Laws of the iterated logarithm for a class of iterated processes

TL;DR

This paper investigates the path behavior of iterated processes involving Brownian motion and stable processes, establishing laws of the iterated logarithm and small ball probabilities through a novel connection with stable subordinators.

Contribution

It extends known connections between iterated Brownian motion and stable subordinators to a broader class of processes, deriving new laws of the iterated logarithm and small ball probability estimates.

Findings

Established laws of the iterated logarithm for $X(E(t))$ and $X(L(t))$.

Derived exact small ball probabilities for $X(E_t)$.

Extended the connection between iterated Brownian motion and stable subordinators.

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}.