General Law of iterated logarithm for Markov processes: Limsup law

Abstract

In this paper, we discuss general criteria and forms of both liminf and limsup laws of iterated logarithm (LIL) for continuous-time Markov processes. We consider minimal assumptions for both LILs to hold at zero (at infinity, respectively) in general metric measure spaces. We establish LILs under local assumptions near zero (near infinity, respectively) on uniform bounds of the first exit time from balls in terms of a function φ and uniform bounds on the tails of the jumping measure in terms of a function ψ. One of the main results is that a simple ratio test in terms of the functions φ and ψ completely determines whether there exists a positive nondecreasing function Ψ such that lim sup |Xt|/Ψ(t) is positive and finite a.s., or not. We also provide a general formulation of liminf LIL, which covers jump processes whose jumping measures have logarithmic tails. Our results cover a large class of subordinated diffusions, jump processes with mixed polynomial local growths, jump processes with singular jumping kernels and random conductance models with long range jumps.

How smooth can the convex hull of a Lévy path be?

“…The upper fluctuations of a Lévy process at zero have been the topic of numerous studies, see [6,32] for the one-sided problem and [22,33,37] for the two-sided problem. Similar questions have been considered for more general time-homogeneous Markov processes [12,23]. The time-homogeneity again plays an important role in these results.…”

“…Proof. By virtue of Theorem 2.9(i), it suffices to verify (11) and (12). By assumption, we have 2 ), which tend to 0 as t ↓ 0, implying (12).…”

“…Similarly, Theorem 2.13 gives a fine class of functions f satisfying the assumption in Lemma 2.19(ii) for a large class of processes. Such conclusions on the fluctuations of the Lévy path of X would not be new as the fluctuations of X at 0 are already known, see [12,32,33]. In particular, the upper functions of X and −X at time 0 were completely characterised in [32] in terms of the generating triplet of X.…”

General Law of iterated logarithm for Markov processes: Liminf laws

“…In particular, our assumptions for liminf law of LIL at zero and the form of liminf LIL are truly local so that we can cover highly space-inhomogenous cases. Our results cover all examples in [12] including random conductance models with long range jumps. Moreover, we show that the general form of liminf law of LIL at zero holds for a large class of jump processes whose jumping measures have logarithmic tails and Feller processes with symbols of varying order which are not covered before.…”

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Continuing from [12], in this paper, we discuss general criteria and forms of liminf laws of iterated logarithm (LIL) for continuous-time Markov processes. Under some minimal assumptions, which are weaker than those in [12], we establish liminf LIL at zero (at infinity, respectively) in general metric measure spaces.

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