Start with a compact set K ⊂ Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in Rd and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

Extended abstract of a paper presented at Microscopy and Microanalysis 2010 in Portland, Oregon, USA, August 1 – August 5, 2010.

We consider a multi-type branching random walk on d-dimensional Euclidian space. The~uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) ‘Perron-Frobenius theory’ for matrices that are smooth functions of a variable λ∈L and are nonnegative when λ∈L−⊂L, where L is an open set in ℂd, and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

A random walk that is certain to visit $(0, \infty)$ has associated with it, via a suitable $h$ -transform, a Markov chain called ‘random walk conditioned to stay positive’, which is defined properly below. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel-3. Let $\phi(x) = \log\log x$ . The main result obtained in this paper, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time for which the random walk conditioned to stay positive is below $x$ ultimately lies between $Lx^2/\phi(x)$ and $Ux^2\phi(x)$ , for suitable (non-random) positive $L$ and finite $U$ , as $x$ goes to infinity. For Bessel-3, the best $L$ and $U$ are identified.

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

The occurrence of certain ‘near-constancy phenomena’ in some aspects of the theory of (simple) branching processes forms the background for the work below. The problem arises out of work by Karlin and McGregor [8, 9]. A detailed study of the theoretical and numerical aspects of the Karlin–McGregor near-constancy phenomenon was given by Dubuc[7], and considered further by Bingham[4]. We give a new approach which simplifies and generalizes the results of these authors. The primary motivation for doing this was the recent work of Barlow and Perkins [3], who observed near-constancy in a framework not immediately covered by the results then known.

The theory of Markov renewal processes is applied to study the occurrence of specific sequences of states in a Markov chain. Çinlar&s (1969) results are used to study both the basic process, and that obtained when the overlap of sequences is not permitted, as in the theory of counters. These results are applied to the fragments formed when DNA is digested using one, or more, restriction enzymes.

If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered.

In this paper the major concern is in investigating various divisibility properties of the random variable W, the limit of the Heyde martingale in supercritical branching processes. Throughout the paper ø(s) will be used to denote the Laplace-Stieltjes transform of W.

There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton–Watson process. Some continuity properties of the distributions of their limits are discussed.

In a supercritical branching random walk on Rp, a Galton–Watson process with the additional feature that people have positions, let be the set of positions of the nth-generation people, scaled by the factor n–1. It is shown that when the process survives looks like a convex set for large n. An analogous result is established for an age-dependent branching process in which people also have positions. In certain cases an explicit formula for the asymptotic shape is given.

If Fn∗ is the n-fold Stieltjes convolution of the increasing function F, then a version of Chernoff's theorem, on the limiting behaviour of (Fn∗(na))1/n, is established for Fn∗. If Z(n)(t) is the number of the nth-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z(n).

A result like the Kesten-Stigum theorem is obtained for certain martingales associated with the branching random walk. A special case, when a ‘Malthusian parameter’ exists, is considered in greater detail and a link with some known results about the Crump-Mode model for a population is established.

If Bn is the time of the first birth in the nth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generation Bn/n converges to a constant; if Dn is the time of the last birth in the nth generation then Dn/n also converges to a constant on the survival set. Analogous results hold for the extreme members of the nth generation in a branching random walk.