Limit Theorems in an Occupancy Problem

The waiting time and related statistics in the multinomial scheme: a survey

“…i/ m (7V,fc) (the level degenerated at the point m) relation (11) is specialized in the following way: (10). Representation (1Γ) was obtained by Bekessy [25] and later was reopened in [31].…”

“…If in Theorem 3 g(x) = x, then a x (g) = *(A), a\(g) = λλ(*'(λ)) 2 -t(X) and we have the assertion on asymptotic normality of the waiting time v(N,k) [1]; the degenerate level case was first studied in [25], and in more detail it was analyzed in [10]. In particular, in [10] it is established that if m > 2 is fixed, then the relation…”

“…'s are asymptotically (as N -> oo) independent and identically distributed with the distribution function e~e (see (18)); besides, E(i/j(JV)) r -> (-l)T< r >(l) (see also [32]) In [10] it is proved that, in the region of mean values of the parameter k, relation (22) remains true also for m -> oo if m -O(ln N). But for extremal values of k the situation turns out considerably more complicated: the limit relation of type (18) remains true as m -> oo only for m = ο ((In TV/In In N) l/2 ), and relation (17) does not hold as m -» oo.…”

“…The formula for Dv m (N,N) was obtained in [26], and the formulae for the moments for k = XN were proved in [25]. In other cases the formulae were proved in [10].…”

“…Growing levelNow let m = m(N) -> oo as A^ -> oo. In this case the asymptotic behaviour of the waiting time v m (N,k) was considered in[10,13]. Let us give the main results from there.Brought to you by | University of Glasgow Library Authenticated Download Date | 6/25/15 8:12 AM…”