A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn(x)} converges weakly to a distribution G(x) [9]. We define a metric e1[Fn;G], which is analogous to the functional e[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove that e1[Fn; G] is monotone non-increasing as n → ∞, and also the convergence of the sequence {Fn(x)} under the weaker assumption that for some a > 1 the initial distribution has an ath moment.

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable Xn,q: the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for Xn,q. If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.

Important parameters of particle size distributions in dispersed systems in engineering and related fields are ratios of moments and inverse powers of these ratios, known as mean sizes. The variation in these parameters is examined for the simplest growth model in which the size distribution is translated, and the results for this process considered in relation to the problems of models of other growth processes. For initial size distributions with monotone hazard rate, the results are particularly significant, and the properties of the normalised moments of other distributions are also considered.

We consider a percolation model on the d-dimensional Euclidean space which consists of spheres centred at the points of a Poisson point process of intensity ?. The radii of the spheres are random and are chosen independently and identically according to a distribution of a positive random variable. We show that the percolation function is continuous everywhere except perhaps at the critical point. Further, we show that the percolation functions converge to the appropriate percolation function except at the critical point when the radius random variables are uniformly bounded and converge weakly to another bounded random variable.

This paper illustrates a simple method of deriving distributions for samples from cell populations in exponential growth. The distribution of the elapsed proportion of the DNA-synthetic phase of the cell cycle is derived and used to find the distribution of DNA content in a random sample of cells. Allowing for a normally-distributed measurement error, the resulting distribution agrees well with DNA distributions observed empirically. The statistical analysis of DNA distributions, and the interpretation of the distributional parameters in terms of cell population kinetics, are discussed and illustrated with an example.

This paper is concerned with the determination of the optimal release rate for a continuous storage system. The input to the store is assumed to be a compound Poisson process (with non-negative jumps), and its content is released either at a constant rate or at a linear rate. Rewards are collected at an output-dependent rate and are continuously discounted at a constant rate. The problem consists in finding the optimal release rate which maximizes the infinite-horizon expected discounted return. This problem is solved by deriving an explicit solution for the expected discounted return, from which the optimal release rate can be found.

The object of this paper is to study the actual waiting time of a customer in a GI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer than w in a given time interval (0, T) are calculated.

The problem discussed in this paper arose from the study of the effects of unresolved resonances on neutron cross-sections, but it is considered here in more general terms.

Events in a modified renewal process occur at successive intervals x1, x2, …, and the ith event has associated with it a parameter Γi. The random variables xi and Γi are all independent, and their probability distribution functions are known. Each event contributes to two quantities F(u) and G(v) measured at u and v respectively. The value of the total contribution of all events to G(v) is assumed to be known from observation: this paper gives formulae for the mean value of F(u) conditional on this known value of G(v).

We consider the problem of the distinguishability between Gaussian processes observed through non-linear systems. For a broad class 𝒜 of zero-memory time-dependent non-linear transformations, the following uniqueness relationship is obtained. If Xi, i = 1, 2 are two normalized jointly Gaussian processes, then they are indistinguishable if and only if their transformed processes Yi = A[Xi], i = 1, 2, A ∈ 𝒜, are indistinguishable. Some interesting examples of functions in 𝒜 are presented.

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.

Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.

Some simple methods of estimating inbreeding from tests on a random sample of individuals are discussed. These are mostly iterative procedures, converging to the maximum likelihood estimate, with algorithms for finding standard errors. But there are problems of interpretation, both from the statistical and biological points of view.

In more than four decades of prolific scientific activity, Lajos Takács has produced so much that no one contributor to this Festschrift can hope to cover all of it in a balanced way. I thus propose to make a virtue of necessity, and concentrate on those aspects of Takács' work which have particularly interested or influenced me, and on the impact Takács' ideas in these areas have had on the subsequent development of the subject.

Bartlett and Williams have shown that the hypothesis of the goodness-of-fit of one or more assigned discriminant functions in the case of several groups can be tested using Wilks's A criterion. By factorizing this A criterion suitably, it is also possible to test whether the directions of the given functions are inadequate or whether the number of proposed functions is inadequate. Hotelling's statistic is a competitor of Wilks's A in multivariate analysis; previously its use for this hypothesis and also for the dimensionality and direction aspects of this hypothesis was unknown. The present paper derives the appropriate type statistics for this and also gives their exact null distributions.

A stationary stochastic process must satisfy various requirements to make it a realistic model for a phenomenon in the real world. Some of these requirements are quantitative, such as agreement of distribution or moments. Other, more qualitative requirements deal with the general behavior of the process. Two such requirements are non-singularity and asymptotic independence. Each will be discussed from a variety of points of view, and given precise definition in a succession of progressively stronger forms.

Computer scientists have introduced ‘paging algorithms' which are a special class of Markov chains on permutations known, in probability theory, as ‘libraries': books being placed on a shelf T (T is an infinite interval of the set Z of the integers) and a policy ρ : T → T such that ρ (t) < t being chosen, a book b placed at t ∊ T is selected with probability pb, it is removed and replaced at ρ (t) prior to next removal. The different arrangements of books on the shelf are the states of the Markov chain. In this paper we prove that, if the shelf is not bounded on the left, any library (i.e. for any policy ρ and any probability ρ on the books) is transient.

A waiting time without memory, or age-independent residual life-time, is a positive-valued random variable T with the property that for any x, y > 0, given that T >x, the conditional probability of T >x +y is the same as the unconditional probability of T >y; in other words, the physical process operates as if it has no memory concerning the successive occurrences of a certain event. The paper investigates the consequences of defining the property of lack of memory on more general time-domains than the positive reals. As a side issue, there is discussion of a stochastic variation of Cauchy's functional equation.

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.