Poincaré and log-Sobolev inequality for stationary Gaussian processes and moving average processes

BLAISE PASCAL

Guangfei Li, Yu Miao, Huiming Peng, Liming Wu Poincaré and log-Sobolev inequality for stationary Gaussian processes and moving average processes

Volume 12, n^o2 (2005), p. 231-243.

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Poincaré and log-Sobolev inequality for stationary Gaussian processes and moving

average processes

Guangfei Li Yu Miao Huiming Peng

Liming Wu

Abstract

For stationary Gaussian processes, we obtain the necessary and sufficient conditions for Poincaré inequality and log-Sobolev inequality of process-level and provide the sharp constants. The extension to moving average processes is also presented, as well as several concrete examples.

1 Introduction and Main Results

LetX := (X_n)_n∈Z be a real valued stationary Gaussian process with

EX₀²=σ²>0, EX₀X_n=σ²ρ(n), ∀n∈Z. (1.1) It is a very important class of stochastic processes both in theory and appli- cations. The limit theorems about stationary Gaussian processes are abun- dant, see Avram [1] for the central limit theorem, Donsker and Varadhan [3]

and L. Wu [9] for the large deviations, H.Djellout and al. [2] for moderate deviations, and the references therein.

In this note we are mainly interested in Poincaré inequality and Logarith- mic Sobolev inequality on the product space R^Z for the law of the process X. As developed by Ledoux [5] and many other authors, the Poincaré or log- Sobolev yield sharp concentration inequalities, which are much more robust than the limit theorems quoted above.

We begin by describing those two inequalities on E:=R^Z. Regarding l²(Z) :={h:= (h_n)n∈Z|h_n∈R,|h|:=

s X

n∈Z

|h_n|²<+∞}

as a tangent space of E=R^Z, let∇ be the corresponding gradient, i.e., for a functionF onE, derivable at each coordinate x_i, i.e., ∂_xiF exists, let

∇F(x) = (∂_xiF(x))i∈Z.

IfF :E→Rand ∇F :E→l²(Z) are continuous, we say thatF ∈C¹(E).

Definition 1.1: We say that a R-valued stochastic process X = (X_n)n∈Z

satisfies the Poincaré inequality, if there is some best constant c_P(X)∈R⁺ such that

V ar_P(F(X))≤c_P(X)E X

i∈Z

|∂_xiF|²(X), ∀F ∈C¹(E)\

C_b(E), (1.2) andX satisfies the log-Sobolev inequality (LSI in short) if there is some best constant c_LS(X)∈R⁺such that

Ent_P(F²(X))≤2c_LS(X)E X

i∈Z

|∂_xiF|²(X), ∀F ∈C¹(E)\

C_b(E). (1.3) HereEnt_P(F(X)) :=E^PF(X) log ^F(X)

E^PF(X) forP-integrable nonnegativeF(X), is the Kullback entropy.

The same definition and notation apply also to the random vector inRⁿ. It is well known thatc_P(X)≤c_LS(X) (cf. [5]).

For the stationary Gaussian process given in (1.1), ifρ(m) = 0,∀m6= 0, it becomes an i.i.d. sequence for which it is now well known (due to Gross [6], see [5])

c_P(X) =c_LS(X) =σ². (1.4) To state our main result, let us introduce the (nonnegative and bounded) spectral measureµon the torus Tidentified as[−π, π], which is determined by (Bochner’s theorem)

σ²ρ(n) = 1 2π

Z π

−π

e^−intdµ(t), ∀n∈Z (1.5) The main result of this note is:

Theorem 1.2: For the stationary Gaussian process in (1.1), we have

c_P(X) =c_LS(X) =

(kfk_∞, if µdt, f :=dµ/dt;

+∞, otherwise (1.6)

where kfk∞ = esssup_tf(t). In particular, X satisfies the Poincaré or log- Sobolev inequality iffµdtand the density f :=dµ/dtis bounded.

When ρ(·)∈l²(Z), the spectral density f =dµ/dtexists and belongs to L²([−π, π]) :=L²([−π, π], dt)and

f(t) =σ²X

k∈Z

ρ(k)e^ikt =σ² 1 + 2X

k≥1

ρ(k) cos(kt)

!

. (1.7)

From the result above we have immediately

Corollary 1.3: Ifρ(n)≥0for all n or if (−1)ⁿρ(n)≥0for all n, then c_P(X) =c_LS(X) =σ² 1 + 2X

n≥1

|ρ(n)|

! .

In particular, X satisfies the Poincaré or log-Sobolev inequality iff P

n≥1|ρ(n)|<+∞.

In the literature, P

n≥1|ρ(n)| < +∞ is often called short range depen- dence, cf. Taqqu [8].

Remark 1.4: For stationary Gaussian processX = (X_k)k∈Zwith lawP (on R^Z), one can consider the abstract Wiener space(R^Z, H, P), whereH ⊂R^Zis the Cameron-Martin space associated withP. It is not difficult (but already more difficult than the proof of Theorem 1.2) to check that for any smooth F :R^Z→Rdepending only on a finite number of variables,

k∇_HFk²_H =h∇F,Γ∇Fi_l²_(Z)

where∇_H is the Malliavin gradient, andΓ = (σ²ρ(k−l))_k,l∈Zis the covariance matrix ofX. By the Gross theorem,

Ent_P(F²(X)) =Ent_P(F²)≤2 Z

k∇_HFk²_HdP = 2Eh∇F,Γ∇Fi_l2(Z)(X) (1.8) (an easy derivation of it is given in the proof of Lemma 2.1). This log-Sobolev inequality, though involving the covariance structure of X, is however less convenient (than Theorem 1.2) for the derivation of concentration inequali- ties. WhenΓ :l²(Z)→l²(Z)is bounded, the right hand side (r.h.s. in short) of (1.8) is bounded by

λ_max(Γ)E|∇F|²_l2(Z)(X).

By (2.2) below, λ_max(Γ) =kfk∞.

This note is organized as follows. The next section is devoted to the proof of Theorem 1.2 and its counterpart for continuous time Gaussian processes.

In §3, we present an extension to the moving average processes and provide several examples.

2 Proof of Theorem 1.2 and the continuous time counterpart

2.1 Proof of Theorem 1.2

It is based on the following (known) observation:

Lemma 2.1: For a d-dimensional random vector X of law N(0,Γ), where Γis the covariance matrix, then

c_P(X) =c_LS(X) =λ_max(Γ) where λ_max(Γ) denotes the maximal eigenvalue ofΓ, i.e.,

λ_max(Γ) = sup

x∈R^d

< x,Γx >

< x, x > .

We give its proof for the convenience of the reader and especially for its simplicity.

Proof: Letξ be a random vector of lawN(0, I)onR^d. Then X and√ Γξ have the same law. Therefore by (1.4), we have for any boundedC¹function F onR^d,

Ent(F²(X)) =Ent(F²(√

Γξ))≤2E|∇_ξF(√ Γξ)|²

= 2E|√

Γ(∇F)(

√

Γξ)|²= 2EhΓ(∇F)(X),(∇F)(X)i

≤2λ_max(Γ)E|∇F(X)|²

where it follows that c_P(X) ≤ c_LS(X) ≤λ_max(Γ). Furthermore, letting x₀ be an unit eigenvector ofΓassociated with λ_max(Γ)and F(x) :=hx, x₀i, we see that|∇F(x)|=|x₀|= 1and F(X)is centered, Gaussian with variance

V ar(F(X)) =hx₀,Γx₀i=λ_max(Γ) =λ_max(Γ)E|∇F(x)|²

where it follows thatc_P(X)≥λ_max(Γ). The lemma is proved.

Proof: (Proof of Theorem 1.2) Considering (X_k/σ) if necessary, we may assume thatσ= 1without loss of generality. LetX⁽ⁿ⁾:= (X_k)_−n≤k≤n, which is centered, Gaussian with the covariance matrix given by the Toplitz matrix

Γ_n= (ρ(k−l))−n≤k,l≤n.

In the definition 1.1, one can take only bounded C¹-function F depending on a finite number of variables (by approximation, the detail is left to the reader). In other words we always have

c_P(X) = sup

n

c_P(X⁽ⁿ⁾), c_LS(X) = sup

n

c_LS(X⁽ⁿ⁾). (2.1) Then in the present situation we get by Lemma 2.1,

c_P(X) = sup

n

c_P(X⁽ⁿ⁾) = sup

n

c_LS(X⁽ⁿ⁾) = sup

n

λ_max(Γ_n) =c_LS(X).

We divide the proof into two cases.

Case 1. ρ(·)∈/ l²(Z). In this case, we have λ_max(Γ_n)≥ sup

x∈R²ⁿ⁺¹;|x|=1

|(Γ_nx)₀|= sup

x∈R²ⁿ⁺¹;|x|=1 n

X

k=−n

x_kρ(k) = v u u t

n

X

k=−n

ρ(k)²

and thusc_LS(X) =c_P(X) = +∞.

Case 2. ρ(·) ∈ l²(Z). In this case µ dt and the spectral density f =dµ/dtis in L²([−π, π]). It remains to show that

sup

n

λ_max(Γ_n) =kfk∞. (2.2)

The following simple proof is due to the referee. By Rayleigh’s principle, and noting thatρ(k−l) = _2π¹ R

e^−i(k−l)tf(t)dt, we have for any n≥1, λ_max(Γ_n)≤ sup

|x|≤1

hx,Γ_nxi= sup

|x|≤1

1 2π

Z π

−π

n

X

k=−n

x_ke^ikt

2

f(t)dt,

which is obviously bounded from above bykfk∞by Parseval’s equality. Con- versely, using the equality above and the denseness of trigonometric polyno- mials in the Banach space C_bTof complex valued continuous and bounded

functions onT, we have furthermore sup

n

λ_max(Γ_n) = sup

g

1 2π

Z π

−π

|g(t)|²f(t)dt

where the supremum runs over all complex-valuedg∈C_bTsuch that 1

2π Z π

−π

|g|²(t)dt≤1.

This supremum equals tokfk_∞.

2.2 Continuous time stationary Gaussian processes

Let now X = (X_t)t∈R be a real-valued stationary centered Gaussian pro- cess, defined on the probability space (Ω,F,P), with continuous covariance function onR,

γ(t) :=EX₀X_t, ∀t∈R. Letµbe the spectral measure of X onR, determined by

γ(t) = 1 2π

Z

R

e^−itsdµ(s), ∀t∈R

(Bochner’s theorem). It is nonnegative and bounded (indeed µ(R) =γ(0)).

We can and will assume that for each T >0, the sample paths ofX[−T,T]= (X_t)_t∈[−T,T] are a.s. inL²[−T, T] :=L²([−T, T], dt) (such version exists for its covariance operator is of trace class, see the proof of Theorem 2.2).

Let E = L²_loc(R), the space of all real-valued locally (dt−) square inte- grable functions onR, equipped with the project limit topology ofL²[−T, T] as T → +∞. Regarding L²(R) := L²(R, dt) as the tangent space ofE, for any Gateaux-differentiable functionF onEandx∈Esuch that|D_hF(x)| ≤ C_xkhk₂for allh∈L²(R), where khk₂:=khk_L²_(R,dt) and

D_hF(x) := lim

ε→0

1

ε(F(x+εh)−F(x)), we can define the gradient∇F(x) = (∇_tF(x))_t∈R∈L²(R) by

Z

R

∇_tF(x)h(t)dt=D_hF(x), ∀h∈L²(R).

In the variational calculus, we have formally∇_tF(x) = _δx(t)^δ F(x).

WhenF :E→Rand ∇F :E→L²(R) are continuous and bounded, we say thatF ∈C_b¹(E). Similarly as in Definition 1.1, let c_P(X)∈[0,+∞] be the best constant for the following Poincaré inequality

V ar_P(F(X))≤c_p(X)E Z

R

|∇_tF|²(X)dt, ∀F ∈C_b¹(E),

and c_LS(X) ∈ [0,+∞] be the best constant for the following log-Sobolev inequality

Ent_P(F²(X))≤2c_LS(X)E Z

R

|∇_tF|²(X)dt, ∀F ∈C_b¹(E).

These functional inequalities with respect to the L²-metric (instead of the Cameron-Martin metric) have been investigated by M. Gourcy and the fourth author [4] for diffusions. We have the following counterpart of Theorem 1.2.

Theorem 2.2: We have c_P(X) =c_LS(X) =

(kfk_∞, if µdt, f :=dµ/dt;

+∞, otherwise.

Proof: 1) We begin with an extension of Lemma 2.1: let X be a cen- tered Gaussian random variable valued in a separable Hilbert space H of lawN(0,Γ), where the self-adjoint nonnegative definite covariance operator Γ :H →H, determined by

Ehh₁, Xihh₂, Xi=hh₁,Γh₂i.

It is well known thatΓis of trace class (and conversely if Γis of trace class, then N(0,Γ) is a probability measure on H). Then using the usual pre- Dirichlet formE|∇F|²_H(X) and lettingc_P(X), c_LS(X) be the best constants for the corresponding Poincaré and log-Sobolev inequalities, respectively, we have

c_P(X) =c_LS(X) =λ_max(Γ). (2.3) Indeed, let (e_n)_n∈N be an orthonormal basis of H such that Γe_n = λ_ne_n where the sequence of eigenvalues (λ_n)_n∈N is ranged as non-increasing. As

{hX, e_ni;n ∈ N} are independent with laws {N(0, λ_n);n ∈N}, we have by the independent tensorization ([5]),

c_P(X) = sup

n

c_P(hX, e_ni), c_LS(X) = sup

n

c_LS(hX, e_ni), c_P(hX, e_ni) =c_LS(hX, e_ni) =λ_n

where (2.3) follows.

2)By approximation, it is easy to check that c_P(X) = sup

T >0

c_P(X[−T,T]), c_LS(X) = sup

T >0

c_LS(X[−T,T])

wherec_P(X[−T,T]), c_LS(X[−T,T])are the best constants defined in Step 1) with H =L²[−T, T]. The covariance operator ofX_[−T,T] are given by

(Γ_Th)(t) = Z

[−T,T]

γ(t−s)h(s)ds, ∀h∈L²[−T, T].

By Step 1), we have only to prove that sup

T >0

λ_max(Γ_T) =

(kfk_∞, if µdt, f :=dµ/dt;

+∞, otherwise.

Let L²

C[−T, T], L²

C(R) be the spaces of complex-valued L²-integrable func- tions on [−T, T]and R, respectively. For anyh∈L²_C(R), let

ˆh(t) = 1

√2π Z

R

e^itsh(s)ds

be the Fourier transform of h. It is well known that h → ˆh is unitary on L²_C(R). Regarding h∈ L²[−T, T] as an element of L²(R) by puttingh = 0 out ofL²[−T, T], we have

λ_max(Γ_T) = sup

h∈L²

C[−T,T],khk2≤1

hh,Γ_Thi

= sup

h∈L²

C[−T,T],khk2≤1

1 2π

Z

R

Z

R

e^itsh(s)ds

2

dµ(t)

= sup

h∈L²

C[−T,T],khk2≤1

Z

R

|ˆh(t)|²dµ(t).

Since anyh∈L²_C(R)T

L¹_C(R), h1\[−T,T]→ˆhin L²_C(R)and also uniformly on R, we have

sup

T >0

λ_max(Γ_T) = sup

h∈L² C(R)T

L¹

C(R),khk2≤1

Z

R

|ˆh(t)|²dµ(t).

Since the familyA of all ˆh withh ∈L²_C(R)T

L¹_C(R) constitutes an algebra separating the points ofR, then by monotone class theorem, for any complex- valued measurable and bounded function g on R, say g ∈ b_CB(R), we can find a sequence(g_n= ˆh_n∈ A)such thatg_n→g inL²(R, dt+µ). Thus

sup

T >0

λ_max(Γ_T) = sup

g∈b_CB(R),kgk2≤1

Z

R

|g|²(t)dµ(t).

This implies easily the desired result.

3 Extension and several examples

In this section we extend Theorem 1.2 to general moving average processes and provide some concrete examples.

3.1 Extension to moving average processes

Let (ξ_k)_k∈Z be a sequence of i.i.d. real-valued random variables such that Eξ₀ = 0 and Eξ²₀ = 1, and (a_k)k∈Z ∈ l²(Z). Consider the moving average process

X_n:=

+∞

X

k=−∞

a_kξ_n+k, n∈Z. (3.1) It is a well defined stationary process with covariance function

γ(n) =EX₀X_n=

+∞

X

k=−∞

a_ka_k−n, ∀n∈Z.

Its spectral density function is given by f(θ) =

+∞

X

n=−∞

γ(n)e^inθ =

+∞

X

n=−∞

a_ne^inθ

2

.

A stationary Gaussian process with spectral density f ∈L²([−π, π])can be always written as a moving average process with driven noises (ξ_k) being i.i.d. with lawN(0,1). So the following result partially generalizes Theorem 1.2.

Theorem 3.1: Let X = (X_n) be the moving average process given above with the spectral density f. Then

c_P(X)≤ kfk∞c_P(ξ₀), c_LS(X)≤ kfk∞c_LS(ξ₀). (3.2)

Remark 3.2: Bobkov and Götze [7] have found necessary and sufficient conditions for both characterizingc_P(ξ₀)<+∞and c_LS(ξ₀)<+∞.

Proof: By (2.1), it is enough to prove that for any n ≥ 1 and for any smooth and bounded functionF :R^In →Rwhere I_n= [−n, n]T

Z, V ar(F(X_In))≤ kfk∞c_P(ξ₀)E|∇F(X_In)|²;

Ent(F²(X_In))≤ kfk_∞c_LS(ξ₀)E|∇F(X_In)|².

We prove here only the first Poincaré inequality (the proof of the LSI is completely similar). Let A_n = (a_k−l)_k∈In,l∈Z : l²(Z) → R^In and ξ = (ξ_l)_l∈Z as a column vector. We have X_In = Aξ. Now for each N ≥ 1, let P¯_Nξ = (1_k∈INξ_k)_k∈Z as before. Since F(A_nP¯_Nξ)is a bounded and smooth function ofξ_IN, by usingc_P(ξ) =c_P(ξ₀)we have

V ar(F(A_nP¯_Nξ))≤c_P(ξ₀)E|∇_ξF(A_nP¯_Nξ)|²

=c_P(ξ₀)E|(A_nP¯_N)^∗(∇F)(A_nP¯_Nξ)|²

≤c_P(ξ₀)λ_max((A_nP¯_N)(A_nP¯_N)^∗)E|(∇F)(A_nP¯_Nξ)|² where A^∗ denotes the adjoint matrix or operator. Letting N go to infinity, asA_nP¯_Nξ→A_nξ=X_In, a.s., so we get by dominated convergence

V ar(F(X_In))≤c_P(ξ₀) sup

N≥1

λ_max((A_nP¯_N)(A_nP¯_N)^∗)E|(∇F)(X_In)|². Furthermore, since

h(A_nP¯_N)(A_nP¯_N)^∗x, xi=|P_N^∗A^∗_nx|²≤ |A^∗_nx|²=hA_nA^∗_nx, xi, ∀x∈R^In, we haveλ_max((A_nP¯_N)(A_nP¯_N)^∗)≤λ_max(A_nA^∗_n)for allN. ButA_nA^∗_ncoincides with the covariance matrixΓ_n= (ρ(k−l))_k,l∈In ofX_In, hence we obtain

V ar(F(X_In))≤c_P(ξ₀)λ_max(Γ_n)E|(∇F)(X_In)|². Now the desired Poincaré inequality follows by (2.2).

3.2 Several examples

Example 3.3: (Fractional Brownian Motion) Let (B^H(t))t≥0 be the real fractional Brownian Motion with Hurst index H ∈ (0,1), i.e., a centered Gaussian process with

EB^H(s)B^H(t) = 1

2 s^2H+t^2H − |t−s|^2H

, ∀s, t≥0.

Let X_n = B^H(n+ 1)−B^H(n), n ∈N, which is stationary with covariance function

ρ(n) =Cov(X₀, X_n) = 1

2 (n+ 1)^2H + (n−1)^2H−2n^2H

, n≥0 andρ(n) =ρ(−n), n≤0. Note thatρ(n)∼2H(2H−1)n^2(H−1), asn→+∞.

1) If H > 1/2, thenρ(n) > 0 for all n and P

n≥0ρ(n) = +∞. Thus by Corollary 1.3, X does not satisfy the Poincaré inequality, neither the log-Sobolev inequality.

2) IfH = 1/2(the trivial case), then c_P(X) =c_LS(X) = 1.

3) If H <1/2, then ρ(n) < 0 for all n and P

n≥1|ρ(n)| = 1/2. Thus the spectral density f exists and it is continuous on the torus T. Conse- quently c_P(X) =c_LS(X) =kfk_∞ ≤2.

Example 3.4: (ARMA model) Consider the autoregressive process X_n+1=θX_n+σξ_n+1, n≥0

whereθ∈(−1,1),(ξ_k)_k∈Z is a sequence of i.i.d. r.v. withEξ₀= 0andEξ₀²= 1,σ²>0is the strength of noise, andX₀is independent of(ξ_n)_n≥1. Its unique invariant measure is the law ofP+∞

k=0θ^kξ_−k. Below we takeX₀=P+∞

k=0θ^kξ_−k. In that case,(X_n)_n≥0is a moving average process witha_n = 1_n≤0θ^|n|and with covariance function γ(n) =σ²(1−θ²)⁻¹θ^|n|. Its spectral density function is given by

f(t) = σ² 1−θ²

X

n∈Z

θ^|n|e^int= σ²

1 +θ²−2θcost, ∀t∈T.

Thenkfk∞=σ²(1− |θ|)⁻². Consequently by Theorem 3.1, c_P(X)≤ σ²

(1− |θ|)²c_P(ξ₀), c_LS(X)≤ σ²

(1− |θ|)²c_LS(ξ₀).

In practice the following noises are the most often used:

1) ξ₀ is Gaussian N(0,1). We have c_P(ξ₀) = c_LS(ξ₀) = 1 and then by Theorem 1.2, c_P(X) =c_LS(X) =σ²(1− |θ|)⁻².

2)ξ₀ is of law uniform on [−a, a]where a= (3/2)^1/3 (so thatEξ₀²= 1). In this case it is well known ([5]) that

c_P((π/a)ξ₀) =c_LS((π/a)ξ₀) = 1.

Then c_P(ξ₀) =c_LS(ξ₀) =a²π⁻². Thus we get c_P(X)≤c_LS(X)≤

3 2π³

2/3

σ² (1− |θ|)².

3) ξ₀ is of density e^−2|x|dx (symmetric exponential law). Again it is well known that c_P(2ξ₀) = 1 butc_LS(2ξ₀) = +∞([5]). Hence

c_P(X)≤ σ² 4(1− |θ|)².

Since for any λ >0, by Jensen’s inequality,Ee^λ|X0|2 ≥Ee^λ|ξ0|2 = +∞, we have c_LS(X) = +∞ by [5].

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[9] L. Wu. on large deviations for moving average processese. Probability, Finance and Insurance, pp.15-49, the proceeding of a Workshop at the University of Hong-Kong (15-17 July 2002 ). Eds: T.L. Lai, H.L. Yang and S.P. Yung. World Scientific 2004, Singapour.

Guangfei Li Wuhan University Dep. of Mathematics Hubei, MA 430072 CHINA

hawkfeilee@yahoo.com.cn

Liming Wu

Université Blaise Pascal Lab. de Mathématiques CNRS-UMR 6620 63177 Aubière France

Li-Ming.Wu@math.univ-bpclermont.fr

Huiming Peng Wuhan University Dep. of Mathematics Hubei, MA 430072 CHINA

phm821@sina.com.cn

Yu Miao

Wuhan University Dep. of Mathematics Hubei, MA 430072 CHINA

and Henan Normal University College of Mathematics and Information Science Henan, MA 453007 CHINA

yumiao728@yahoo.com.cn