Lower Tail Probabilities and Related Problems

Transcription

1 Lower Tail Probabilities and Related Problems Qi-Man Shao National University of Singapore and University of Oregon

2 . Lower Tail Probabilities Let {X t, t T } be a real valued Gaussian process indexed by T with E X t = 0. P X t X t0 x t T as x 0 where t 0 T. Examples: a Csaki, Khoshnevisan and Shi 2000: Let W s, t be the two dimensional Brownian sheet. Then for x > 0 small ln P W s, t x ln 2 /x ln P 0 s,t 0 s,t W s, t x ln2 /x ln ln/x. b Capture time of Brownian pursuits Bramson and Griffeath 99: Let W 0, W,, W n be independent standard Brownian motions. Define { } τ n = inf t > 0 : max W kt = W 0 t +. When is E τ n finite? Note that for any a > 0, by Brownian scaling, Pτ n > t = P max = P max W k s W 0 s < 0 s t W k s W 0 s < t. /2 0 s Thus the problem is really a lower tail probability problem. DeBlassie 987: P{τ n > t} ct γn as t.

3 Bramson and Griffeath 99: E τ 3 = Conjecture: E τ 4 <. Li and Shao 200: E τ 5 <. c 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 = n b+o i=0 where n is even, Z i are i.i.d. N0,, and b = 4 lim T T ln P where X t is a centered stationary Gaussian process with E X s X t = 2e t s /2 + e t s X t 0 A General Result 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 /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 and define For θ = Q, define Q = h>0 Bt, h = {s T : dt, s h} t T 0 ln NBt, h, εh /2 dε A = {t T : dt, t 0 θ x}, A k = {t T : θ k x < dt, t 0 θ k x}, 2

4 where 0 k L, L = + [ln θ D/x] and D = t T dt, t 0. Let N k x = NA k, θ k 2 x for k = 0,,, L Nx = + N k x. 0 k L Li and Shao 2003: Assume that Q < and E X s X t0 X t X t0 0 for s, t T Then P X t X t0 x t T e Nx For x > 0, let s i T, i =,..., M be a sequence such that for every i M CorrX si X t0, X sj X t0 5/4 j= and Then ds i, t 0 = E X si X t0 2 /2 x/2. P X t X t0 x t T e M/0. Some Special Cases Let {Xt, t [0, ] d } be a centered Gaussian process with X0 = 0 and stationary increments, that is t, s [0, ] d, E X t X s 2 = σ 2 t s. If there are 0 < α β < such that σh/h α, σh/h β 3

5 Then there exist 0 < c c 2 < depending only on α, β and d such that for 0 < x < /2 c 2 ln x ln P Xt σx c ln t [0,] x. d 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 L α t x t [0,] d ln x. Let {Xt, t [0, ] d } be a centered Gaussian process with X0 = 0 and E X t X s = d i= 2 σ2 t i + σ 2 s i σ 2 t i s i. If there are 0 < α β < such that σh/h α, σh/h β Then ln P Xt σ d x t [0,] d ln d x. In particular, for d-dimensional Brownian sheet W t ln P W t x t [0,] d ln d x and more generally ln P B α t x t [0,] d ln d x Open question: Can the assumption be replaced by c σh σ2h c 2 σh for some c 2 c >? 4

6 2. Lower Tail Probabilities for Stationary Gaussian Processes Let {W t, t 0} be the Brownian motion and {Ut, t 0} be the Ornstein-Uhlenbeck process. It is known that {Ut, t 0} and {W e t /e t/2, t 0} have the same distribution. Moreover P W t x = P W x 2/π /2 x as x 0 and as T. 0 t P Ut 0 = exp T/2 + ot Is there a connection between these two types of lower tail probabilities? Li and Shao 2003: Let {Y t, t 0} be an almost surely continuous stationary Gaussian process with E Y t = 0 and E Yt 2 = for t 0. Put ρt = E Y 0 Y t. Assume that ρt 0. We have i The limit exists, left continuous, and for every x R. ii If ρt is decreasing and px := lim T T ln P px = T ln P T >0 a 2 ρθt ρt h,θ := inf 0<t h ρt Y t x Y t x > 0 for every 0 < h < and 0 < θ <, then px is continuous. Let B α be a fractional Brownian motion of order α 0 < α < 2 and put X α t := B αe t e tα/2. 5

7 Li and Shao 2003: We have c α := lim T T ln P X α t 0 exists. Moreover, 0 < c α < and P B α t x = x 2cα/α+o as x 0 0 t Molchan 999: c α = α/2 Similarly, we have an alternative representation for the constant b in Example c. Let Y 0 = 0 and Y t = 2t 2 W ue ut du for t > 0, where W is the Brownian motion. Then E Y t = 0 and 0 E Y ty s = 2st s + t for s, t > 0. Hence {X t } in Example c and {Y e t /e t/2 } have the same distribution. Li and Shao 2002: We have P as x 0. Furthermore, 0.5 < b <. 0 t Y t x = x b/2+o Open questions:. If {X t, t 0} is a differentiable stationary Gaussian process with positive correlation, what is the limit lim T T ln P X t 0? 2. What is b? 3. Capture Time of the Fractional Brownian Motion Pursuit 6

8 Let {B k,α t; t 0}k = 0,, 2,..., n be independent fractional Brownian motions of order α 0, 2. Put { } τ n := τ n,α = inf t > 0 : max B k,αt = B 0,α t +. and When is E τ n finite? Note that Pτ n > s = P Let = P max max B k,α t B 0,α t < 0 t s B k,α t B 0,α t < s. α/2 0 t X k,α t = e tα/2 B k,α e t, k = 0,,, n γ n,α := lim T T ln P max X k,αt X 0,α t 0 Li and Shao 2003: P as x 0 Kesten 992: max 0 t B k,α t B 0,α t < x = x 2γn,α/α+o 0 < lim inf n Conjecture: lim n γ n / ln n exists. Li and Shao 2002: γ n,/ ln n lim γ n, / ln n /4 n γ n,α lim inf d α n ln n lim n γ n,α ln n <, where d α = 2 0 exα + e xα e x e x α dx. In particular, lim n γ n ln n = 4 Conjecture: γ n,α lim n ln n =. d α 7

9 4. Some Comparison Inequalities Li and Shao 2002: Let n 3, and let ξ j, j n and η j, j n be standard normal random variables with covariance matrices R = r ij and R 0 = r 0 ij, respectively. Assume Then n P {η j u j } j= j= r ij r 0 ij 0 for all n P {ξ j u j } P { exp i<j n for any u i 0, i =, 2,, n satisfying ln n j= i, j n {η j u j } π 2 arcsinr 0 ij π 2 arcsinrij exp u2 i + u 2 } j 2 + rij r l ki r l ijr l kju i + r l kj r l ijr l kiu j 0 for l = 0, and for all i, j, k n. Note: Condition ** is satisfied if u i = u 0. Open question: Does the result remain valid without assuming **? Shao 2003: Let X,..., X n be jointly Gaussian random variables with mean zero. Then P max X i x 2 mink,n k/2 P max X i x P max X i x, i n i k k<i n Let B α be the fractional Brownian motion of order α. Then there exists c α > 0 such that P 0 s a B α t x, a t b c α P B α t x P 0 s a for any 0 < a < b, x > 0 and y > 0. 8 B α t B α a y a t b B α t B α a y

10 Assume X = X,..., X n N0, Σ, and Y = Y,..., Y n N0, Σ 2. If Σ 2 Σ is positive semidefinite, then C R n, P Y C Σ / Σ 2 /2 PX C. 9

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