Law of the iterated logarithm for pure jump Lévy processes

Transcription

1 Law of the iterated logarithm for pure jump Lévy processes Elena Shmileva, St.Petersburg Electrotechnical University July 12, 2010

2

3 limsup LIL, liminf LIL Let X (t), t (0, ) be a Lévy process. There are two types of the Law of the Iterated Logarithm (LIL): lim sup T X (T ) ϕ(t ) = c a.s., where c [0, ], (1) here ϕ(t) as t. Denote by M the sup-process corresponding to the Lévy process X, i.e., M(t) = X (t ), t (0, ), where x( ) = sup s [0,1] x(s). lim inf T M(T ) h(t ) = c a.s., where c [0, ], (2) here h(t) as t.

4 What kinds of techniques are used to obtain the LILs? Limsup LIL proof uses large deviation estimates: P{ X ( ) > r} = ψ(r)(1 + o(1)) as r, ψ(r) 0. There is a series of recent articles by Bertoin, Savov, Maller and Doney on limsup LIL for general Lévy processes.

5 What kinds of techniques are used to obtain the LILs? Liminf LIL proof uses small deviation estimates: P{ X ( ) < ε} = exp { C F (ε)(1 + o(1))} as ε 0, here F (ε) = O(1) as ε 0, C (0, ). A comprehensive method for the first order asymptotics (without constants) in the Small Deviation estimates for any Lévy process is found in 2008 by F. Aurzada and St. Dereich. It is based on searching an EMM by the Esscher transform and on martingale inequalities.

6 Liminf LIL for general Lévy process. Direct connection to Small Deviation estimates F. Aurzada and M.Savov (2010) established a connection between the first order of Small Deviation (SD) asymptotics and liminf LIL: Short time Liminf LIL: Fact ( Consider b c (t) = F 1 log log t ct ), where F (ε) as ε 0 corresponds to the Small Deviation order. If C is the Small Deviation constant, then lim inf T 0 X (T ) b C (t) = 1 a.s.

7 Examples of SD and liminf LIL. SαS Lévy process Let X α be a symmetric α-stable (SαS) Lévy process. It is a well known fact that P { X α ( ) < ε} = exp { K α ε α (1 + o(1)) } as ε 0, here 0 < K α <, for which there is still no implicit expression. We see that F (ε) = ε α, F 1 (x) = x 1/α. Then lim inf T 0 X (T ) 1/α = K (T / log log T ) 1/α α and by the self-similarity property we have lim inf T X (T ) 1/α = K (T / log log T ) 1/α α a.s. a.s.

8 Examples of SD and liminf LIL. Variance Gamma process Let X be a Variance Gamma (VG) process, i.e., X (t) = σw (S(t)) + µs(t), where σ 0, µ R, W is a Wiener process and S is a gamma subordinator independent of W. If µ = 0, then there exists K (0, ) such that P { X ( ) < ε} = exp { K log ε (1 + o(1))} as ε 0. ( ) We have F 1 (x) = e x and F 1 log log t log log t Kt = e Kt, therefore lim inf T 0 exp{ X (T ) log log t Kt } (0, ) a.s. We see that the correct normalizing function is still not known, because the unknown constant K participates as a power, not as a multiplier.

9 Let X be a Lévy process. Consider { a family of scalings of the process X (T t) ϕ(t ) }T, t [0, 1], where ϕ(0) = 0 and ϕ(t ) as >0 T. X (T t) For each T > 0 the scaling ϕ(t ), t [0, 1] is a random element of the Skorokhod space D[0, 1]. Let us introduce a set C := {f C[0, 1] : f (0) = 0}. The Functional LIL states that the family of scalings of X properly renormalized has an a.s. cluster set (convergence is uniform) in C (endowed with the uniform topologie). We denote this as follows: { } X (T t), t [0, 1] S a.s. ϕ(t ) where S C is the cluster set. T >0

10 Functional LIL includes the following statement: for any f S lim inf X (T ) T ϕ(t ) f ( ) = 0 a.s. If you put f 0, then you will see a liminf LIL statement for X.

11 Functional LILs for the Wiener process Baldi, Rayonette 1992: Let W be a Wiener process, then { } W (T t) c, t [0, 1] c 2 S a.s., 2T log log T where S := T >0 { f : f (0) = 0, f AC[0, 1], } 1 0 f (t) 2 dt 1. If γ(t ) = o(1), then { } W (T t) γ(t ), t [0, 1] {0} a.s. 2T log log T T >0 And...

12 Functional LILs for the Wiener process And... If γ(t ) and γ(t ) = o(log log T ), then { } W (T t) γ(t ), t [0, 1] 2T log log T T >0 C a.s. If γ(t) and there exists c 0 > 0 s.t. γ(t ) c 0 log log T, then for any f C lim inf W (T ) T γ(t ) f ( ) 2T log log T c 0π 4 a.s. If you put γ(t ) = log log T and f 0, you will see liminf LIL for W.

13 My results: the case of empty cluster set Let X α be a SαS process, α (1, 2). Theorem Let h : R + R + s.t. h(0) = 0 and there exists c > 0 s.t. h(t ) c(log log T ) 1/α, then for any f C the following holds lim inf T X α (T ) T 1/α h(t ) f ( ) c 1 Kα 1/α where K α is the Small Deviations constant. a.s.,

14 My results: the case of C cluster set Theorem For any f C, if δ (0, 1], we have lim inf (log log T )δ X α (T ) T T 1/α f ( ) (log log T ) δ 1/α = K α 1/α a.s. This yields that for δ (0, 1] { } X α (T t) T 1/α, t [0, 1] (log log T ) δ 1/α T >0 C a.s.

15 My results: shifted Small Deviations for SαS process. The proof is based on shifted small deviation estimates: for all f C and λ > 0, r > 0 such that λr α 1 0, r 0 we have P { X α ( ) λ f ( ) < r} = exp { K α r α (1 + o(1)) } and if λr α 1, r 0 there exists C = C(f, α) { P { X α ( ) λ f ( ) < r} = exp C λ } r log(λr α 1 )(1 + o(1)).

16 For bigger functions the result is not obtained yet. The only thing that is easy to show is: if h( ) is such that dt 1 t h(t) <, the a.s. cluster set is equal to α {0}, because X α (T ) lim T T 1/α h(t ) = 0 a.s. This is due to the following statement (cf. J. Bertoin, 1996) : lim sup T X α (T ) T 1/α h(t ) = 0 or = a.s. according as 1 dt < or =. t h(t) α

17 The limsup LIL and liminf LIL for general Lévy processes are recently studied. The first one uses the LD estimates, the second uses the SD estimates.

18 The limsup LIL and liminf LIL for general Lévy processes are recently studied. The first one uses the LD estimates, the second uses the SD estimates. The functional LIL generalizes the both LILs. The results are known just for SαS processes and the Wiener process.

19 The limsup LIL and liminf LIL for general Lévy processes are recently studied. The first one uses the LD estimates, the second uses the SD estimates. The functional LIL generalizes the both LILs. The results are known just for SαS processes and the Wiener process. The a.s. cluster set for the SαS process could be empty, equal to the set of all continuous function, or just to the zero-function, depending on the normalizing function.

20 The limsup LIL and liminf LIL for general Lévy processes are recently studied. The first one uses the LD estimates, the second uses the SD estimates. The functional LIL generalizes the both LILs. The results are known just for SαS processes and the Wiener process. The a.s. cluster set for the SαS process could be empty, equal to the set of all continuous function, or just to the zero-function, depending on the normalizing function. Functional LIL results use the shifted SD estimates.

21 Thank you for your attention!