ASYMPTOTIC NORMALITY IN A COUPON COLLECTORS PROBLEM

Let {as, s=1, 2, ..., N} be a set of reals and {ps, s=1, 2, ..., N} be a set of probabilities, i.e. ps≧0 and p1+p2+...+pN=1. Let I1I2,... be independent random variables, all with the distribution P(I=s)=ps, s=1, 2, ..., N. Put Uv=l if Iv∋{I1, I2, ..., Iv-1} and Uv=0 otherwise, v=1, 2, .... The random variable Zn= {Mathematical expression} is called the bonus sum after ncoupons for a coupon collector in the situation {(ps, as), s=1, 2, ..., N}. Consider a sequence {(pks, aks), s=l, 2, ..., Nk}, k=1, 2, ..., of collector situations, and let {Zn(k), n=1, 2, ...}, k=1, 2, ..., be the corresponding sequence of bonus sum variables. Let d be an arbitrary natural number and let {Mathematical expression}, k=1, 2, ..., where 1 ≦nk(1)<nk(2)<⋯< nk(d).We assume that N(k)→t8 and that {Mathematical expression}. It is shown that the random vector V(k)is, under general conditions, asymptotically (as k→t8) normally distributed. An asymptotic expression for the covariance matrix of V(k)is derived. © 1969 Springer-Verlag.