Branching Processes: Proceedings of the First World Congress by Harry Cohn (auth.), C. C. Heyde (eds.)

By Harry Cohn (auth.), C. C. Heyde (eds.)

This quantity provides the edited complaints of the 1st international Congress on Branching tactics. The contributions current new examine and surveys of the present study job during this box. hence, all these venture examine within the topic will locate this a well timed and top of the range quantity to have on their cabinets.

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S. == 1, "'2t == 0 and It == I? Nagaev and Han(1980) and Han(1980) studied the case 0 :::; Nt, "'2t == 0 and It == I? s. v. Some results for the processes with == 0 and It == I? s. were announced by Yaney and Yanev(1991). e. p = Pt, q = qt and r = rt were investigated by Yaney and Mitov(1985). An important characteristic of the process {tt} is the life-period T = T(T) started at the moment T 2: 0, and defined by YT-l = 0, YT+n > 0, 0 :::; n < T, YT+ T = O. Further on we will also consider the process stopped at zero {Zt} defined by :"'It "'It "'It Zt d = YT+tI{Zt_l > O}' t "'It = 1,2, ...

2) t-SN(I) -XN(I)+1 <0. r, F(t,s)=(F;(t,s),F2 (t,s), ... F r (t,s», E{ s[,t(t) Sf(') ... S;/(I)}= g(F(t,s» is the probability generating function of the process Zi(t),t;;::: 0, «D(t,s)=E{SZ(')}, Q(t,s)=l-~(t,s), ~(t,s)=E{szdl)} = Q(t)=Q(t,O), R(t,s)= I-«D(t,s), R(t)=R(t,O). g. 3) o It is clear that P{ 'ti:::;;t} = P{ Zi (t) = O}= ~(t,O) and J ~(t-y,O)dK(y), I D(t) = P{TJi :::;;t}=P{Xi +'t i :::;;t}= D(O+)=O. f. «D(t,s) satisfies the integral equation: I I o 0 «D(t,s) = 1- K(t)- D(t) + J~t- y,s)dK(y) + J«D(t - y,s)dD(y).

KjlKl + ... + Kj ~ 0) ~ C for all j. (8) Theorems 1 and 2 are proved in Quine and Szczotka (1994). In the sequel, we first consider two examples where (7) holds for non-iid rv's, but (6) andlor (4) fails. Then we give two examples involving iid rv's where (7) and (6) both hold and give a more general iid result. Finally we give a non-iid extension of Theorem 2. 1 The critical case Non-iid examples We first give two examples of what can happen when (7) holds in the non-iid case. Example 1 Suppose Kl,K2 ,··· are independent with P(Kj = 1) = I-liP and P (Kj = j) = liP.

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