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

Let E be a recurrent event, aₙ be the probability that E occurs at the n-th trial and pₙ be the probability that E occurs for the first time at the n-th trial. A. N. Kolmogorov [2] proved that as n → ∞\[ B_n = a_n - 1/μ → 0, \] where μ = ∑ k 1 kpₙ and then W. Feller [3] estimated the remainder term Bₙ under some addition-conditions. In this note a more exact estimate of Bₙ under more general conditions as compared to Feller’s is given.