In dealing with Markov chains, we are sometimes forced by circumstances to make observations on lumped states. We may wish, however, to deal with a random variable which is a function of the numbers in the various (unlumped) states at various times and whose moments, therefore, will involve the probabilities of transition between the unlumped states.

This paper demonstrates that, although these probabilities are immeasurable directly, we can, provided that we restrict our attention to a certain well-defined class of functions as our random variables, reduce all moments and product-moments of such random variables to functions of lumped state probabilities.

Let (S n ) n≥0 be a correlated random walk on the integers, let M 0 ≥ S 0 be an arbitrary integer, and let M n = max{M 0, S 1,…, S n }. An optimal stopping rule is derived for the sequence M n - nc, where c > 0 is a fixed cost. The optimal rule is shown to be of threshold type: stop at the first time that M n - S n ≥ Δ, where Δ is a certain nonnegative integer. An explicit expression for this optimal threshold is given.

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

An approximate model for the study of platoon formation on two-lane highways is discussed in detail. The model assumes that the two-lane highway is divided in each traffic direction into alternating road sections of fixed lengths. The passing in one type of section is unrestricted and the passing in the other one is prohibited. It is assumed that there are slow and fast vehicles on the highway and that inputs follow independent Poisson processes. The results include the distribution of the number of vehicles in a platoon and the average speed of a typical fast vehicle.

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

Define a random variable

We refer to Yn as a sustained maximum. Under the assumption that the Xi's are i.i.d. the exact distribution of Yn is obtained. Under certain conditions on the underlying distribution F we obtain weak limit results for the Yn as well. Also a combinatorial extreme-value problem is solved.

We consider the variation in size of raw silk in the fixed size raw silk reeling process, and the relations between occurrences of feeding ends and some properties of cocoons from the stochastic viewpoint. The results obtained by stochastic analysis appear reasonable for the reeling experiment using three groups of cocoons.

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

A stochastic theory is given for speculative industrial chemical research. It is shown how the theory may be used to suggest a suitable balance between long-term and short-term projects.

It is well known (Lee 1960), especially to communications and control systems engineers, that the weighting function (unit impulse function) of a linear causal system may be obtained by cross correlating its output with its input. In general a Wiener-Hopf integral equation must be solved for the weighting function; but if the input were “white noise” of unit spectral density then, as shown by Lee (1950), the weighting function would be equal to this cross correlation function for positive values of its argument. Although a perfectly “white” input cannot be obtained in practice, a number of modifications of this technique have been used; Anderson, Buland and Cooper (1959), for example, describe the use of specially selected samples of “discrete-interval binary noise” in a problem in adaptive control.

This note presents a new “uncertainty function” of finite probability distributions that arises in sequential search theory. The function arises as the finite term in an asymptotic expression for an expected search time as the allowable “error probability” approaches zero. This function is not a function of entropy; it may be of more relevance than entropy to workers in certain fields for describing the complexity of a finite distribution.

This paper continues the study (initiated in [2]) of the GI/G/K system in heavy traffic. We give a Berry–Esseen type theorem for the rate of convergence of the queue length.

The stochastic model consists of a two-dimensional random walk with unit steps, each step being a Bernoulli trial. Two sets of possible conditions arising from astronomical theory give rise to two different barriers in the plane of the walk which cause a return to the origin. Another barrier indicates the formation of a planet and also causes a return to the origin. Consideration of the composition of the planets formed at this barrier allows their orbital distances and direction of rotation to be calculated. The predictions of one of the two models were found to be in close agreement with the observed planetary distances and masses in the solar system.

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.

For a Brownian bridge from 0 to y, we prove that the mean of the first exit time from the interval $\left( -h,h \right),h>0$ , behaves as ${\mathrm{O}}(h^2)$ when $h \downarrow 0$ . Similar behaviour is also seen to hold for the three-dimensional Bessel bridge. For the Brownian bridge and three-dimensional Bessel bridge, this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to provide a detailed proof of an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We also give some elementary but nonetheless appealing relationships between a discrete determinantal process and the well-known CS decomposition.

We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).

From the Poisson–Dirichlet diffusions to the Z-measure diffusions, they all have explicit transition densities. We show that the transition densities of the Z-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are the same as the coefficients in the Poisson–Dirichlet diffusions. This fact will be uncovered by a dual process method in a special case where the Z-measure diffusions are established through an up–down chain in the Young graph.

The notion of ordered system signature, originally defined for independent and identical coherent systems, is first extended to the case of independent and non-identical coherent systems, and then some key properties that help simplify its computation are established. Through its use, a dynamic ordered system signature is defined next, which facilitates a systematic study of dynamic properties of several coherent systems under a life test. The theoretical results established here are then illustrated through some specific examples. Finally, the usefulness in the evaluation of aging used systems of the concepts introduced is demonstrated.