Laws of the iterated logarithm for a class of iterated processes

Abstract

Let $X=\{X(t), t\geq 0\}$ be a Brownian motion or a spectrally negative stable process of index $1<\a<2$. Let $E=\{E(t),t\geq 0\}$ be the hitting time of a stable subordinator of index $0<\beta<1$ independent of $X$. We use a connection between $X(E(t))$ and the stable subordinator of index $\beta/\a$ to derive information on the path behavior of $X(E_t)$. This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin \cite{bertoin}. Using this connection, we obtain various laws of the iterated logarithm for $X(E(t))$. In particular, we establish law of the iterated logarithm for local time Brownian motion, $X(L(t))$, where $X$ is a Brownian motion (the case $\a=2$) and $L(t)$ is the local time at zero of a stable process $Y$ of index $1<\gamma\leq 2$ independent of $X$. In this case $E(\rho t)=L(t)$ with $\beta=1-1/\gamma$ for some constant $\rho>0$. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper \cite{MNX}. We also obtain exact small ball probability for $X(E_t)$ using ideas from \cite{aurzada}.Comment: 13 page

Small ball probabilities for a class of time-changed self-similar processes

“…Here, E β (z) = ∞ n=0 z n /Γ(nβ +1) is the Mittag-Leffler function with parameter β. Nane [21] also extended the result to a time-changed process X • E β , where the outer process X is a self-similar process possessing a certain small ball probability, which particularly includes the case of a fractional Brownian motion. However, the exact small deviation constant cannot be specified unlike the situations considered in [2]; see Remark 12 for details of this point.…”

“…Let W be a one-dimensional standard Brownian motion and let E β be the inverse of a stable subordinator D β of index β ∈ (0, 1), independent of W . Nane [21] established that the small ball probability of the time-changed Brownian motion W • E β is given by…”

“…The proof of (1) provided in [21] essentially relies on the following expression for the Laplace transform of the random variable E β (1) and its asymptotic behavior along the negative real axis (see e.g. Proposition 1(a) of [5] and Theorem 1.4 of [22]):…”

Small deviations of iterated processes in the space of trajectories

“…In the same way, one may also apply results on small ball probabilities for the Wiener process in weighted sup-norms and L -norms, the Bessel processes in the sup-norm and weighted sup-norms (see [4]). Further examples may be constructed for self-similar processes from [2,15,16] and references therein.…”

“…Results on asymptotic behaviour of small deviation and small ball probabilities have been obtained by Nane [15], Frolov [5,7], Martikainen, Frolov and Steinebach [13], Aurzada and Lifshits [1], Nane [16], Baumgarten [2] for ξ( ) and Λ( ) from various classes of stochastic processes (see also references therein). One can find results on logarithmic asymptotics in these papers.…”