We consider a non-zero-sum game in which two searchers (player I and II) compete with each other for quicker detection of an object hidden in one of n boxes. Let p (q) be the prior location distribution of the object for player I (II). Exponential detection functions are assumed for both players. Each player wishes to maximize the probability that he detects the object before the opponent detects it. In the general case, a Nash equilibrium point is obtained in the form of a solution of simultaneous differential equations. In the case of p = q, we obtain an explicit solution showing the surprising result that both players have the same equilibrium strategy even though the detection rates are different.

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

A number of stochastic properties for a model of traffic flowing in a single lane are derived. The arrival process used is one which models (a) the bunching phenomenon of traffic and (b) the fact that vehicles have a spatial dimension and travel at minimum temporal spacing. Weak assumptions about the desired speeds of vehicles are made.

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

Move-to-front and move-to-back heuristics are considered for adaptively reordering routing tables in communications networks. Limiting properties are established for Markov chain models of these heuristics. The limiting distributions of the number of path attempts per call for the two heuristics and for a fixed random permutation are stochastically ordered.

Denoting by v(t) the residual life of a component in a renewal process, Çinlar and Jagers (1973) and Holmes (1974) have shown that if E(v(t)) is independent of t for all t, then the process is Poisson. In this note we prove, under mild conditions, that if E(G(v(t))) is constant, then the process is Poisson. In particular if E((v(t))r) for some specific real number r ≧ 1 is independent of t, then the process is Poisson.

A particle travels along the half-line [0,∞) in such a way that it has probability λδu + o(δu) of generating an event in any small element (u,u + δu) of its track. The particle is observed only in the line segment 0 ≦ u ≦ x and successive events occur at X1, X2, …, Xn (0 ≦ X1 ≦ X2 ≦ … ≦ Xn ≦ x) where X1, …,Xn are random variables and the number n of events in [0, x] is also random. The distances constitute a finite univariate population process as defined by Moyal [1], the individuals being the distances Yi with state space [0, x].

A methodology for an analytical derivation of visibility probabilities of n stationary target points in the plane is developed for the case when shadows are cast by a Poisson random field of obscuring elements. In addition, formulae for the moments of a measure of the total proportional visibility along a star-shaped curve are given.

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.

Yu. V. Linnik showed that certain transformations, given by Formulae (1.1), (1.6) and (1.7) transform a normal sample into itself. The transformations (1.1) and (1.7) apply to samples of size 2 while (1.6) admits an arbitrary sample size. It is also assumed that the population mean is zero.

In the present paper the converse theorems are proven so that characterizations of the normal distribution are obtained. The problem leads to the functional equations (2.3) and (2.13) whose solution yields the desired results.

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.

We establish the absolute continuity of the limit random variables of two supercritical Galton-Watson branching processes, one allowing unrestricted immigration and the other having a state dependent immigration component.

Olivier and Walrand (1994) claimed that the departure process of an MMPP/M/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. They also conjectured that the departure process of an MAP/PH/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. We show that their proof of the first result has an algebraic error, which leaves open the above question of whether the departure process of an MMPP/M/1 can be an MAP.

We consider a sequence, of random length M, of independent, continuous observations Xi, 1 ≤ i ≤ M, where M is geometric, X1 has cumulative distribution function (CDF) G, and Xi, i ≥ 2, have CDF F. Let N be the number of upper records and let Rn, n ≥ 1, be the nth record value. We show that N is independent of F if and only if G(x) = G0(F(x)) for some CDF G0, and that if E(|X2|) is finite then so is E(|Rn|), n ≥ 2, whenever N ≥ n or N = n. We prove that the distribution of N, along with appropriately chosen subsequences of E(Rn), characterize F and G and, along with subsequences of E(Rn - Rn-1), characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.

We propose an estimator for the cumulative distribution function G of the sojourn time in a steady-state M/G/∞ queueing system, when the available data consists of the arrival and departure epochs alone, without knowing which arrival corresponds to which departure. The estimator generalizes an estimator proposed in Brown (1970), and is based on a functional relationship between G and the distribution function of the time between a departure and the rth latest arrival preceding it. The estimator is shown to outperform Brown's estimator, especially when the system is heavily loaded.

We study the best achievable performance (in terms of the average queue size and delay) in a stochastic and dynamic version of the bin-packing problem. Items arrive to a queue according to a Poisson process with rate 2ρ, where ρ ∈ (0, 1). The item sizes are independent and identically distributed (i.i.d.) with a uniform distribution in [0, 1]. At each time unit, a single unit-size bin is available and can receive any of the queued items, as long as their total size does not exceed 1. Coffman and Stolyar (1999) and Gamarnik (2004) have established that there exist packing policies under which the average queue size is finite for every ρ ∈ (0, 1). In this paper we study the precise scaling of the average queue size, as a function of ρ, with emphasis on the critical regime where ρ approaches 1. Standard results on the probabilistic (but static) bin-packing problem can be readily applied to produce policies under which the queue size scales as O(h2), where h = 1 / (1 - ρ), which raises the question of whether this is the best possible. We establish that the average queue size scales as Ω(hlogh), under any policy. Furthermore, we provide an easily implementable policy, which packs at most two items per bin. Under that policy, the average queue size scales as O(hlog3/2h), which is nearly optimal. On the other hand, if we impose the additional requirement that any two items packed together must have near-complementary sizes (in a sense to be made precise), we show that the average queue size must scale as Θ(h2).

In this paper, we study a monotone process maintenance model for a multistate system with k working states and ℓ failure states. By making different assumptions, we can apply the model to a multistate deteriorating system as well as to a multistate improving system. We show that the monotone process model for a multistate system is equivalent to a geometric process model for a two-state system. Then, for both the deteriorating and the improving system, we analytically determine an optimal replacement policy for minimizing the long-run average cost per unit time.

Let {X(s, t), s = (s1, s2) ∈ ℝ2, t ∈ ℝ} be a stationary random field defined over a discrete lattice. In this paper, we consider a set of domain of attraction criteria giving the notion of extremal index for random fields. Together with the extremal-types theorem given by Leadbetter and Rootzen (1997), this will give a characterization of the limiting distribution of the maximum of such random fields.

We consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process. This hedging strategy is called robust if it yields a superhedge as soon as the local volatility model overestimates the market volatility. We show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function. In a genuine local volatility model the situation is shown to be less stable: robustness can break down for many relevant convex payoffs including average-strike Asian options, lookback puts, floating-strike forward starts, and their aggregated cliquets. Furthermore, we prove that a sufficient condition for the robustness in every local volatility model is the directional convexity of the payoff function.