An Estimate of the Remainder Term in a Limit Theorem for Recurrent Events

Let $\mathcal{E}$ be a recurrent event, $a_n$ be the probability that $\mathcal{E}$ occurs at the $n$-th trial and $p_n$ be the probability that $\mathcal{E}$ occurs for the first time at the $n$-th trial. A. N. Kolmogorov [2] proved that as $n\to\infty$
$$ B_n=a_n-\frac{1}{\mu}\to 0, $$
where $\mu=\sum_{k\geqq 1}kp_n$ and then W. Feller [3] estimated the remainder term $B_n$ under some addition-conditions. In this note a more exact estimate of $B_n$ under more general conditions as compared to Feller's is given.