Consider, initially, a time series regression model of the simplest kind, namely Assume that x(t) is second-order stationary with zero mean and absolutely continuous spectrum with density f(ω) so that The y(t) are taken to be part of a sequence generated entirely independently of x(t) and will be treated as constants. Let βN be the least squares estimate of β and Call the numerator and denominator of b(N), respectively, c(N), d(N). We shall use K for a positive finite constant, not always the same one. We have the following result, which is Menchoff's inequality [3].

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi(t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi(t)/EZi(t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough tγ exp (αt) is the asymptotic behavior of EZi(t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

It is shown that every measure of expected information which has the branching property is of the form where J is a given information measure which is compositive under a regular binary operation and the Ψn are antisymmetric, bi-additive functions. In a probability space, such measures (entropies) take the form

The nature and historical development of both stochastic and deterministic models for binary graphs are discussed. Here the focus of applications is sociological and emphasizes representations of networks of interpersonal relations as directed graphs. Models from the natural sciences and from the social sciences are examined and suggestions for future research are given.

The calculation of probabilities on pedigrees of arbitrary complexity is discussed for a basic model of transmission and penetrance (encompassing Mendelian inheritance, and certain environmental influences).

The structure of pedigrees, and the types of loops occurring, is discussed. Some results in graph theory are obtained and, using these, a recurrence relation derived for certain probabilities. The recursive procedure enables the successive peeling off of certain members of the pedigree, and the condensation of the information on those individuals into a function on a subset of those remaining. The underlying theory is set out, and examples given of the utilization of the resulting algorithm.