B. A. Sevastyanov
Convergence in distribution of random mappings of finite sets to branching processes
A countable set X = X(t) is partitioned into pairwise disjoint finite layers X(t), the cardinalities |X(t)| of the sets X(t), t = 0, 1, 2, . . . , are finite.
Each layer is partitioned into r disjoint sets Xi (t), i = 1, . . . ,r, so that X(t) = Xi
(t), Ni
(t) = |X
i
(t)|
and Ni
(t) ∼ Nθi
(t) as
N → ∞. We set X′ = XX(0).
We consider random mappings y = f (x) of the set X′ into the set X. We assume
that for any pairwise unequal xi
, i = 1, . . . , k, the random variables yi
= f (x
i
), i = 1, . . . ,k, are independent and f (X(t)) ? X (t
− 1), t = 1, 2, . . . . Let Yi
(0) ? Xi
(0) be some fixed subsets and
Yi(t) = f
−1(Y(t − 1))∩Xi
(t), t =1,
2, . . . , be the sequence of preimages of Yi
(0)in these random mappings. It is shown that μi
(t, N) = |Yi
(t)|, i = 1, . . . r, converges in distribution as N →
∞ to a non-homogeneous in time branching process with r types of particles.
Discrete Mathematics and Applications, Walter de Gruyter
Print ISSN: 0924-9266
Volume: 15, 04/2005
Pages: 105 - 108
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