Laplace transforms of L 2 ball, Comparison Theorems and Integrated Brownian motions. Jan Hannig

Transcription

1 Laplace transforms of L 2 ball, Comparison Theorems and Integrated Brownian motions. Jan Hannig Department of Statistics, Colorado State University hannig@stat.colostate.edu Joint work with F. Gao, University of Idaho T.-Y. Lee a, University of Maryland F. Torcaso, Johns Hopkins University a We regret to inform you that Prof. Lee has died on Tuesday this week. He will be missed dearly.

2 m-times integrated Brownian motion Let B t be a standard Brownian motion. Define X m (t) = Z t Z sm Z s2 B(s ) ds ds 2... ds m. X t has the same distribution as t Integrated BM are found: (t x) m m! db(x).... Random spline (used in modeling climate) Example of a (infinite dimensional) prior for which the frequentists and Bayesian analysis gives totally different answers. (Freedman (998)) Randomly driven wave equations Study of random polynomials (work of Wenbo Li) 2

3 Background Let X(t) be centered (mean zero) Gaussian process on [, ]. The covariance kernel K(s, t) = EX(s)X(t). The covariance operator Af(s) = Eigenvalues and eigenfunctions: K(s, t)f(t) dt. Solutions to λf = Af. For this particular operator (positive, compact) the eigenvalues satisfy λ λ 2 >, lim n λ n = The eigenfunctions f n are an orthonormal basis of L 2 [, ]. Karhunen-Loève expansion: X(t) = λ n f n (t)ξ n where ξ n are i.i.d. N(, ). 3

4 Background cont. Karhunen-Loève expansion implies: Z X 2 2 = X 2 (t) dt = X λ n ξ 2 n Laplace transform of the squared L 2 ball L(t) E exp{ t X 2 2} = Y ( + 2tλn ) /2. Sytaja Tauberian theorem (974) a P X 2 2 ε 2 (2πt 2 h (t)) /2 exp{tε + h(t)} where h(t) = log L(t) and lim ε tε+th (t) t 2 h (t) =. a This result has an interesting history. It appears to have been independently reproved in the literature over and over again by, e.g., Hoffmann- Jorgensen, Shepp & Dudley (979), Ibragimov (982), Dembo, Mayer-Wolf & Zeitouni(995). This particular form is due to Lifshits (997) who proved a similar result for very general ξ n. 4

5 Previous results Khoshnevisan & Shi (998) got Laplace transform of the squared L 2 small ball in the case m =. Chen & Li (2) obtained only an approximation to the Laplace transform which is sufficient to give the following the logarithmic asymptotics for general m: log P Z where D m = 2m+ 2 X 2 m (t) dt ε2 (2m + 2) sin D m ε 2 2m+ as ε +, 2m+2 π 2m+. 2m+2 Other exciting contributions were concurently done by Prof. Nazarov, Prof. Nikitin and coauthors. 5

6 Our contributions Obtained a closed form expression for the Laplace transform of the squared L 2 ball using a method that works for a wide range of Gaussian processes. Generalize the comparison theorem of Li (992). Define and study a new class of integrated Brownian motions. Extend the small ball asymptotics beyond the logarithmic term P Z Xm(t) 2 dt ε 2 C m ε 2m+ exp{ D m ε 2 where D m is on the previous slide,! /2 " Cm = (2m + 2)(m+)/2 det U 2m + 2 (2m + )π (2m + 2) sin π 2m + 2 2m+ } # (m+)/(2m+) and U = Vandermonde(, exp(i m+ π),..., exp(i m m+ π)). 6

7 Our approach Consider all the random processes to be defined on [, ]. In order to use this we need to find R a defining equation for eigenvalues of the covariance operator Af(s) = K(s, t)f(t)dt, where R K(s, t) = EX m (s)x m (t) = s t (m!) 2 (s u) m (t u) m du. By successive differentiation we get a higher order Sturm-Liouville problem: with the boundary conditions λf (2m+2) (t) = ( ) m+ f(t), < t < f() = f () = = f (m) () = f (m+) () = = f (2m+) () = 7

8 Our approach (cont.) The solution is in the form: f(t) = P 2m+ j= c j e α j t with α j = λ /(2m+2) iω j and ω j = exp( jπ m+ i). Thus we need to solve det M(λ) = where M(λ) = ω ω ω 2m+ ω m ω m ω2m+ m ω m+ e α ω m+ e α ω m+ 2m+ eα 2m+ ω 2m+ e α ω 2m+ e α ω 2m+ 2m+ eα 2m+ is the characteristic determinant of the differential operator. C A 8

9 Our approach first option. Scale M(λ) to N(λ). det N = C cos(λ /(2m+2) )+O From here λ k ( ) 2m+2. (k +k 2 )π ( ( exp λ /(2m+2) sin ( π m+ The desired exact small ball result will follow using a slightly improved L 2 small ball comparison theorem Li (992) coupled with Rouché s and Jensen s theorems GHLT (22). Nazarov (23) also obtained exact small ball constant using complex analysis tools. ))) 9

10 Our approach second option. Simple observation: Hadamard s factorization theorem implies Corollary. Let X be a Gaussian process whose covariance operator has nonzero eigenvalues λ n, repeated according to their multiplicity. Suppose there is an entire function f(z) of order ρ <, such that, z n = /λ n, n, are the only zeros, counting multiplicities, of f(z). Then E exp{ t X 2 2 } = f( 2t) /2. f() Set ρ = /λ and take f(ρ) = det M. Theory of Linear Differential Operators guarantees that f is entire and multiplicities are correct. L(t) = E ( exp{ t X m 2 2} ) = ( ) /2 f( 2t). f() The small ball probability follows from Sytaja s theorem.

11 Example Consider m-times integrated Brownian motion (X m ) and Brownian bridge (Y m ): E exp{ t X 2 2 } = E exp{ t Y 2 2 } E exp{ t X } = cos(2 3/4 t /4 ) + cosh(2 3/4 t /4 ) 2 7/4 t /4 A /2 sin(2 3/4 t /4 ) + sinh(2 3/4 t /4, )! /2, cos(3 /2 2 5/6 t /6 ) cosh(2 5/6 t /6 ) + 8 cosh(2 /6 t /6 ) E exp{ t Y } + 2 cos(2 /6 3 /2 t /6 ) cosh(2 /6 t /6 ) + cosh(2 7/6 t /6 ) /2, =3 2(2t) /2 4 3 cosh(2 5/6 t /6 ) sin(3 /2 2 5/6 t /6 ) + 3 cosh(2 /6 t /6 ) sin(2 /6 3 /2 t /6 ) + 4 cos(3 /2 2 5/6 t /6 ) sinh(2 5/6 t /6 ) + 4 sinh(2 /6 t /6 ) + cos(2 /6 3 /2 t /6 ) sinh(2 /6 t /6 ) + 2 cosh(2 /6 t /6 ) sinh(2 /6 t ) /6 /2.

12 General Comparison Theorems Recall Li s comparison theorem (Li, 992): Let ξ n be i.i.d. standard Gaussian, {a n } and P {b n } be two positive non-increasing summable sequences such that n= a n/b n <. Then as X P n= anξ 2 n ε2 Y bn/ana /2 P n= n= bnξ 2 n ε2 A. We replace the condition P n= a n/b n < by Q n= (a n/b n ) converges. This solves a conjecture of Li (992) and implicitly the conjecture of Linde (994). When evaluating the infinite product the following consequence of Jensen s theorem could be useful: Let f(z) and g(z) be entire functions with only positive real simple zeros. Denote the zeros of f by α < α 2 < α 3 < and the zeros of g by β < β 2 < β 3 <. If lim k z =r max k f(z) g(z) = where the sequence of radii r k tending to are chosen so that β k < r k < β k+ for large k, then Y n= αn βn = f() g(). Conjecture of Li: If using ξ p the comparison theorem remains valid with /p in the exponent. 2

13 General Comparison Theorems (cont.) Take a sequence of i.i.d. random variables ζ n with a distribution function F (x). Define the index of variation α := lim x log F (/x). log x Using Dunker, Lifshits, Linde (998) we can prove: Theorem (GHT (23)). Let {ζ n } be a sequence of i.i.d. positive random variables satisfying some regularity conditions related to the behavior of F (/x) at infinity. Let {a n } and {b Q n } be positive, non-increasing, summable sequences such that n= (a n/b n ) converges. Then as ε + Pr X n= a n ζ n ε! Y n=! α b n Pr a n X n= b n ζ n ε!. 3

14 General Integrated Processes What process has λ k = ( ) 2m+2? (k 2 )π We know that X(t) = [(k 2 )π] m 2 sin((k 2 )πx)ξ k. The small ball probability is simpler where P Z X 2 (t) ε 2 C m ε C m = 2 (m+)/2 2m + 2 (2m + )π 2m+ exp n D m ε 2m+ 2 /2 (2m + 2) sin π 2m + 2 o, (m+)/(2m+). What happens if we permute the boundary condition in the equation? 4

15 Permuted conditions Consider Sturm-Liouville equation: with the boundary conditions λf (2m+2) (t) = ( ) m+ f(t), < t < f(t ) = f (t ) = f (t 2 ) = = f (2m+) (t 2m+ ) =, where t j {, } for each j, and P 2m+ j= t j = m +. We say that the boundary conditions are antisymmetric if t j = t 2m+ j. R By integrating we get λf(t) = Af(t) = K(s, t)f(s) ds. (K(s, t) is a positive definite for antisymmetric conditions only.) 5

16 Permuted conditions (cont.) Antisymmetric conditions are determined by the first m +. Denote them {i m, i m,..., i } {, } m+. R R t Define T f(t) = f(s) ds and T f(t) = t f(s) ds. Notice: Covariance operator corresponding to our equation with {i m, i m,..., i } is Af(t) = T im T i T i T im f(t). If X(t) is a Gaussian process with covariance operator A then the covariance operator of T i X(t) is T i AT i. Covariance operator of Brownian motion B (t) is T T. Covariance operator of B (t) = B ( t) is T T. Conclusion: The process corresponding to the permuted conditions is X {i m,i m,...,i } (t) = T im T i B i. Nazarov and Nikitin (23) generalized this idea to other integrated random processes. 6

17 Euler-integrated Brownian motion We get the exact λ k = (k )π 2m 2 2 by setting {i m, i m,..., i } = {,,,,... }. The covariance kernel of X {,,,,... } is K m (s, t) = Z Z X = ( ) m m+ j= = ( ) m+ 2 2m (s s )(s s 2 ) (s m t) ds ds 2 ds m 2 2(m+ j) (2j )![2(m + j)]! (s t)2j E 2(m+ j) ( s t 2 ) (2m + )! E 2m+ t s 2 where E n (x) is the n-th Euler polynomial ( 2ext e t + = P t + s E 2m+, 2 n= E n (x) tn n! ). 7

18 Open questions Euler-integrated and usual integrated B.M. are extremes among the generally integrated Brownian motions in the sense of stochastic domination. Small ball ordering answered in GHLT (22) using Hadamard (893). Other properties of generally integrated processes. Is there any significant difference? Other norms such as L p and L. 8