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ON INVARIANT DISTRIBUTION ESTIM FOR A CLASS OF CONTINUOUS-TI STATIONARY PROCESSES Dominique Dehay
[email protected] Rennes 2 Haute-Bretagne
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SASP V, Le Mans, january 7th
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I - Introduction II - Asymptotic independence III - Convergence of the empirical distribution function IV - Asymptotic normality V - Asymptotic variance estimation VI - Weak convergence
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I - Introduction © ª X := Xt, t ≥ 0 – continuous-time, – real-valued – stationary, – µ invariant marginal distribution, law of Xt, ∀t ≥ 0 £
F (x) := P X0 ≤ x 1 b FT (x) := T
Z 0
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¤
T
I{Xs≤x} ds
Example : diffusion process dXt = S(Xt)dt + dWt
where S : R −→ R Borelian with a polynomial majorant lim sup sgn(x)S(x) < 0. |x|→∞
Then – existence and unicity of the invariant law µ with de ´ √ ³ ¡ 2 loi b – T FT (x) − F (x) −→ N 0, σ (x)) Z ∞³ ´ σ 2(x) = 2 Ft(x, x) − F (x)2 dt 0 "µ ¶2 # F (ξ ∧ x) − F (ξ)F (x) 2 = 4f (x) E f (ξ) L(ξ) = µ 4
– Estimation of the asymptotic variance Z TÃb bT (Xt)FbT (x 1 F (X ∧ x) − F T t 2 2 σ bx := 4fb(x) T 0 fbT (Xt) + T −1/4 Then
(D. & Kutoyants 2004) P
σ bx2 −→ σx2
– TLC fonctionnel n√ ¡ o ¢ law T FbT (x) − F (x) : x ∈ R −→ G, n o G = G(x) : x ∈ R Gaussian process
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¡ C0(R),
(Negri 1998).
Discrete-time process ¡ ¢ 1) Yn n i.i.d. r.v. distribution F ( · ). n
X 1 Fbn(x) := I{Yk ≤x}. n k=1
Then
´ ¡ 2 √ ³ loi b n Fn(x) − F (x) −→ N 0, σ (x)) ¡ ¢ 2 σ (x) = F (x) 1 − F (x) ¡ ¢ 2 b b σ bn(x) := Fn(x) 1 − Fn(x) −→ σ 2(x) n√ ¡ o ¢ law n Fbn(x) − F (x) : x ∈ R −→ G,
G Gaussian process
(voir Dudley 1984).
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a.s. ¡
D(R),
¡ ¢ 2) Yn n strongly mixing stationary process with P n αn < ∞ Then
´ ¡ 2 √ ³ law b n Fn(x) − F (x) −→ N 0, σ (x))
∞ X ¢ ¡ 2 σ (x) = F (x) 1 − F (x) + 2 Fk (x, x) − F
¡
k=1
Futhermore P 2 1/2 If n n ϕn < ∞ (Billingsley 1968) or if F ( · ) is continuous and αn = O(n−a), a > 1 (Rio 2000) then n√ ¡ o ¡ ¢ law b D(R), n Fn(x) − F (x) : x ∈ R −→ G, G Gaussian process.
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II - Asymptotic independence (CL*) Castellana Leadbetter (1986) ΓT (A) −→ Γ(A), ∀A ∈ B(R2) where Z T³ ´ (2) ΓT (A) := µt (A) − µ ⊗ µ(A) dt. 0
(SM) Strong mixing (α-mixing) (Rosenblatt 1956) © ª © s σ (X) := σ X(u) : u ≤ s et σs(X) := σ X(u) : n¯ ¯ ¯ α(t) := sup P[A ∩ B] − P[A] P[B]¯ : A ∈ σ s(X), B ∈ σs+t( X is strongly mixing when lim α(t) = 0. t→∞
Remark :
α( · ) ∈ L1([0, ∞))
=⇒
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(CL∗).
Examples 1) Diffusion process
(Veretennikov 1998)
¡ ¢ α(t) ≤ exp −λt
2) Exponentially ergodic Markov process Markov X with ¢ ¡ process – P t t≥0 transition semi-group – kP t(x, · ) − µkvar ≤ M (x)ρt – 0 < ρ < 1, M ∈ L1(µ). Then X is strongly mixing and α( · ) ∈ L1([0, ∞)) with Z α(t) ≤ M (x) µ(dx) ρt. R
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III - Convergence of the empirical distribution £ ¤ b X stationary =⇒ E Ft(x) = F (x) and Z T³ ´³ ´ 2 t var FbT (x) = 1− Ft(x, x) − F (x)2 dt T 0 T Proposition 1 Under (CL*),
(Asymptotic covariance)
C(x, y) := lim T cov[FbT (x), FbT (y)] Z T∞→∞ ³ ´ = Ft(x, y) + Ft(y, x) − 2F (x)F (y) dt. 0
(Asymptotic variance) £ ¤ 2 b σ (x) := lim T var Ft(x) T →∞ Z ∞³ ´ ³ =2 Ft(x, x) − F (x)2 dt = 2 Γ (−∞, x] × 0
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Proposition 2 (Almost-sure convergence) Assume that one of the two following conditions is satisfi (i) α(t) = O(t−γ ) for some 0 < γ < 1, (ii) (CL*). Then
¯ ¯ ¯ ¯ lim T δ ¯FbT (x) − F (x)¯ = 0
T →∞
a.s.
for each 0 ≤ δ < γ/3, in the case (ii) we take γ = 1.
The process is ergodic and satisfies Glivenko-Cantelli pr · ¸ ¯ ¯ ¯ ¯ P lim sup¯FbT (x) − F (x)¯ = 0 = 1. T →∞ x
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IV - Asymptotic variance estimation Z ∞³ ´ Ft(x, x) − F (x)2 dt. σ 2(x) = 2 0
Let 0 < η < 1/3 fixed 1 Fbt,T (x) := T Z σ bT2 (x) := 2
Z
T
0 T η³
I{Xs≤x} I{Xs+t≤x} ds Fbt,T (x) − FbT (x)
2
´
dt.
0
Proposition 3
If α( · ) ∈ L1([0, ∞)) then lim σ bT2 (x) = T →∞
Assume that α(t) = O(t−γ ) for some γ © 2 ª δ 2 Then lim T σ bT (x) − σ (x) = 0 T →∞ n in quadratic mean for 0 < η < 13 and δ < min 21 (1 − 3η) n almost-surely for 0 < η < 41 , γ > 32 and δ < min 31 (1 − 4 Proposition 4
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Let
Z
bT (x, y) := C
T η³
0
1 Fbt,T (x, y) := T
Z 0
Fbt,T (x, y) + Fbt,T (y, x) − 2 FbT (x)FbT (
T
I{Xs≤x} I{Xs+t≤y} ds
0 < η < 1/3, then bT (x, y) = C(x, y) q.m. lim C
T →∞
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V - Asymptotic normality Proposition 5 If α( · ) ∈ L1([0, ∞)) then ´ √ ³ law ¡ 2 b b GT (x) := T FT (x) − F (x) −→ N 0, σ (
If in addition σ(x) 6= 0 we deduce that √ ³ ´ b GT (x) T law ¡ b := 2 FT (x) − F (x) −→ N 0, 2 σ bT (x) σ bT (x)
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Sketch for the proof of Proposition 5 –
–
x fixed
¯√ © ª p © ª¯¯ ¯ ¯ T FbT (x) − F (x) − [T ] Fb[T ](x) − F (x) ¯ ≤ Z
–
n
Yn(x) = n−1
I{Xt≤x} dt − F (x)
–
strongly-mixing stationary sequence of bounded zer
–
CLT (Ibraginov Linnik 1971)
–
N X √ © ª ¡ 1 b N FN (x)−F (x) = √ Yk (x) −→ N 0, σ N k=1
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VI - Weak convergence Notations : (Pollard 1990, Rio 2000) – X = R with pseudo-metric ρ(x, y) := |F (x) − F (y)| – B(X) space of bounded measurable functions f : X with the uniform norm kf ( · )k∞ = supx |f (x)|.
Theorem 6 Assume that O(t−γ ) for some γ > 1. Then n√ ¡ o ¢ ¡ law b b GT = T FT (x) − F (x) : x ∈ R −→ G, in B – G := {G(x) : x ∈ R} Gaussian process £ ¤ – cov G(x), G(y) = C(x, y) – P-almost all the paths of G are uniformly continuous
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Corollary 7 n√ ¡ o ¢ law bT = G T FbT (x) − F (x) : x ∈ R −→ G,
Corollaire 8 If in addition F ( · ) is continuous, then n√ ¡ o ¢ law bT = G T FbT (x) − F (x) : x ∈ R −→ G,
¡
¡
D(R
Cb(R
P-almost all paths of G are uniformly continuous in R.
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Sketch for the proof of Theorem 6 ¯√ © ª p © ª¯¯ ¯ i) ¯ T FbT (x) − F (x) − [T ] Fb[T ](x) − F (x) ¯ ii)
(Rio 2000)
Theorem (Pollard 1990) Let (X, ρ) be aªtotally-bounded pseudo-metric space, a © bn(x) : x ∈ X , n ≥ 0 a sequence of measurable proce G ¡ ¢ b b – Gn(x1), . . . , Gn(xk ) converges in law ∀k, x1, . . . , –
∀² > 0, ∀η > 0, ∃δ > 0, " lim sup P n→∞
∗
#
¯ ¯ b b ¯ sup Gn(x) − Gn(y)¯ > η
