Laws of the iterated logarithm for a class of iterated processes

Abstract

Let $X=\{X(t),t\ge 0\}$ be a Brownian motion or a spectrally negative stable process of index $1<\a<2$. Let $E=\{E(t),t\ge 0\}$ 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 $β/\a$ 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 $1<\gamma \le 2$ 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}.

13 pages

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Cited by in corpus (4)

• Correlation structure of time-changed fractional Brownian motion
• Mixed Generalized Fractional Brownian Motion
• Small ball probabilities for a class of time-changed self-similar processes
• On the long range dependence of time-changed mixed fractional Brownian motion model