Meeting time distributions in Bernoulli systems

Meeting time is defined as the time for which two orbits approach each other within distance $\epsilon$ in phase space. We show that the distribution of the meeting time is exponential in $(p_1,...,p_k)$-Bernoulli systems. In the limit of $\epsilon\to0$, the distribution converges to exp(-\alpha\tau), where $\tau$ is the meeting time normalized by the average. The exponent is shown to be $\alpha=\sum_{l=1}^{k}p_l(1-p_l)$ for the Bernoulli systems.