A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1−F is regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1 and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.

Conditions that have been widely used to establish stochastic ordering of random vectors include those given by Kamae et al. (1977), Barlow and Proschan (1975) and Langberg (1988). In this note we offer conditions that are weaker than those above, thus providing potential application to a wider class of problems.

We investigate the stationarity of minification processes when the marginal is a discrete distribution. There is a close relationship between the problem considered by Arnold and Isaacson (1976) and the stationarity in minification processes. We give a necessary and sufficient condition for a discrete distribution to be the marginal of a stationary minification process. Members of the Poisson and negative binomial families can be the marginals of stationary minification processes. The geometric minification process is studied in detail, and two characterizations of it based on the structure of the innovation process are given.

In this note we introduce recurring-with-carry sequences which generalize add-with-carry and subtract-with-borrow sequences introduced in [1], and describe periods of admissible recurring-with-carry sequences which include add-with-carry and subtract-with-borrow sequences with a few exceptions.

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numéraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numéraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself.

The key theorem of general numéraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.

Nash (1980) demonstrated that index policies are optimal for a class of generalised bandit problem. A transform of the index concerned has many of the attributes of the Gittins index. The transformed index is positive-valued, with maximal values yielding optimal actions. It may be characterised as the value of a restart problem and is hence computable via dynamic programming methodologies. The transformed index can also be used in procedures for policy evaluation.

Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.

We present a method of deriving the limiting distributions of the number of occurrences of success (S) runs of length k for all types of runs under the Markovian structure with stationary transition probabilities. In particular, we consider the following four bestknown types. 1. A string of S of exact length k preceded and followed by an F, except the first run which may not be preceded by an F, or the last run which may not be followed by an F. 2. A string of S of length k or more. 3. A string of S of exact length k, where recounting starts immediately after a run occurs. 4. A string of S of exact length k, allowing overlapping runs. It is shown that the limits are convolutions of two or more distributions with one of them being either Poisson or compound Poisson, depending on the type of runs in question. The completely stationary Markov case and the i.i.d. case are also treated.

We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of the nth composition of these maps in the Wasserstein metric via a contraction argument. The contraction condition involves the operator norm of the expectation of a bilinear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to check. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.

We consider a discrete-time stochastically perturbed dynamical system on the Polish space given by the recurrence formula Xn = S(Xn–1, Yn), where Yn are i.i.d. random elements. We prove the existence of unique stationary measure and versions of classical limit theorems for the process (Xn).

Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.

Records from are analyzed, where {Yj} is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

This paper presents a software release policy based on a two-person game of timing. Existing release policies depend solely on cost factors and ignore the element of competition between rival producers, whereas in our policy both of these factors are taken into consideration. Through a series of preliminary results, it is shown that an optimal release policy exists as a Nash equilibrium point in the space of mixed strategies. We also present numerical examples of this optimal policy applied to software reliability growth models which are based on the non-homogeneous Poisson process.

A class of non-linear stochastic models is introduced. Phase transitions, critical points and the domain of attraction are discussed in detail for some concrete examples. For one of the examples the explicit expression for the domain of attraction and the rates of convergence are obtained.

The paper deals with processes in (or ), one of whose components is skip-free. We obtain identities for distributions of hitting times for the components of the process generalizing the well-known one for the one-dimensional case. These relations reflect the fact that in this case spatial and time coordinates play, in some sense, symmetric roles. They turn out to be useful for solving several problems. For example, they allow us to find the distribution of the number of jumps of the process, which fall in a fixed set before the skip-free component of the process hits a fixed level. Examples are given showing how our results can be applied to models in branching processes, queueing, and risk theory.

We introduce a model for the healing process of a destroyed region and use some relations to the coverage models. The restoring model depends on the region which is hit n times at random. Each random part of the region which is destroyed is restored at a certain fixed speed, but the restoring process may start after a random delay. We focus mainly on the total healing time until the whole region is restored, and analyse its limit distribution as n tends to ∞. The dependence of this limit distribution on the different ingredients is of interest. We consider two cases with different geometrical influence. One case is restricted to a one-dimensional region, that means the circumference of a circle or the unit interval as region. The results follow from extreme value theory. We discuss also some particular cases and examples.

A new partial ordering among life distributions in terms of their uncertainties is introduced. Our measure of uncertainty is Shannon information applied to the residual lifetime. The relationship between this ordering and various existing orderings of life distributions are discussed. Various properties of our proposed concept are examined. Based on our proposed ordering and various existing orderings, the notion of a ‘better system' is introduced.

Availability is an important characteristic of a system. Different types of availability are defined. For the case when a sequence of bivariate random variables of lifetime and repair time are i.i.d. certain properties have been established previously. In practice, however, we need to consider the situation where these bivariate random variables are independent but not identically distributed. Properties of two measures of availability for the i.i.d. case are extended to this more general case.