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.

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn = O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.

By analogy with statistical mechanics we consider a random collision process with discrete time wand continuous states x ∈ [0, ∞). We assume three conditions (i), (ii) and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments.

We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

The Markovian arrival process generalizes the Poisson process by allowing for dependent and nonexponential interarrival times. We study the autocorrelation function of the two-state Markovian arrival process. Our findings show that the correlation structure of such a process has a very specific pattern, namely, it always converges geometrically to zero. Moreover, the signs of the autocorrelation coefficients are either constant or alternating.

The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

We consider a compound Poisson process whose jumps are modelled as a sequence of positive, integer-valued, dependent random variables, W1,W2,…, viewed as insurance claim amounts. The number of points up to time t of the stationary Poisson process which models the claim arrivals is assumed to be independent of W1,W2,…. The premium income to the insurance company is represented by a nondecreasing, nonnegative, real-valued function h(t) on [0,∞) such that limt→∞h(t) = ∞. The function h(t) is interpreted as an upper boundary. The probability that the trajectory of such a compound Poisson process will not cross the upper boundary in infinite time is known as the infinite-horizon nonruin probability. Our main result in this paper is an explicit expression for the probability of infinite-horizon nonruin, assuming that certain conditions on the premium-income function, h(t), and the joint distribution of the claim amount random variables, W1,W2,…, hold. We have also considered the classical ruin probability model, in which W1,W2,… are assumed to be independent, identically distributed random variables and we let h(t)=u + ct. For this model we give a formula for the nonruin probability which is a special case of our main result. This formula is shown to coincide with the infinite-horizon nonruin probability formulae of Picard and Lefèvre (2001), Gerber (1988), (1989), and Shiu (1987), (1989).

We study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

Probability theory, and its dynamic aspect stochastic process theory, is both a venerable subject, in that its roots go back to the mid-seventeenth century, and a young one, in that its modern formulation happened comparatively recently - well within living memory. The year 2003 marked the seventieth anniversary of Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung, usually regarded as having inaugurated modern (measure-theoretic) probability theory. It also marked the fiftieth anniversary of Doob's Stochastic Processes. The profound and continuing influence of this classic work prompts the present piece.

The eigentime identity for one-dimensional diffusion processes on the halfline with an entrance boundary at ∞ is obtained by using the trace of the deviation kernel. For the case of an exit boundary at ∞, a similar eigentime identity is presented with the aid of the Green function. Explicit equivalent statements are also listed in terms of the strong ergodicity or the uniform decay for diffusion processes.

For an order statistic (X1:n,…,Xn:n) of a collection of independent but not necessarily identically distributed random variables, and any i ∈ {1,…,n}, the conditional distribution of (Xi+1:n,…,Xn:n) given Xi:n > s is shown to be stochastically increasing in s. This answers a question by Hu and Xie (2006).

We use embedding techniques to analyse the error of approximation of an optimal stopping problem along Brownian paths when Brownian motion is approximated by a random walk.

Regenerative stopping problems are stopping problems which recommence from the initial state upon stopping. An algorithm is presented which solves semi-Markov regenerative stopping problem with a finite number of continue actions by solving a sequence of stopping problems. New results for the optimal stopping problem are obtained as well as for the regenerative stopping problem. A model in the literature is used as a detailed example of the algorithm.

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

For a non-negative separable random field Z(t), $t\in \mathbb{R}^d$ , satisfying some mild assumptions, we show that $ H_Z^\delta =\lim_{{T} \to \infty} ({1}/{T^d}) \mathbb{E}\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) }\} <\infty$ for $\delta \ge 0$ , where $0 \mathbb{Z}^d\,:\!=\,\mathbb{R}^d$ , and prove that $H_Z^0$ can be approximated by $H_Z^\delta$ if $\delta$ tends to 0. These results extend the classical findings for Pickands constants $H_{Z}^\delta$ , defined for $Z(t)= \exp( \sqrt{ 2} B_\alpha (t)- \lvert {t} \rvert^{2\alpha })$ , $t\in \mathbb{R}$ , with $B_\alpha$ a standard fractional Brownian motion with Hurst parameter $\alpha \in (0,1]$ . The continuity of $H_{Z}^\delta$ at $\delta=0$ is additionally shown for two particular extensions of Pickands constants.

The deleting or delay effect which a recurrent point process exerts upon another recurrent point process (or process with stationary independent increments) is treated. The concept arises in the statistical analysis of neuronal firing patterns, but may also be useful in applications to storage systems in operations research.

Is the Ewens distribution the only one-parameter family of partition structures where the total number of types sampled is a sufficient statistic? In general, the answer is no. It is shown that all counterexamples can be generated via an urn scheme. The urn scheme need only satisfy two general conditions. In fact, the conditions are both necessary and sufficient. However, in particular, for a large class of partition structures that naturally arise in the infinite alleles theory of population genetics, the Ewens distribution is the only one in this class where the total number of types is sufficient for estimating the mutation rate. Finally, asymptotic sufficiency for parametric families of partition structures is discussed.

In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.

In this paper we consider a diffusive stochastic predator–prey model with a nonlinear functional response and the randomness is assumed to be of Gaussian nature. A large deviation principle is established for solution processes of the considered model by implementing the weak convergence technique.