The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.

In order to describe the evolution of the attainable structures of a homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space as they are changing in time and we find the value of the volume asymptotically. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures; extensions are obtained concerning results from the Perron–Frobenius theory referring to Markov systems.

A particular random walk on the integers leads to a new, tractable Markov chain. Of the stationary probabilities, we discuss the existence, some analytic properties and a factorization which leads to an algorithmic procedure for their numerical computation. We also consider the positive recurrence of some variants which each call for different mathematical arguments. Analogous results are derived for a continuous version on the positive reals.

We consider the two-dimensional process {X(t), V(t)} where {V(t)} is Brownian motion with drift, and {X(t)} is its integral. In this note we derive the joint density function of T and V(T) where T is the time at which the process {X(t)} first returns to its initial value. A series expansion of the marginal density of T is given in the zero-drift case. When V(0) and the drift are both positive there is a positive probability that {Χ (t)} never returns to its initial value. We show how this probability grows for small drift. Finally, using the Kontorovich–Lebedev transform pair we obtain the escape probability explicitly for arbitrary values of the drift parameter.

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.

We prove a generalization of Sanov's theorem in which the state space S is arbitrary and the set of probability measures on S is endowed with the τ -topology.

The purpose of this paper is to provide explicit formulas for a variety of probabilistic quantities associated with an asymmetric random walk on a finite rectangular lattice with absorbing barriers. Quantities of interest include probabilities that a walker will exit the lattice onto some particular set of boundary states, the expected duration of the walk, and the expected number of visits to one state given a start in another. These quantities are shown to satisfy two-dimensional recurrence relations that are very similar in structure. In each case, the recurrence relations may be represented by matrix equations of the form X = AX + XB + C, where A and B are tridiagonal Toeplitz matrices. The spectral properties of A and B are investigated and used to provide solutions to this matrix equation. The solutions to the matrix equations then lead to solutions for the recurrence relations in very general cases.

Exact and ordinary lumpability in finite Markov chains is considered. Both concepts naturally define an aggregation of the Markov chain yielding an aggregated chain that allows the exact determination of several stationary and transient results for the original chain. We show which quantities can be determined without an error from the aggregated process and describe methods to calculate bounds on the remaining results. Furthermore, the concept of lumpability is extended to near lumpability yielding approximative aggregation.

Krafft and Schaefer [14] considered a two-parameter Ehrenfest urn model and found the n-step transition probabilities using representations by Krawtchouk polynomials. For a special case of the model Palacios [17] calculated some of the expected first-passage times. This note investigates a generalization of the two-parameter Ehrenfest urn model where the transition probabilities pi,i+1 and pi,i+1 are allowed to be quadratic functions of the current state i. The approach used in this paper is based on the integral representations of Karlin and McGregor [9] and can also be used for Markov chains with an infinite state space.

Let A be a stochastic matrix and ε a positive number. We consider all stochastic matrices within ε of A and their corresponding stochastic eigenvectors. A convex polytope containing these vectors is described. An efficient algorithm for computing bounds on the components of these vectors is also given. The work is compared to previous such work done by the author and by Courtois and Semai.

At which (random) sample size will every population element have been drawn at least m times? This special coupon collector's problem is often referred to as the Dixie cup problem. Some asymptotic properties of the Dixie cup problem with unequal sampling probabilities are described.

Let Y1, Y2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

This paper considers a variation of the classical secretary problem; this variation allows for the interviewer to make an offer to previously interviewed applicants and for the applicant to reject an offer. In this version, the process ends after any offer is made. Two special cases are considered, where the optimal procedure and its probability of success are given for each. Furthermore, the problem is discussed in general and conditions for optimal strategies are given.

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t−1 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

We consider the problem of conditioning a non-explosive birth and death process to remain positive until time T, and consider weak convergence of this conditional process as T → ∞. By a suitable almost sure construction we prove weak convergence. The almost sure construction used is of independent interest but relies heavily on the strong monotonic properties of birth and death processes.

Let A1, A2, …, An and B1, B2,. ., BN be two sequences of events on the same probability space. Let mn(A) and mN(B), respectively, be the number of those Aj and Bj which occur. Let Si,j denote the joint ith binomial moment of mn(A) and jth binomial moment of mN(B), 0 ≤ i ≤ n, 0 ≤ j ≤ N. For fixed non-negative integers a and b, we establish both lower and upper bounds on the distribution P(mn(A) = r, mN(B) = u) by linear combinations of Si,j, 0 ≤ i ≤ a, 0 ≤ j ≤ b. When both a and b are even, all mentioned S¡,j are utilized in both the upper and the lower bound. In a set of remarks the results are analyzed and their relation to the existing literature, including the univariate case, is discussed.

In this paper we review conditions under which Wald's equation holds, mainly if the expectation of the given stopping time is infinite. As a main result we obtain what is probably the weakest possible version of Wald's equation for the case of independent, identically distributed (i.i.d.) random variables. Moreover, we improve a result of Samuel (1967) concerning the existence of stopping times for which the expectation of the stopped sum of the underlying i.i.d. sequence of random variables does not exist. Finally, we show by counterexamples that it is impossible to generalize a theorem of Kiefer and Wolfowitz (1956) relating the moments of the supremum of a random walk with negative drift to moments of the positive part of X1 to the case where the expectation of X1 is —∞. Here, the Laplace–Stieltjes transform of the supremum of the considered random walk plays an important role.

The classical exponential smoothing procedure to be applied to a sequence of observations is modified into some adaptive variants and also generalized to the recursive scheme

This means that given the history of observations and smoothed values up to time n, the updated smoothed value Wn +1 is given by a function u of Wn and Xn, which satisfies only certain structural properties. We study these procedures within the framework of random systems with complete connections and exploit results from Markov process theory to study the sequence . A broad class of ‘smoothing functions' u, which induce sequences with nice statistical properties, is presented. A method to estimate the limit of the expected smoothed values is also developed.,

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.