In this paper we consider the problem of first-crossing from above for a partial sums process m+St, t ≥ 1, with the diagonal line when the random variables Xt, t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

Let Yk(ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z(ω) := ∑k≥0Yk(ω) is the total progeny of ω. In this paper, we will prove various statistical properties of Z and Yk. We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that Yk(ω) := ∑j=0kYj(ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree ω. We then proceed to study the joint probability distribution of Z and Ykk, and show that, as n → ∞, Yk/nk is asymptotically Gaussian under the conditional distribution P(· | Z = n).

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.

Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.