  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}.
