Asymptotic normality in a coupon collector's problem

Summary Let {a_s, s=1, 2, ..., N} be a set of reals and {p_s, s=1, 2, ..., N} be a set of probabilities, i.e. p_s0 and p₁+p₂+...+p_N=1. Let I₁I₂,... be independent random variables, all with the distribution P(I=s)=p_s, s=1, 2, ..., N. Put U_v=l if I_v{I₁, I₂, ..., I_v–1} and U_v=0 otherwise, v=1, 2, .... The random variable Z_n= is called the bonus sum after ncoupons for a coupon collector in the situation {(p_s, a_s), s=1, 2, ..., N}.Consider a sequence {(p_ks, a_ks), s=l, 2, ..., N_k}, k=1, 2, ..., of collector situations, and let {Z_n^(k), n=1, 2, ...}, k=1, 2, ..., be the corresponding sequence of bonus sum variables. Let d be an arbitrary natural number and let , k=1, 2, ..., where 1 n_k⁽¹⁾<n_k⁽²⁾<< n_k^(d).We assume that N^(k)t8 and that .It is shown that the random vector V^(k)is, under general conditions, asymptotically (as kt8) normally distributed. An asymptotic expression for the covariance matrix of V^(k)is derived.Research supported in part at Stanford University, Stanford, California under contract N0014-67-A-0112-0015.