHD). The idea is to detect the most powerful harmonics that are located at f c2 , 2f c1 and 2f c2 . They correspond to the carrier frequency of the unwanted signal and the frequencies of the non-linear harmonics due to the wanted signal power and the interfering signal.

C. Practical Implementation

Let us now develop the feedback loop to adjust automatically the RX gain. The considered scenario is the linearity scenario, which means that a powerful interferer is present. We propose the scheme represented in fig. 1. In this figure, the CNHD bloc stands for Cyclostationary Nonlinear Harmonics Detector (see section III.B). Let us define the input signal vector x N (k) as the block of N samples such as x N (k) = x(kN ), . . . , x(kN + N -1) . N is considered large enough to perform the detection with confidence. Let also define Φ N (k) = x N (k); x 3 N (k) a 2×N matrix, W = α 1 , α 3 a vector of gain and w N (k) = w(kN ), . . . , w(kN +N -1) a 1×N complex AGWN samples vector such as p N (k) is defined as in (11) (equivalent to (4)).

p N (k) = W Φ N (k) + w N (k) (11)
For each bloc of N samples the CNHD estimates ĴN,L (4f c2 ), the detection criterion. If ĴN,L (4f c2 ) is greater than Γ (the detection threshold), it decides that the saturation is due to the interferer presence. So, the

f nl x N (k), α k CNHD x N (k) w N (k) p N (k) ĴL,N(4fc2) Γ ê(k) - + f nl x N (k), α k
ê(k) = ĴN,L (4f c2 ) -Γ (12)
This formulation is equivalent to maintaining the RX at the limit of detectable harmful harmonics. It allows to take advantage of the largest affordable gain while decreasing the power of unwanted components. However, one can notice that the chosen method is not adaptive filtering. Indeed, original samples of x N (k) before its processing through RX analog components remain unknown. Hence, a minimization of the quadratic error in the Wiener sense can not be done.

The best way to adjust the gain value is to realize a gradient descent minimizing the quadratic error ê(k) 2 . It requires to assume that ê(k) 2 is convex, which implies that ĴN,L (4f c2 ) depends on α 1 and is monotone. The ideal gradient descent updating relation is:

α1,tot (k + 1) |dB = α1,tot (k) |dB -µ ∂ê(k) 2 ∂α 1 ( 13
)
Here µ is a step size that is chosen in function of the algorithm complexity. Indeed, it can be constant, function of k or optimal depending on the needs. One can notice that the gradient descent in (13) implies to use the theoretical expression of ĴN,L (4f c2 ) as a function of α 1,tot .

As we lack such an expression, we propose an empirical solution. We propose to use (12) as it represents the distance to the ideal gain value, to adjust the amplification of the whole circuit. This approach is valid since the value of ĴN,L (4f c2 ) is linked to the gain thanks to [START_REF] Kenington | High-linearity RF Amplifier Design, ser. Artech House microwave library[END_REF]. So, we propose an approximated solution inspired from the NLMS algorithm that takes into account the value of ê(k) in ( 14). Here, sgn . stands for the sign function and µ(k) = 10/k is a variable step size that allows a convergence to a fixed value. Then, the VIIP 3,tot value has to be updated with the relation [START_REF] Grollier | Enhanced spectrally aware rf front end receiver under non-linearity[END_REF] to update the nonlinear model. Now, two remarks can be made:

• The first point is that ê(k) directly depends on Γ, which is set to satisfy a particular false alarm probability (P f a ). So, changing the false alarm probability directly impacts the estimated gain α1 (k +1).

When we consider ( 14), one can observe that decreasing P f a means less detection. So it leads to decrease α1 (k +1) less frequently and still working in a nonlinear regime. • The second point is that (14) is not optimal in the Wiener sense. There is no guaranty that the relations converge to a value that minimizes the MMSE problem. Hence, we have to check if the final value actually decreases the nonlinear harmonic power. To answer these two points, we propose to examine the power of the nonlinear term that lays in the SOI bandwidth.

We saw in (6) that the theoretical expression of the blocking term, denoted as s(t), is:

s(t) = 3/2α 3 z 1 (t)|z 2 (t)| 2 .
From this relation, s(t) theoretical power is developed in (15).

P s = E |s(t)| 2 = 3 2 α 3 β 1 β 2 2 2 (15)
(15) is used to define a new comparison metric between the power of s(t) and the power of α 1 y 1 (t) (the ideally amplified SOI): the Signal to In-band Harmonic Ratio (SIHR).

SIHR = P α1y1 P s (16)
This definition is different from the SIR, which is the ratio β 1 /β 2 (SOI power over interferer power). The relation in ( 16) is represented graphically in fig. 2. The Let us now study the -30 dB SIR curve as an example. Considering α 1,tot equals 32 dB, the corresponding SIHR is -11.4 dB. It means that the in-band harmonic power is much higher than the SOI power. For now, we assume that a SIHR close to 20 dB implies that the in band unwanted harmonic has no influence on the decoding process. This assumption will be verified thanks to simulations and the fig. 3 analysis in the next section. So, one can use fig. 2 to check the best value of α 1 for each SIR values. That way, for a -30 dB SIR the gain has to be set at 21 dB. Knowing the nonlinear gain, signal and interferers powers, this method gives a gain value without any adaptive algorithm. However, in this application, we have no access to such informations.

In tab.I, we sum-up several values of gain after a few algorithm iterations. According to this table, in the previous example a 21 dB gain is obtained for a P f a set at 5%.

IV. SIMULATION AND NUMERICAL RESULTS

A. Experimental Conditions

To respect the Shannon's condition for the interferer signal, we set a sampling rate f s = 10f c1 . The shaping filter h(t) is defined as a square-root raised cosine of period T = 2.5T 1 , a roll-off at 0.8 and span at 6 symbols. Blocker and useful signal are in adjacent channels, and the interferer carrier frequency is defined as T 2 = T 1 /2.4. Both SOI and interferer are 4PSK modulated, with independent and identically distributed symbols. As discussed in [START_REF] Grollier | On carrier qpsk signal detector based on second order cyclic-moments[END_REF], we assume that 8 symbols are enough to perform a detection of such signals using the CNHD method. We defined the SOI power β 1 to -106.0 dBm. The detection threshold Γ is fixed for a P f a of 5%, as it represents a good tradeoff between detection and gain adjustment.

The RX gain is initialized at α 1,tot (1) = 32dB, when the corresponding V IIP 3,tot (1) = -12.7dBm. For different α 1,tot (k) values the relation in [START_REF] Grollier | Enhanced spectrally aware rf front end receiver under non-linearity[END_REF] has to be developed. The V IIP 3,tot value can be measured for a given α 1,tot thanks to a two-tone test. For the particular RX front-end we considered, we tested several linear gain values and performed a two-tone test for each one. That way, we obtained an experimental relation between α 1,tot and V IIP 3,tot .

(V IIP 3,tot (k)) |dB = -(α 1,tot (k)) |dB + 10.7 (17)
Nonlinear model coefficients are then adapted thanks to relations ( 14) and (17).

B. Simulation Results

Let us first consider the assumption that a SIHR close to 20 dB is enough to have a BER close to the "no interferer" case. To make sure that the unwanted harmonic power in the SOI band is small compared to the SOI power, we provide fig. 3. For several noise levels, we measured the influence of the SIHR value. We see that for each E b /N 0 level, a negative SIHR (dB) implies a degraded BER. When the SIHR comes positive, the BER improves to get closer of its value when no interferer is present. One can see that the BER decreases to a stable value when the SIHR increases. So, past a SIHR of approximately 20 dB, even a larger gain decrease may not lead to a significant BER improvement. This behavior is verified in the following simulations. Let now tackle the effect of the gain adaptation through a BER analysis. In fig. 4, we defined a SIR such as the RX is faintly nonlinear (i.e. -30 dB). As it can be seen, the uncorrected BER is severely degraded: for a -10 3 BER value, there is a 5 dB E b /N 0 gap when compared to the "no interferer" case. Nonetheless, the RX is still able to decode a part of the received signal as the curve decreases when the E b /N 0 ratio increases. It means that the unwanted harmonic in the SOI bandwidth is powerful, but not dominant. However, our gain adjustment mechanism allows a significant BER improvement. We represent several BER curves for which α 1 is different. In compliance with the SIHR analysis, the BER is improved by a gain decrease, while the SIHR is smaller than 20 dB. For a BER of 10 -4 , the improvement is close to BER when no interferer is present. Nonetheless, there is indeed a small difference between the 21 dB and the 12 dB BER curves. If we compare now the 21 dB BER curve with the no interferer case, for a 10 -7 BER the improvement margin is about 0.85 dB, which is small. Whatever the gain, there is still a small gap between the theoretical bound and gain adapted BER curves.

Finally, let us assume a SIR equals -40 dB such as the RX works in a highly nonlinear regime. Such a powerful interferer makes the RX saturates, as shown in fig. 5. We clearly see that without the gain adaptation mechanism, the BER is constant even when the noise power decreases. Our method allows the BER to be significantly improved. When compared with the 5 dB gain BER curve, the gap is very small, accordingly with fig. 3 remarks. However, as the interferer is very powerful (even after the adapted filter), its contribution is too important to be removed completely. That is why a large gap exists between the corrected gain BER curve and the theoretical bound.

V. CONCLUSION In this work, we first recalled the method principles to make the RX works in its linear region thanks to a feedback loop. Based on results in [START_REF] Grollier | Enhanced spectrally aware rf front end receiver under non-linearity[END_REF], [START_REF] Grollier | On carrier qpsk signal detector based on second order cyclic-moments[END_REF] we proposed an adaptive mechanism to maintain nonlinear harmonics power small. To our best knowledge, this concept is not tackled yet in the state of the art. Then, we analyzed the effects of a gain adaptation on in-band IP3 harmonics. This analysis allows to confirm that a gain maximization is possible while minimizing the nonlinear harmonics power. Important BER improvement is noticed in blocking and saturating situation. This proof of concept will soon be evaluated in a real device. Besides previous developments concerning PMR radios, the same method can be applied in every situation where cochannel interference make the RX works in its nonlinear region.

Fig. 1 .

 1 Fig. 1. Adaptive Feedback Loop with a Cyclostationary Nonlinear Harmonics Detector (CNHD)

Fig. 2 .

 2 Fig. 2. Comparison of theoretical power of in-band components for several SIR values as a function of the RX Gain

Fig. 3 .

 3 Fig. 3. Influence of SIHR on the BER for several E b /N 0 values at fixed SIR = -30 dB

Fig. 4 .

 4 Fig. 4. Comparison of BER curves for uncorrected and corrected RX gain, for a SIR = -30 dB

Fig. 5 .

 5 Fig. 5. Comparison of BER curves for uncorrected and corrected RX gain, for a SIR = -40 dB

ACKNOWLEDGMENT

This work was funded in the scope of the FITNESS project (see [1]). The authors would like to thank all FITNESS partners and specifically the CEA for their cooperation.