Small ball probabilities and metric entropy

Transcription

1 Small ball probabilities and metric entropy Frank Aurzada, TU Berlin Sydney, February 2012 MCQMC

2 Outline 1 Small ball probabilities vs. metric entropy 2 Connection to other questions 3 Recent results for concrete examples

3 Outline 1 Small ball probabilities vs. metric entropy 2 Connection to other questions 3 Recent results for concrete examples

4 Small ball probabilities Let (X t ) t 0 be a stochastic process with X 0 = 0 Goal: find asymptotic rate of P?, with ε 0 ε ε X 1 In many examples, P with γ > 0 und κ > 0. = e κε γ (1+o(1)), with ε 0

5 Small ball probabilities Let (X t ) t 0 be a stochastic process with X 0 = 0 Goal: find asymptotic rate of P?, with ε 0 ε ε X 1 In many examples, P with γ > 0 und κ > 0. = e κε γ (1+o(1)), with ε 0

6 Small ball probabilities Let (X t ) t 0 be a stochastic process with X 0 = 0 Goal: find asymptotic rate of P?, with ε 0 ε ε X 1 Therefore, we study φ X (ε) := log P the so-called small ball function of X. γ = κε γ (1 + o(1)), with ε 0

7 Entropy numbers Let X be a centred Gaussian random variable with values in a sep. Banach space (E,. ): i.e. X, g Gaussian g E.

8 Entropy numbers Let X be a centred Gaussian random variable with values in a sep. Banach space (E,. ): i.e. X, g Gaussian g E. There is a linear operator u : L 2 0, 1 E belonging to X such that ( Ee i X,g = exp 1 ) 2 u (g) 2 2, g E. Note: u(l 2 0, 1) is the RKHS of X

9 Entropy numbers Let X be a centred Gaussian random variable with values in a sep. Banach space (E,. ): i.e. X, g Gaussian g E. There is a linear operator u : L 2 0, 1 E belonging to X such that ( Ee i X,g = exp 1 ) 2 u (g) 2 2, g E. Note: u(l 2 0, 1) is the RKHS of X Example: X BM in E = C0, 1 (uf )(t) = t 0 f (s)ds; u : L 2 0, 1 C0, 1.

10 Entropy numbers / small ball function On the one hand, we consider the small ball function: ( φ X (ε) = log P X E ε = log P )

11 Entropy numbers / small ball function On the one hand, we consider the small ball function: ( φ X (ε) = log P X E ε = log P ) On the other hand, the entropy numbers of u: e n (u) := inf{ε > 0 ε-net of 2 n 1 points of u(b L2 0,1) in E}, where B L2 0,1 is the unit ball in L 2 0, 1 (inverse of covering numbers).

12 Asymptotics We use the following notation Weak asymptotics: a(ε) b(ε), ε 0 means lim sup ε 0 a(ε) b(ε) < a(ε) b(ε), ε 0 means a(ε) b(ε) and b(ε) a(ε)

13 Asymptotics We use the following notation Weak asymptotics: a(ε) b(ε), ε 0 means lim sup ε 0 a(ε) b(ε) < a(ε) b(ε), ε 0 Strong asymptotics: means a(ε) b(ε) and b(ε) a(ε) a(ε) b(ε), ε 0 means lim sup ε 0 a(ε) b(ε) = 1 a(ε) b(ε), ε 0 Similarly for n means a(ε) b(ε) and b(ε) a(ε)

14 The small ball entropy connection Theorem (Kuelbs/Li 93, Li/Linde 99, A./Ibragimov/Lifshits/van Zanten 08) For r > 0 and δ R: φ X (ε) ε r log ε δ e n (u) n 1/2 1/r (log n) δ/r φ X (ε) ε r log ε δ e n (u) n 1/2 1/r (log n) δ/r where the first requires φ X (ε) φ(2ε). Further, for δ > 0 and κ > 0, φ X (ε) κ log ε δ log e n (u) κ 1/δ n 1/δ φ X (ε) κ log ε δ log e n (u) κ 1/δ n 1/δ. small ball pr. entropy numbers (probabilistic) (functional analytic)

15 The small ball - entropy connection Example: X Riemann-Liouville process in C0, 1 (uf )(t) = t 0 (t s) H 1/2 f (s)ds; u : L 2 0, 1 C0, 1. one has φ X (ε) ε 1/H In particular for X BM, H = 1/2 φ X (ε) ε 2 e n (u) n 1/2 H e n (u) n 1

16 Outline 1 Small ball probabilities vs. metric entropy 2 Connection to other questions 3 Recent results for concrete examples

17 Connections of small ball prob. to other questions In the setup of Gaussian processes, there are various connections to: entropy of function classes convergence rate of series representations coding quantities for the process approximation quantitites for the process Chung s law of the iterated logarithm statistical problems... Generally: the small ball rate increases the slower the better the process can be approximated the smoother the process is

18 Connections of small ball prob. to other questions approximation of stochastic processes n X (n) t = ξ i ψ i (t) X t i=1 error X (n) X 0 coding, quantisation, quadrature N Ef(X) f( ˆX i)q i i=1 law of the iterated logarithm lim inf t 0 φ X (ε) = log P = κε γ (1 + o(1)) sup s t X s = c b(t) path regularity Gaussian process n-times differentiable γ 1/n functional analysis entropy numbers of linear operators between Banach spaces PDE problems other approximation quantities such as Kolmogorov widths, etc.

19 Connections of small ball prob. to other questions approximation of stochastic processes n X (n) t = ξ i ψ i (t) X t i=1 error X (n) X 0 coding, quantisation, quadrature N Ef(X) f( ˆX i)q i i=1 law of the iterated logarithm lim inf t 0 φ X (ε) = log P = κε γ (1 + o(1)) sup s t X s = c b(t) path regularity Gaussian process n-times differentiable γ 1/n functional analysis entropy numbers of linear operators between Banach spaces PDE problems other approximation quantities such as Kolmogorov widths, etc.

20 Connections of small ball prob. to other questions approximation of stochastic processes n X (n) t = ξ i ψ i (t) X t i=1 error X (n) X 0 coding, quantisation, quadrature N Ef(X) f( ˆX i)q i i=1 law of the iterated logarithm lim inf t 0 φ X (ε) = log P = κε γ (1 + o(1)) sup s t X s = c b(t) path regularity Gaussian process n-times differentiable γ 1/n functional analysis entropy numbers of linear operators between Banach spaces PDE problems other approximation quantities such as Kolmogorov widths, etc.

21 Connection to smoothness of process Theorem (A. 11) Let (X t ) t 0,1 be a centred Gaussian process and n an integer. If (a modif. of) X is n-times differentiable with X (n) L 2 0, 1 then φ X (ε) = log P ε 1/n. ε ε X 1

22 Connection to smoothness of process Theorem (A. 11) Let (X t ) t 0,1 be a centred Gaussian process and n an integer. If (a modif. of) X is n-times differentiable with X (n) L 2 0, 1 then φ X (ε) = log P ε 1/n. ε ε X 1 Now, what happens when n (above) is non-integer?

23 Connection to smoothness of process Define fractional differentiation: Let γ > 0 (recall X 0 = 0) X (γ) t = x(t) if X t = t 0 (t s) γ 1 x(t)dt.

24 Connection to smoothness of process Define fractional differentiation: Let γ > 0 (recall X 0 = 0) Theorem (A. 11) X (γ) t = x(t) if X t = t 0 (t s) γ 1 x(t)dt. Let (X t ) t 0,1 be a centred Gaussian process and γ > 1/2. If X (γ) exists and X (γ) L 2 0, 1 then φ X (ε) = log P ε 1/γ.

25 Connection to smoothness of process Define fractional differentiation: Let γ > 0 (recall X 0 = 0) Theorem (A. 11) X (γ) t = x(t) if X t = t 0 (t s) γ 1 x(t)dt. Let (X t ) t 0,1 be a centred Gaussian process and γ > 1/2. If X (γ) exists and X (γ) L 2 0, 1 then φ X (ε) = log P ε 1/γ. Example : Brownian motion X is γ-times differentiable (Hölder), γ < 1 2. log P ε 2 = ε 1 1/2.

26 Connection to smoothness of process Define fractional differentiation: Let γ > 0 (recall X 0 = 0) Theorem (A. 11) X (γ) t = x(t) if X t = t 0 (t s) γ 1 x(t)dt. Let (X t ) t 0,1 be a centred Gaussian process and γ > 1/2. If X (γ) exists and X (γ) L 2 0, 1 then φ X (ε) = log P ε 1/γ. Example : Brownian motion X is γ-times differentiable (Hölder), γ < 1 2. log P Similar results for different norms ε 2 = ε 1 1/2.

27 Connection to smoothness of process Corollary (A. 11) Let (X t ) t 0,1 be a centred Gaussian process. If X has a C -modif. then for any δ > 0 ( ) lim ε 0 εδ log P = 0.

28 Connection to smoothness of process Corollary (A. 11) Let (X t ) t 0,1 be a centred Gaussian process. If X has a C -modif. then for any δ > 0 ( ) lim ε 0 εδ log P = 0. Let X and Y be (not nec. indep.) centred Gaussian, s.t. φ X (ε) = log P ε γ, and one has Y (α) L 2 0, 1 with α > 1/γ and α > 1/2. Then φ X+Y (ε) = log P sup X t + Y t ε ε γ.

29 Outline 1 Small ball probabilities vs. metric entropy 2 Connection to other questions 3 Recent results for concrete examples

30 The entropy method: recent results several recent results using the entropy connection in the case of slowly varying φ X, i.e. exp. decreasing e n (u) Kühn 11: L 2 and L case EX t X s = e σ2 t s 2, t, s R d Result: φ X (ε) log ε d+1 (log log ε ) d. A./Gao/Kühn/Li/Shao 11+: L 2 and L case EX t X s = 22β+1 (ts) α, t, s > 0 (t + s) 2β+1 Result: φ X (ε) log ε 3.

31 The entropy method: recent results cont d The spectral measure F of stationary Gaussian process is given by: EX t X s = EX t s X 0 = k(t s) = e i(t s)u df(u).

32 The entropy method: recent results cont d The spectral measure F of stationary Gaussian process is given by: EX t X s = EX t s X 0 = k(t s) = e i(t s)u df(u). A./Ibragimov/Lifshits/van Zanten 08: spectral measure df(u) = e u ν du, F = e k ν δ 2πk Result for L norm: φ X (ε) k Z log ε 2, ν > 1, F; log log ε φ X (ε) log ε 1+1/ν, 0 < ν 1, F; or ν > 0, F.

33 The entropy method: recent results cont d The spectral measure F of stationary Gaussian process is given by: EX t X s = EX t s X 0 = k(t s) = e i(t s)u df(u). A./Ibragimov/Lifshits/van Zanten 08: spectral measure df(u) = e u ν du, F = e k ν δ 2πk Result for L norm: φ X (ε) k Z log ε 2, ν > 1, F; log log ε φ X (ε) log ε 1+1/ν, 0 < ν 1, F; or ν > 0, F. Karol /Nazarov 11+: rather general spectral measure, R d indexed, L 2 case df(u) = e G(u) du relate behaviour of G at with small ball probabilities

34 Thank you for your attention! Frank Aurzada Technische Universität Berlin page.math.tu-berlin.de/ aurzada/