Communications of the
Korean Mathematical Society CKMS

The convergence rate of the expectation of the logarithm of the first return time $R_n$ with block length $n$ has been investigated for Bernoulli processes. This idea is applied to check the randomness of the digits of the decimal expansion of $\pi$, $e$ and $\sqrt{2}$.

Keywords: the first return time, entropy, digits of $\pi$, ergodicity

MSC numbers: Primary 37M25, secondary 94A17