Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Law of the iterated logarithm for pure jump L´ evy processes Elena Shmileva, St.Petersburg Electrotechnical University July 12, 2010 Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion limsup LIL, liminf LIL Let X ( t ) , t ∈ (0 , ∞ ) be a L´ evy process. There are two types of the Law of the Iterated Logarithm (LIL): | X ( T ) | lim sup ϕ ( T ) = c a.s., where c ∈ [0 , ∞ ] , (1) T →∞ here ϕ ( t ) ր ∞ as t → ∞ . Denote by M the sup-process corresponding to the L´ evy process X , i.e., M ( t ) = � X ( t · ) � , t ∈ (0 , ∞ ), where � x ( · ) � = sup s ∈ [0 , 1] | x ( s ) | . M ( T ) lim inf h ( T ) = ˜ c a.s., where ˜ c ∈ [0 , ∞ ] , (2) T →∞ here h ( t ) ր ∞ as t → ∞ . Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion 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 L´ evy processes. Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion 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 L´ evy 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. Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Liminf LIL for general L´ evy 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 | , where F ( ε ) ր ∞ as ε → 0 ct corresponds to the Small Deviation order. If C is the Small Deviation constant, then � X ( T · ) � lim inf = 1 a . s . b C ( t ) T → 0 Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Examples of SD and liminf LIL. S α S L´ evy process Let X α be a symmetric α -stable (S α S) L´ evy process. It is a well known fact that − K α · ε − α (1 + o (1)) � � P {� X α ( · ) � < ε } = exp 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 � X ( T · ) � ( T / log | log T | ) 1 /α = K 1 /α lim inf a . s . α T → 0 and by the self-similarity property we have � X ( T · ) � ( T / log log T ) 1 /α = K 1 /α lim inf a . s . α T →∞ Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Examples of SD and liminf LIL. Variance Gamma process Let X be a Variance Gamma (VG) process, i.e., X ( t ) = σ W ( S ( t )) + µ S ( 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 � � = e − log | log t | log | log t | , therefore Kt Kt � X ( T · ) � lim inf ∈ (0 , ∞ ) a . s . exp {− log | log t | T → 0 } Kt We see that the correct normalizing function is still not known, because the unknown constant K participates as a power, not as a multiplier. Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Let X be a L´ evy process. Consider a family of scalings of the process � � X ( T t ) ϕ ( T ) , t ∈ [0 , 1] T > 0 , where ϕ (0) = 0 and ϕ ( T ) ր ∞ as T → ∞ . For each T > 0 the scaling X ( T t ) 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 ) � ϕ ( T ) , t ∈ [0 , 1] →→ S a . s . T > 0 where S ⊆ C is the cluster set. Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Functional LIL includes the following statement: � X ( T · ) � � � for any f ∈ S lim inf ϕ ( T ) − f ( · ) � = 0 a . s . � � T →∞ � If you put f ≡ 0, then you will see a liminf LIL statement for X . Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Functional LILs for the Wiener process Baldi, Rayonette 1992: Let W be a Wiener process, then � W ( T t ) � →→ c 2 S a . s ., √ 2 T log log T · c , t ∈ [0 , 1] T > 0 � 1 � � 0 f ′ ( t ) 2 dt ≤ 1 where S := f : f (0) = 0 , f ∈ AC [0 , 1] , . If γ ( T ) = o (1), then � � W ( T t ) √ 2 T log log T · γ ( T ) , t ∈ [0 , 1] →→ { 0 } a . s . T > 0 And... Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion Functional LILs for the Wiener process And... If γ ( T ) → ∞ and γ ( T ) = o (log log T ), then � � W ( T t ) √ 2 T log log T γ ( T ) , t ∈ [0 , 1] →→ C a . s . T > 0 If γ ( t ) → ∞ and there exists c 0 > 0 s.t. γ ( T ) ≥ c 0 log log T , then � � W ( T · ) � ≥ c 0 π � � for any f ∈ C lim inf √ 2 T log log T γ ( T ) − f ( · ) a . s . � � 4 T →∞ � If you put γ ( T ) = log log T and f ≡ 0, you will see liminf LIL for W . Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S α S processes Conclusion My results: the case of empty cluster set Let X α be a S α S 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 � � X α ( T · ) � � � ≥ c − 1 K 1 /α lim inf T 1 /α h ( T ) − f ( · ) a . s ., � � α T →∞ � where K α is the Small Deviations constant. Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes
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