Statistica Sinica 7(1997), 157-166

A CENTRAL LIMIT THEOREM FOR THE NUMBER

OF SUCCESS RUNS: AN EXAMPLE OF

REGENERATIVE PROCESSES

S. G. Kou and Y. S. Chow

University of Michigan and Columbia University

Abstract: For each n≥1, let {X_n,j,  j≥1} be i.i.d. Bernoulli random variables with P{X_n,1=1}=p_n=1-q_n=1-P{X_n,1=0} ,0<p_n<1. Define , where R_n,j=inf{k≥0: X_n,j+k=0} is the number of success runs starting at j, m_n is a sequences of positive integers, and a_n=p_n/q_n . We show that, under the condition m_np_n→∞ , the central limit theorem

holds if and only if m_nq_n→∞ as n→∞, where .A key observation here is that {R_n,j, 1≤j≤m_n} forms a regenerative process so that some useful techniques from renewal theory can be utilized here.

Key words and phrases: Central limit theorem, number of success runs, regenerative process, renewal theory.