Law of the iterated logarithm for pure jump L evy processes Elena - - PowerPoint PPT Presentation
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for S S processes Conclusion Law of the iterated logarithm for pure jump L evy processes Elena Shmileva,
Outline Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for SαS processes Conclusion
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 SαS processes Conclusion
Two types of the Law of the Iterated Logarithm Generalization of the LILs. Functional LIL My results on the Functional LIL for SαS 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 SαS processes Conclusion
Let X(t), t ∈ (0, ∞) be a L´ evy 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) 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(·) = sups∈[0,1] |x(s)|. lim inf
T→∞
M(T) h(T) = ˜ c a.s., where ˜ c ∈ [0, ∞], (2) 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 SαS processes Conclusion
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 SαS processes Conclusion
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 SαS processes Conclusion
the first order of Small Deviation (SD) asymptotics and liminf LIL: Short time Liminf LIL:
Fact
Consider bc(t) = F −1
log | log t| ct
corresponds to the Small Deviation order. If C is the Small Deviation constant, then lim inf
T→0
X(T·) bC(t) = 1 a.s.
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 SαS processes Conclusion
Let Xα be a symmetric α-stable (SαS) L´ evy process. It is a well known fact that P {Xα(·) < ε} = exp
ε → 0, here 0 < Kα < ∞, 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·) (T/ log | log T|)1/α = K 1/α
α
a.s. and by the self-similarity property we have lim inf
T→∞
X(T·) (T/ log log T)1/α = K 1/α
α
a.s.
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 SαS processes Conclusion
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| Kt
Kt
, therefore lim inf
T→0
X(T·) exp{−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.
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 SαS processes Conclusion
Let X be a L´ evy process. Consider a family of scalings of the process
ϕ(T) , t ∈ [0, 1]
T → ∞. For each T > 0 the scaling X(T t)
ϕ(T) , t ∈ [0, 1] is a random element
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. 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 SαS processes Conclusion
Functional LIL includes the following statement: for any f ∈ S lim inf
T→∞
ϕ(T) − f (·)
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 SαS processes Conclusion
Baldi, Rayonette 1992: Let W be a Wiener process, then
√2T log log T · c, t ∈ [0, 1]
→→ c2S a.s., where S :=
1
0 f ′(t)2 dt ≤ 1
If γ(T) = o(1), then
√2T log log T · γ(T), t ∈ [0, 1]
→→ {0} a.s. 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 SαS processes Conclusion
And... If γ(T) → ∞ and γ(T) = o(log log T), then
√2T log log T γ(T), t ∈ [0, 1]
→→ C a.s. If γ(t) → ∞ and there exists c0 > 0 s.t. γ(T) ≥ c0 log log T, then for any f ∈ C lim inf
T→∞
√2T log log T γ(T) − f (·)
4 a.s. 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 SαS processes Conclusion
Let Xα 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→∞
T 1/αh(T) − f (·)
α
a.s., where Kα is the Small Deviations constant.
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 SαS processes Conclusion
Theorem
For any f ∈ C, if δ ∈ (0, 1], we have lim inf
T→∞ (log log T)δ
T 1/α(log log T)δ−1/α − f (·)
α
a.s. This yields that for δ ∈ (0, 1]
T 1/α(log log T)δ−1/α , t ∈ [0, 1]
→→ C a.s.
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 SαS processes Conclusion
The proof is based on shifted small deviation estimates: for all f ∈ C and λ > 0, r > 0 such that λrα−1 → 0, r → 0 we have P {Xα(·) − λ f (·) < r} = exp
P {Xα(·) − λ f (·) < r} = exp
r log(λrα−1)(1 + o(1))
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 SαS processes Conclusion
For bigger functions the result is not obtained yet. The only thing that is easy to show is: if h(·) is such that ∞
1 dt t h(t)α < ∞, the a.s. cluster set is equal to
{0}, because lim
T→∞
|Xα(T)| T 1/αh(T) = 0 a.s. This is due to the following statement (cf. J. Bertoin, 1996) : lim sup
T→∞
|Xα(T)| T 1/αh(T) = 0
according as ∞
1
dt t h(t)α < ∞
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 SαS processes Conclusion
◮ The limsup LIL and liminf LIL for general L´
evy processes are recently studied. The first one uses the LD estimates, the second uses the SD estimates.
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 SαS processes Conclusion
◮ The limsup LIL and liminf LIL for general L´
evy 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.
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 SαS processes Conclusion
◮ The limsup LIL and liminf LIL for general L´
evy 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.
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 SαS processes Conclusion
◮ The limsup LIL and liminf LIL for general L´
evy 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.
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 SαS processes Conclusion
Elena Shmileva, St.Petersburg Electrotechnical University Law of the iterated logarithm for pure jump L´ evy processes