In this article we define the s-order equilibrium distribution of a life distribution and establish some interesting properties regarding moments for s-order equilibrium distributions. Results are expected to be useful in reliability and renewal processes.

Let denote the class of local subexponential distributions and F∗ν the ν-fold convolution of distribution F, where ν belongs to one of the following three cases: ν is a random variable taking only a finite number of values, in particular ν ≡ n for some n ≥ 2; ν is a Poisson random variable; or ν is a geometric random variable. Along the lines of Embrechts, Goldie, and Veraverbeke (1979), the following assertion is proved under certain conditions: This result is applied to the infinitely divisible laws and some new results are established. The results obtained extend the corresponding findings of Asmussen, Foss, and Korshunov (2003).

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.

Approximate and asymptotic distributional results are obtained for the likelihood ratio test of the hypothesis that a time series is composed from s sinusoidal components, at unknown frequencies, with additive Gaussian white noise, against the hypothesis that there are an additional r sinusoidal components at unknown frequencies. The work extends that of Fisher (1929), and contains a number of simulations illustrating the results.

We study the measure of a typical cell in a Voronoi tessellation defined by n independent random points X1, . . ., Xn drawn from an absolutely continuous probability measure μ with density f in ℝd. We prove that the asymptotic distribution of the measure – with respect to dμ = f(x)dx – of the cell containing X1 given X1 = x is independent of x and the density f. We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as d becomes large. In particular, we show that the variance converges to 0 exponentially fast in d. We also obtain a bound independent of the density for the rate of convergence of the diameter of a typical Voronoi cell.

We investigate the nature of the set of emptiness times of a dam whose release rate depends on the content and whose cumulative input process is a pure-jump Lévy process. Detailed results are obtained for stable input processes and release functions of the form r(x) = xβ I(o,∞)(x).

Jump points in curves arise when the conditions under which data are generated change suddenly, for example because of an unplanned change in a treatment. This paper suggests bootstrap methods for quantifying the error in estimates of jump points, and for constructing confidence intervals for jump points and confidence bands for the curve. These problems have the unusual feature that the sampling error of the jump-point estimator often has a highly non-normal distribution, which depends intimately on the distribution of regression errors. The methods are illustrated by a simulation study as well as by an application to data on the annual flow volume of the Nile river.

In this paper recent results by Weiner [10] on Mn:= max{Z0, · ··, Zn} are strengthened and generalized, where (Zn)n is a critical Galton–Watson branching process with finite and positive offspring variance and Z0 ≡ 1. It is shown that Explicit asymptotic bounds are given for with . If (Zn)n has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t)t. Using martingale methods one obtains thereof.

These results generalize to the case Z0 ≡ k.

We consider a general homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such processes comes from modeling human and animal mobility patterns, restart processes in communication protocols, and from application of restarting random walks in information retrieval. We provide a connection between the transition probability functions of the original Markov process and the modified process with restarts. We give closed-form expressions for the invariant probability measure of the modified process. When the process evolves on the Euclidean space, there is also a closed-form expression for the moments of the modified process. We show that the modified process is always positive Harris recurrent and exponentially ergodic with the index equal to (or greater than) the rate of restarts. Finally, we illustrate the general results by the standard and geometric Brownian motions.

An asymptotic expression for an absorption probability for the Ornstein-Uhlenbeck process is presented along with an application of the result to a problem in optional stopping. The relation of this result to the asymptotic behavior of a weighted Kolmogorov-Smirnov statistic is also discussed. Sweet and Hardin (1970) derive an exact solution (not in closed form) for this same problem.

We consider a monotone case of the non-zero-sum stopping game with discrete time parameter which is called the Dynkin problem. Marner (1987) has investigated a stopping game with general monotone reward structures, but his monotonicity is too strong to apply our problem. We establish that there exists an explicit equilibrium point in our monotone case. We also give a simple example applicable to a duopolistic exit game.

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.

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.

For an aperiodic, irreducible Markov chain with the non-negative integers as state space, a criterion for ergodicity is given. This criterion generalizes the criteria appearing in Foster (1953), Pakes (1969) and Marlin (1973), in the sense that any test function (Liapunov function) which satisfies their conditions also satisfies ours. Applications are presented through some examples.

In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. The value of this sum has become known as Kemeny’s constant. Various proofs have been given over time, some more technical than others. We give here a very simple physical justification, which extends without a hitch to continuous-time Markov chains on a finite state space. For Markov chains with denumerably infinite state space, the constant may be infinite and even if it is finite, there is no guarantee that the physical argument will hold. We show that the physical interpretation does go through for the special case of a birth-and-death process with a finite value of Kemeny’s constant.

This paper extends the rate conservation principle to cadlag processes whose jumps form a non-stationary point process with a time-dependent intensity. It is shown that this is a direct consequence of path integration and the strong law of large numbers for local martingales. When specialized to mean rates a non-stationary version of Miyazawa's result is obtained which is recovered in the stationary case. Some applications of the result are also given.

We consider the problem of finding a betting strategy for an infinite sequence of wagers where the optimality criterion is the minimization of the expected exit time of wealth from an interval. We add the side constraint that the right boundary is hit first with at least some specified probability. The optimal strategy is derived for a diffusion approximation.