Lower Tail Probabilities and Normal Comparison Inequalities. In Memory of Wenbo V. Li s Contributions

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1 Lower Tail Probabilities and Normal Comparison Inequalities In Memory of Wenbo V. Li s Contributions Qi-Man Shao The Chinese University of Hong Kong

2 Lower Tail Probabilities and Normal Comparison Inequalities In Memory of Wenbo V. Li s Contributions Qi-Man Shao The Chinese University of Hong Kong

3 Outline Lower tail probabilities Non-stationary processes Stationary processes Special cases Capture time of Brownian pursuits Random polynomials Normal comparison inequalities

4 1. Lower Tail Probabilities Let {X t, t T} be a real valued Gaussian process indexed by T with E X t = 0. Lower tail probability refers to P supx t X t0 x t T as x 0, t 0 T or Examples: P sup X t x t T a Capture time of Brownian pursuits b Random polynomials as T

5 A general result for non-stationary Gaussian processes Let X = {X t, t T} be a real valued Gaussian random process indexed by T with mean zero. Define the L 2 -metric ds, t = E X s X t 2 1/2, s, t T. For every ε > 0 and a subset A of T, let NA, ε denote the minimal number of open balls of radius ε for the metric d that are necessary to cover A. For t T and h > 0, let Bt, h = {s T : dt, s h} and define Q = sup h>0 sup t T 0 ln NBt, h, εh 1/2 de

6 For θ = Q, define A 1 = {t T : dt, t 0 θ 1 x}, A k = {t T : θ k 1 x < dt, t 0 θ k x}, where 0 k L, L = 1 + [ln θ D/x] and D = sup t T dt, t 0. Let N k x = NA k, θ k 2 x for k = 0, 1,, L Nx = 1 + N k x. 0 k L

7 Theorem Li and Shao 2004 i Assume that Q < and E X s X t0 X t X t0 0 for s, t T Then P sup X t X t0 x t T e Nx ii For x > 0, let s i T, i = 1,..., M be a sequence such that for every i M CorrX si X t0, X sj X t0 5/4 j=1 and ds i, t 0 = E X si X t0 2 1/2 x/2. Then P sup X t X t0 x t T e M/10.

8 Special Cases: Let {Xt, t [0, 1] d } be a centered Gaussian process with X0 = 0 and stationary increments, that is t, s [0, 1] d, E X t X s 2 = σ 2 t s. If there are 0 < α β < 1 such that σh/h α, σh/h β Then there exist 0 < c 1 c 2 < such that for 0 < x < 1/2 c 2 ln 1 x ln P sup Xt σx c 1 ln 1 t [0,1] d x. In particular, for the fractional Levy s Brownian motion L α t of order α, i.e. L α 0 = 0 and E L α t L α s 2 = t s α, ln P sup L α t x ln 1 t [0,1] d x.

9 Let {Xt, t [0, 1] d } be a centered Gaussian process with X0 = 0 and E X t X s = d i=1 If there are 0 < α β < 1 such that Then 1 2 σ2 t i + σ 2 s i σ 2 t i s i. σh/h α, σh/h β ln P sup Xt σ d x t [0,1] d ln d 1 x. In particular, for d-dimensional fractional Brownian sheet B α t i.e., σh = h α ln P sup B α t x ln d 1 t [0,1] d x

10 Lower tail probabilities for stationary Gaussian processes Let {Bt, t 0} be the Brownian motion and {Ut, t 0} be the Ornstein-Uhlenbeck process. It is known that {Ut, t 0} and {Be t /e t/2, t 0} have the same distribution. Moreover P as x 0 and as T. sup 0 t 1 P Bt x = P B1 x 2/π 1/2 x sup 0 t T Ut 0 = exp T/2 + ot Is there a connection between these two types of lower tail probabilities?

11 Let {X t, t 0} be a Gaussian process with X 0 = 0, E X t = 0. Assume that A1 E X s X t 0 and E X 2 t = t α for α > 0; A2 {Y t = Xe t /e α/2, t 0} is a stationary Gaussian process; A3 {X at, 0 t 1} and {a α/2 X t, 0 t 1} have the same distribution for each fixed a > 0. A4 ρt := E Y t Y 0 is decreasing and a 2 h,θ := inf ρθt ρt > 0 0<t h 1 ρt for every 0 < h < and 0 < θ < 1

12 By subadditivity and the Slepian lemma, 1 c α,y := lim T T ln P 1 sup Y t 0 = sup 0 t T T>0 T P sup 0 t T Y t 0 exists. Next result shows that the constant c is closely related to the rate of the lower tail probability P sup 0 t 1 X t x. Li and Shao 2004: Under conditions A1 A4, we have P sup X t x = x 2c α,y/α+o1 0 t 1 as x 0. Molchan 1999: For fractional Brownian motion {B α t, t 0} of order α 0 < α < 1 and hence c α = 1 α. P sup B α t x = x 1 α/α+o1 0 t 1

13 Explicit bounds of lower tail probabilities Li and Xiao 2013 Let Bt be the Brownian motion. For 0 < θ < 1 and P sup Bt θb1 x 0 t 1 1 3θ 2 1 θ 2 2π x3, P sup Bt θtb1 x 1 θ 2/π x 0 t 1 ln P sup Bt 0 t 1 1 where c is a specified constant. 0 Bsds x x 2 c

14 2. Special cases Capture time of Brownian pursuits Let B 0, B 1,, B n be independent standard Brownian motions. Define { } τ n = inf t > 0 : max B kt = B 0 t k n When is E τ n finite? Note that for any a > 0, by Brownian scaling, Pτ n > t = P max = P 1 k n max 1 k n sup B k s B 0 s < 1 0 s t sup B k s B 0 s < t 1/2. 0 s 1 Thus, the estimate is reduced to a lower tail probability problem.

15 DeBlassie 1987: P{τ n > t} ct γn as t. Bramson and Griffeath 1991: E τ 3 = Li and Shao 2001: E τ 5 <. Ratzkin and Treibergs 2009: E τ 4 <.

16 What is the asymptotic behavior of γ n? Kesten 1992: 0 < lim inf γ n/ ln n lim sup γ n / ln n 1/4 n Conjecture: lim n γ n / ln n exists. n Li and Shao 2002: lim γ n/ ln n = 1/4 n

17 Li and Shao 2002 also consider the capture time of the fractional Brownian motion pursuit. Let {B k,α t; t 0}k = 0, 1, 2,..., n be independent fractional Brownian motions of order α 0, 1. Put { } τ n := τ n,α = inf t > 0 : max B k,αt = B 0,α t k n Let and X k,α t = e tα B k,α e t, k = 0, 1,, n 1 γ n,α := lim T T ln P Li and Shao 2002: 1 γ n,α lim inf d α n ln n sup 0 t T lim sup n max X k,αt 0 1 k n γ n,α ln n <, where d α = 2 0 e2αx + e 2αx e x e x 2α dx. Conjecture: γ n,α lim n ln n = 1. d α

18 The probability that a random polynomial has no real root Dembo, Poonen, Shao and Zeitouni 2002 n P Z i x i < 0 x R 1 = n b+o1 i=0 where n is even, Z i are i.i.d. N0, 1, and 1 b = 4 lim T T ln P sup 0 t T X t 0 where X t is a centered stationary Gaussian process with E X s X t = 2e t s /2 1 + e t s

19 Dembo, Poonen, Shao and Zeitouni 2002: 0.4 < b < Simulation: b = 0.76 ± 0.03 Li and Shao 2002: 0.5 < b < 1

20 Let {Bt, t 0} be the Brownian motion and put B 0 t = Bt, Li and Shao and B m t = t 0 B m 1 sds P sup B m t x = x rm+o1 0 t 1 b 2r m 2m + 1, 2r m 2m + 1 b Open questions: What is the value of b? If {X t, t 0} is a differentiable stationary Gaussian process with positive correlation, what is the limit 1 lim T T ln P sup 0 t T X t 0?

21 3. Comparison Inequalities Let n 2, and let ξ j, 1 j n be standard normal random variables with correlation matrix R 1 = r 1 ij, let η j, 1 j n be standard normal random variables with correlation matrix R 0 = r 0 ij. Set ρ ij = max r 1 ij, r0 ij. Slepian s Lemma: If r 1 ij r0 ij, then n n P {ξ j u j } P {η j u j } j=1 j=1

22 Normal comparison inequality: Berman 1964,1971, Cramer and Leadbetter 1967: n n P {ξ j u j } P {η j u j } j=1 1 2π 1 i<j n Li and Shao 2002: j=1 r 1 ij r 0 ij + 1 ρ 2 ij 1/2 exp n n P {ξ j u j } P {η j u j } j= i<j n j=1 r 1 ij r 0 ij + exp u2 i + u 2 j 21 + ρ ij u2 i + u 2 j 21 + ρ ij

23 Li and Shao 2002: If r 1 ij r 0 ij 0 for all 1 i, j n Then n P {η j u j } j=1 n n P {ξ j u j } P {η j u j } j=1 { exp 1 i<j n ln j=1 π 2 arcsinr 0 ij π 2 arcsinrij 1 exp u2 i + u 2 j } 21 + rij 1 for any u i 0, i = 1, 2,, n satisfying r l ki rl ijr l kj u i + r l kj rl ijr l ki u j 0 for l = 0, 1 and for all 1 i, j, k n. Note: Condition ** is satisfied if u i = u 0.

24 Open questions: Does the result remain valid without assuming *? Yan 2009 gave a partial answer. Can a sharper bound be obtained?

25 Thank you!! Spaseeba!!!!