Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

A critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes. We begin with a stationary setup with some relatively general growth process and observe that, under certain expected conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the cases where an independent and identically distributed (i.i.d.) structure holds and where the growth process is a nondecreasing Lévy process, and in particular linear, and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of one may be directly inferred from the other.

We study the susceptible-infected-susceptible model of the spread of an endemic infection. We calculate an exact expression for the mean number of transmissions for all values of the population size (N) and the infectivity. We derive the large-N asymptotic behavior for the infectivitiy below, above, and in the critical region. We obtain an analytical expression for the probability distribution of the number of transmissions, n, in the critical region. We show that this distribution has an n-3/2 singularity for small n and decays exponentially for large n. The exponent decreases with the distance from the threshold, diverging to ∞ far below and approaching 0 far above.

Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn, Xn)) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn, Xn)). The set-parameterized process {MA} defined by MA = sup {Xn:Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt), where Mt = sup{Xn: Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of the n-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive the n-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.

The approximate two- and three-dimensional solutions for spatial correlations, using the non-linear time-series approach for nearest-neighbour systems developed in my previous paper, are further discussed.

Orthogonal expansions for the correlation functions are also developed which determine with this approach, though so far only in principle, the exact solutions.

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

In this paper we derive a new property of the exponential distribution, closely related to a single-server queueing system with unreliable server. We show that this new property is another characterization of the exponential law.

A particle that is located at the position (0, 0) in the (x, y) plane at the moment x = 0 is subjected to random impulses at the moments of time x = 1, 2, …, a + b. As the result of each impulse, the particle may be displaced either by a positive or by a negative step with respect to the y-axis; it is given that the particle is restricted by the condition that it has exactly a positive and b negative displacements. Hence in the (x, y) plane the path of the particle at each impulse is depicted by a shift of a unit to the right and by one unit upward or downward; that is the particle's path starts at (0, 0) and terminates at (a + b, a – b) after a + b steps. We assume that all possible different paths are equally probable, and assume also, without loss of generality, that a ≧ b.

An exponential inequality is generalized to one involving the tail of a decreasing failure rate distribution. The results are then applied in various situations, notably when the exponential inequality does not apply.

Let denote the simple branching process with Z0 = 1 and let G denote the distribution function of Z1. Suppose G satisfies x−α−γ(x)≦1 − G(x) ≦ x−α+γ(x) for large x, where (i) 0 < α < 1, (ii) γ (x) is non-negative and non-increasing, (iii) xγ(x) is non-decreasing and (iv) Then limn→∞α n log (Zn + 1) converges almost surely to a non-degenerate finite random variable W satisfying P(W = 0) = q = probability of extinction of the process.

On a sequence of Bernoulli trials, the definition of a recurrent event ε involves the occurrence of a unique pattern of successes (S) and failures (F), the final element of which is the result of the nth trial. Success runs are the best known of such recurrent events, but Feller (1959, §13.8) mentions more complicated patterns, among which two types may be distinguished. The simpler involves a single more complex pattern such as SSFFSS; the second type involves a set of alternative events defining ε, which is said to occur when any one of the alternatives occurs at trial number n. Thus if ε stands for “either a success run of length r or a failure run of length ρ”, there are two alternatives in the set; the problem is elementary because the component events are “non-overlapping”.

In this paper we investigate the server's busy period in a single-server queueing situation in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A closed-form expression is obtained for the Laplace transform bn(z) of the joint probability and probability density function of the busy period duration and the number of customers served in it. Some numerical values are given showing the effect of correlation between T and S.

A limit theorem of Kurtz for perturbed operator semigroups is applied to show that the two-locus diffusion model in population genetics (allowing for selection, mutation, and migration) converges to a linkage-equilibrium diffusion model as Nc →∞, where N is the population size and c is the recombination fraction; in fact, with an appropriate change of variables, the limiting diffusion is what has been called the independent-loci diffusion model. This generalizes a result of Littler.

We study sojourn times in M/G/1 multiple vacation systems and multiqueue cyclic-service (polling) systems with instantaneous Bernoulli feedback. Three service disciplines, exhaustive, gated, and 1-limited, are considered for both M/G/1 vacation and polling systems. The Laplace-Stieltjes transforms of the sojourn time distributions in the three vacation systems are derived. For polling systems, we provide explicit expressions for the mean sojourn times in symmetric cases. Furthermore a pseudo-conservation law with respect to the mean sojourn times is derived for a polling system with a mixture of the three service disciplines.