Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

In many applications, such as remote sensing or wave slamming on ships and offshore structures, it is important to have a good model for wave slope. Today, most models are based on the assumption that the sea surface is well described by a Gaussian random field. However, since the Gaussian model does not capture several important features of real ocean waves, e.g. the asymmetry of crests and troughs, it may lead to unconservative safety estimates. An alternative is to use a stochastic Lagrangian wave model. Few studies have been carried out on the Lagrangian model; in particular, very little is known about its probabilistic properties. Therefore, in this paper we derive expressions for the level-crossing intensity of the Lagrangian sea surface, which has the interpretation of wave intensity, as well as the distribution of the wave slope at an arbitrary crossing. These results are then compared to the corresponding intensity and distribution of slope for the Gaussian model.

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

A stochastic model of a dynamic marker array in which markers could disappear, duplicate, and move relative to its original position is constructed to reflect on the nature of long DNA sequences. The sequence changes of deletions, duplications, and displacements follow the stochastic rules: (i) the original distribution of the marker array {…, X−2, X−1, X0, X1, X2, …} is a Poisson process on the real line; (ii) each marker is replicated l times; replication or loss of marker points occur independently; (iii) each replicated point is independently and randomly displaced by an amount Y relative to its original position, with the Y displacements sampled from a continuous density g(y). Limiting distributions for the maximal and minimal statistics of the r-scan lengths (collection of distances between r + 1 successive markers) for the l-shift model are derived with the aid of the Chen-Stein method and properties of Poisson processes.

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let Sn=ξ1+⋯+ξn and Mn=maxk≤nSk. Let τ=min{n≥1: Sn≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

By establishing general relationships between branching transformations (Harris-Sevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

In this paper we consider the problem of first-crossing from above for a partial sums process m+St, t ≥ 1, with the diagonal line when the random variables Xt, t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

In this paper we develop a constructive approach to studying continuously and discretely sampled functionals of Lévy processes. Estimates for the rate of convergence of the discretely sampled functionals to the continuously sampled functionals are derived, reducing the study of the latter to that of the former. Laguerre reduction series for the discretely sampled functionals are developed, reducing their study to that of the moment generating function of the pertinent Lévy processes and to that of the moments of these processes in particular. The results are applied to questions of contingent claim valuation, such as the explicit valuation of Asian options, and illustrated in the case of generalized inverse Gaussian Lévy processes.

We study the perpetual American option characteristics in the case where the underlying dynamics involve a Brownian motion and a point process with a stochastic intensity. No assumption on the distribution of the jump size is made and we work with an arbitrary positive or negative jump. After proving the existence of an optimal stopping time for the problem and characterizing it as the hitting time of an optimal boundary, we provide closed-form formulae for the option value, as well as for the Laplace transform of the optimal stopping time. These results are then applied to the analysis of a real option problem when considering the impact of a fundamental and brutal change in the investment project environment. The consequences of this impact, that can seriously modify, positively or negatively, the project's future cash flows and therefore the investment decision, are illustrated by numerical examples.

We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

Based on the concept of multipower variation we establish a class of easily computable and robust estimators for the integrated volatility, especially including the squared integrated volatility, in Lévy-type stochastic volatility models. We derive consistency and feasible distributional results for the estimators. Furthermore, we discuss the applications to time-changed CGMY, normal inverse Gaussian, and hyperbolic models with and without leverage, where the time-changes are based on integrated Cox-Ingersoll-Ross or Ornstein-Uhlenbeck-type processes. We deduce which type of market microstructure does not affect the estimates.

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

We deal with the problem of seating an airplane's passengers optimally, namely in the fastest way. Under several simplifying assumptions, whereby the passengers are infinitely thin and react within a constant time to boarding announcements, we are able to rewrite the asymptotic problem as a calculus of variations problem with constraints. This problem is solved in turn using elementary methods. While the optimal policy is not unique, we identify a rigid discrete structure which is common to all solutions. We also compare the (nontrivial) optimal solutions we find with some simple boarding policies, one of which is shown to be near-optimal.

We consider a weighted random walk model defined as follows. An n-step random walk on the integers with distribution Pn is weighted by giving the path S=(S0,…,Sn) a probability proportional to where the function f is the so-called fitness function. In the case of power-type fitness, we prove the convergence of the renormalized path to a deterministic function with exponential speed. This function is a solution to a variational problem. In the case of the simple symmetric random walk, explicit computations are done. Our result relies on large deviations techniques and Varadhan's integral lemma. We then study an application of this model to mutation-selection dynamics on the integers where a random walk operates the mutation. This dynamics is the infinite-population limit of that of mutation-selection genetic algorithms. We prove that the population grows to ∞ and make explicit its growth speed. This is a toy model for modelling the effect of stronger selection at ∞ for genetic algorithms taking place in a noncompact space.