We study the bijection between binary Galton-Watson trees in continuous time and their exploration process, both in the subcritical and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to ∞. We thus deduce Delmas' generalization of the second Ray-Knight theorem.

By establishing general relationships between branching transformations (Harris-Sevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Zn with offspring distribution Y, there exists a random variable X such that the probability P(Zn = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X1,…, Xn, where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

A martingale is used to study extinction probabilities of the Galton-Watson process using a stopping time argument. This same martingale defines a martingale function in its argument s; consequently, its derivative is also a martingale. The argument s can be classified as regular or irregular and this classification dictates very different behavior of the Galton-Watson process. For example, it is shown that irregularity of a point s is equivalent to the derivative martingale sequence at s being closable, (i.e., it has limit which, when attached to the original sequence, the martingale structure is retained). It is also shown that for irregular points the limit of the derivative is the derivative of the limit, and two different types of norming constants for the asymptotics of the Galton-Watson process are asymptotically equivalent only for irregular points.

Previous study of the time to a common ancestor of all present-day individuals has focused on models in which each individual has just one parent in the previous generation. For example, ‘mitochondrial Eve’ is the most recent common ancestor (MRCA) when ancestry is defined only through maternal lines. In the standard Wright-Fisher model with population size n, the expected number of generations to the MRCA is about 2n, and the standard deviation of this time is also of order n. Here we study a two-parent analog of the Wright-Fisher model that defines ancestry using both parents. In this model, if the population size n is large, the number of generations, 𝒯n, back to a MRCA has a distribution that is concentrated around lgn (where lg denotes base-2 logarithm), in the sense that the ratio 𝒯n(lgn) converges in probability to 1 as n→∞. Also, continuing to trace back further into the past, at about 1.77 lgn generations before the present, all partial ancestry of the current population ends, in the following sense: with high probability for large n, in each generation at least 1.77lgn generations before the present, all individuals who have any descendants among the present-day individuals are actually ancestors of all present-day individuals.

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks.

A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.

In this paper exchangeable bisexual models with fixed population size and non-overlapping generations are introduced. Each generation consists of N pairs of individuals. The pairs of a generation have altogether 2N children. These individuals form randomly the N pairs of the next generation. The extinction probability of the descendants of a fixed number of pairs of generation 0 is discussed. Under suitable conditions it can be approximately described by the extinction probability of a Galton–Watson process, if the population size is large. Special examples are a bisexual Wright–Fisher model and models with a uniformly bounded number of children of a pair.

The Galton-Watson process with immigration which is time-homogeneous but not permitted when the process is in state 0 (so that this state is absorbing) is briefly studied in the subcritical and supercritical cases. Results analogous to those for the ordinary Galton-Watson process are found to hold. Partly-new techniques are required, although known end-results on the standard process with and without immigration are used also. In the subcritical case a new parameter is found to be relevant, replacing to some extent the criticality parameter.

Results connecting Yaglom's limit and Z1 are obtained. For example, if α > 1, E[Zα1] and converge or diverge together.

We obtain an extension of a theorem of Bingham and Doney connecting the random variables Z1 and W in the supercritical Galton-Watson process. The regular variation of the distribution of Z1 is equivalent to the regular variation of the tail of the distribution of W for integer values of α > 1.

The kin number problem concerns the relationship between the distribution of the number of offspring of a randomly chosen individual in a population, and that of the number of relatives of various degrees of affinity of a randomly chosen individual (referred to as ‘Ego’). This problem is considered in terms of a population consisting of a large number of simultaneously developing Galton-Watson family trees. By time-reversal starting with the epoch at which Ego is sampled, it is shown that the number of offspring of Ego's parent (Ego plus siblings) and the number of offspring of Ego's grandparent (Ego's parent plus Ego's parent's siblings) have the same distribution, and the probability generating function (p.g.f.) is obtained in terms of the reproduction p.g.f. Further development of the method yields the joint p.g.f. of any number of generations prior, and subsequent, to that of Ego. Means and variances are obtained, and numerical examples are given based on data for the reproduction p.g.f. in two contrasting human populations. Various applications including demography and cancer research are discussed.

A branching process is described which contains some better-known simple branching processes as special cases. Variables examined are the rate of development, the number of individuals currently alive, and the duration of the process. The apparent dynamics of collapsing supercritical realisations are isolated.

Under weak conditions, this paper provides a best lower and a best upper bounding fractional linear generating function for any probability generating function when they have the same mean. These bounds can be used to obtain bounds for the expectation and the percentiles of the extinction-time distribution of a Galton-Watson branching process and other parameters of interest. For the special case of the four points probability generating function, the best bounds obtained are better than the bounds derived by Agresti (1974).

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

Limit laws and limiting diffusions are obtained for critical branching processes, either Galton-Watson or age-dependent, conditioned on extinction in the interval (T, cT], 1 < c, as T→∞, and also as T→∞ and c ↓ 1.

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.

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

In this article, we consider maximum likelihood estimators of the initial probabilities and the mean of a supercritical Galton-Watson process; we find for these estimators, as well as for the Lotka-Nagaev estimator of the mean, the asymptotic distributions and deduce confidence intervals. As these results hold even if the independence between individuals of the same generation is not satisfied, application to living populations may be considered.