We consider a critical branching process with immigration controlled by a sequence of random variables {ξk,k≥1}, and prove a fluctuation limit theorem when {ξk} is (N−1)-dependent, improving the result in previous literatures.

Consider the linear nonhomogeneous fixed-point equation R=D∑i=1NCiRi+Q, where (Q,N,C1,C2,…) is a random vector with N∈{0,1,2,3,…}∪{∞},Ci≥0 for all i∈N, P(|Q|0)0, and {Ri}i∈N is a sequence of i.i.d. random variables independent of (Q,N,C1,C2,…) having the same distribution as...

In a rapidly growing population one expects that two individuals chosen at random from the nth generation are unlikely to be closely related if n is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching...

We consider the trace reconstruction problem on a tree (TRPT): a binary sequence is broadcast through a tree channel where we allow substitutions, deletions, and insertions; we seek to reconstruct the original sequence from the sequences received at the leaves. The TRPT is motivated by the...