Information theory, in the strict sense, is a rapidly developing branch of probability theory originating from a paper by Claude E. Shannon in the Bell System Technical Journal in 1948,in which anew mathematical model ofcommunications systems was proposed and investigated.

One of the central innovations of this model was in regarding the prime components of a communications system (the source of messages and the communication channel) as probabilistic entities. Shannon also proposed a quantitative measure of the amount of information based on his notion of entropy and proved the basic theorem of this theory concerning the possi bility of reliable transmission of information over a particular class of noisy channels.

This paper is concerned with the distribution of “types” of individuals in an infinite population after indefinitely many nonoverlapping generations of random mating. The absence of selection and mutation is assumed. The probabilistic law which governs the production of an offspring may be asymmetrical with respect to the “sexes” of the two parents, but the law is assumed to apply independently of the “sex” of the offspring. The question of the existence of a limit distribution of types, the rate at which a limit distribution is approached, and properties of limit distributions are treated.

Some comparisons are made between various characteristics of the genetic structures of populations of the same size and age, which have (i) evolved from a small founder population, and (ii) evolved from a population which has been of constant size throughout the period considered.

We consider a dioecious diploid population of N individuals, N1 males and N2 = N – N1 females. The alleles will be represented by a and A, and the population reproduces according to the Wright scheme, that is, by random mating with non-overlapping generations.

A classical type of problem in population genetics is that of calculating the probability that a line descended from a particular gene will become extinct. In one problem of this sort, dealt with by Fisher (1922) and Haldane (1927), it is assumed that the population being studied is very large and that initially the number of genes of a particular type, say type A, is small. These authors obtained the solution by the use of the theory of branching processes.

One of the earliest known distributions is that of the Binomial distribution which arises from a Bernoulli sequence defined on two symbols (or generators) a and b. The corresponding limit theorem is that of Demoivre and Laplace and states (in an obvious notation) that (r – np)/√npq converges to the standard Normal distribution N(0,1). If the generators do not commute the situation is a good deal more complicated and in order to say very much about the sequences generated we must be able to put them in some simple canonical form. One case in which this can certainly be done is when the two symbols generate a semi-group of nilpotency class two. This means that although ba ≠ ab, we do have ba = ab (b,a) where (b, a) is a symbol which commutes with both a and b. Each sequence can then be expressed in the form aαbβ(b,a)γ. In this paper we examine first the conditional distribution of γ given α and β for Bernoulli sequences in the symbols a and b and obtain central limit theorems when γ is appropriately normed. We then consider the more general problem of the m-generator semi-group of nilpotency class two and obtain the corresponding multi-dimensional central limit theorem in the case where the probability measure is discrete and is distributed over the generators.

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

This paper considers asymptotic properties of increasing population age dependent branching processes which have a limiting age distribution. In Section I, the Bellman-Harris model [2] is altered in accord with a suggestion by Kendall [3]. The effect of this modification on the applicable results of [3] is indicated. An extension to the case where each cell passes through a sequence of states to mitosis or proceeds from state to state in accord with a semi-Markov process to mitosis is considered. Section 1.1 considers asymptotic moments for the total number of cells. Section 1.2 treats moments in a sequence-of-states model [4]. Section 1.3 extends the results of 1.2 to a semi-Markov model. Section 1.4 treats various asymptotic lifetime distributions and fraction of cells in a given state for both the sequence-of-states and semi-Markov models.

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process.

The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

Throughout this paper we shall be interested in a certain integer-valued process n(t) (t ≧ 0) which is encountered in queueing theory and textile research.

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.

If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn(s) = Fn–1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn(s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates limn→∞m−n {1−Fn(s)}, 0 ≦ s ≦ 1 where m = F′(1) < 1 and F′′(1) < ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn(s) − Fn(0)} [1 – Fn(0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

In a previous paper devoted to an application of dynamic programming to pattern recognition [1], we pointed out that some identification problems could be regarded as generalized trajectory processes. The functional equation technique [2] could then be employed to obtain an analytic formulation of the determination of optimal search techniques. In many cases, however, (for example, in chess or checkers), a straightforward use of the functional equation is impossible because of dimensionality difficulties. In circumventing these obstacles to effective computational solution, we employed a decomposition technique which we called “stratification” [1, 3]. In this paper, we present a different way of avoiding the dimensionality problem, based upon the concept of “extended state variable”. To indicate the utility of the concept, we shall apply it to the problem of finding a fault in a complex system.

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y1, y2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

The queueing system studied in this paper is the one in which

1. (i) there are an infinite number of servers,

2. (ii) initially (at t = 0) all the servers are idle,

3. (iii) one server serves only one customer at a time and the service times are independent and identically distributed with distribution function B(t) (t > 0) and mean β(< ∞),

4. (iv) the arrivals are in batches such that a batch arrives during (t, t + δt) with probability λ(t)δt + o(δt) (λ(t) > 0) and no arrival takes place during (t, t + δt) with the probability 1 –λ(t)δt + o(δt),

5. (v) the batch sizes are independent and identically distributed with mean α(< ∞), and the probability that a batch size equals r is given by ar(r ≧ 1),

6. (vi) the batch sizes, the service times and the arrivals are independent.

A particle travels along the half-line [0,∞) in such a way that it has probability λδu + o(δu) of generating an event in any small element (u,u + δu) of its track. The particle is observed only in the line segment 0 ≦ u ≦ x and successive events occur at X1, X2, …, Xn (0 ≦ X1 ≦ X2 ≦ … ≦ Xn ≦ x) where X1, …,Xn are random variables and the number n of events in [0, x] is also random. The distances constitute a finite univariate population process as defined by Moyal [1], the individuals being the distances Yi with state space [0, x].