Symptotic behavior of moments in the allocation scheme with random levels

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

“…Then, we define the event A t (lj) = A t (lJi,...Jk-i) which means that at the time T ( /t) = t in the process Π/ an event happens and ξι(ί) = v\\ at the same time t in the prpcesses Π^,..., Π^_, the numbers of events are not less than the corresponding levels and in the other processes uj, j f l>jii"">jk-i, the same will happen after t. Since the processes are independent, k-l = 1 P(A t (lJ))dt = Pl P{t t (t -0) = ι/, -1} Π Pf .W > ",·.} Π' PfoW < "Λ Λ, (3) where Π' denotes the same product as in (2). Under the condition that the event A t (l, J) happens, the number of events £ij,(0 in the process U ja has the truncated Poisson Brought to you by | University of Glasgow Library Authenticated Download Date | 6/25/15 8:12 AM distribution £«υ·.…”

“…Here π τ (ζ) = e~zz r /rl, r = 0, l,...; Jki(N) means that the summation it taken over all ji,...,jk-i, not equal to /, which satisfy the conditions 1 < j\ < ... < ji*_i < TV; the symbol Π' denotes the product over 1 < j < N, j f lji,...jk-i· A clear probability interpretation by means of the recently wide-used method of embedding the multinomial scheme into the Poisson process can be given to representation (2). Namely, let us consider on (0, oo) N independent Poisson processes Πι,..., ITyv with intensities pi,... ,PN respectively, and let ^(i) be the number of events of the process IIj in the interval (0,0; P{£,(0 = r} = π Γ (ρ^)> r = 0,1,... Then the unified process Π = Πι + ... + Π TV is also the Poisson process with intensity p\ + ... + PN = 1, and one can conceive that whenever in the unified process Π an event happens the multinomial trial is conducted and its result is the outcome j if the event belongs to the process Uj.…”

“…The waiting time v (N,k) in the scheme with random levels was intensively studied then by Ivanov, Ivchenko and Protasov [4], Ivanov [2] and Ivchenko [14][15][16].…”

Decomposable statistics in an inverse allocation problem

“…Many publications have been devoted to investigations of the various aspects of the stopping moment (or waiting time) v m (N,k) and other characteristics of the scheme connected with v m (N, k)\ a survey of the results covering the period up to 1983 is given in [2]. The interest in these problems, caused by their practical applications, has not decreased, as demonstrated by later publications [3][4][5][6][7].…”

“…The first of them, presented in particular in papers [3,4], is connected with the sophistication of the stopping moment of trials (the procedure of allocating particles). These papers consider the so-called schemes with random levels, when before the beginning of the trials a random level is fixed for each cell and the trials last until the moment that k cells whose frequencies are not less than the corresponding levels appear for the first time.…”