In this paper we study efficient simulation algorithms for estimating P(X›x), where X is the total time of a job with ideal time T that needs to be restarted after a failure. The main tool is importance sampling, where a good importance distribution is identified via an asymptotic description of the conditional distribution of T given X›x. If T≡t is constant, the problem reduces to the efficient simulation of geometric sums, and a standard algorithm involving a Cramér-type root, γ(t), is available. However, we also discuss an algorithm that avoids finding the root. If T is random, particular attention is given to T having either a gamma-like tail or a regularly varying tail, and to failures at Poisson times. Different types of conditional limit occur, in particular exponentially tilted Gumbel distributions and Pareto distributions. The algorithms based upon importance distributions for T using these asymptotic descriptions have bounded relative error as x→∞ when combined with the ideas used for a fixed t. Nevertheless, we give examples of algorithms carefully designed to enjoy bounded relative error that may provide little or no asymptotic improvement over crude Monte Carlo simulation when the computational effort is taken into account. To resolve this problem, an alternative algorithm using two-sided Lundberg bounds is suggested.

Large deviation principles and related results are given for a class of Markov chains associated to the ‘leaves' in random recursive trees and preferential attachment random graphs, as well as the ‘cherries’ in Yule trees. In particular, the method of proof, combining analytic and Dupuis–Ellis-type path arguments, allows for an explicit computation of the large deviation pressure.

This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.

Convolutions of long-tailed and subexponential distributions play a major role in the analysis of many stochastic systems. We study these convolutions, proving some important new results through a simple and coherent approach, and also showing that the standard properties of such convolutions follow as easy consequences.

We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks (X, Y) such that P(X + Y > x) ∼ (constant) P(X > x). With the further assumption of nonnegativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of X and Y are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored, leading to an approximate solution of an optimization problem which is applied to portfolio design.

In this paper we introduce the transformed two-parameter Poisson–Dirichlet distribution on the ordered infinite simplex. Furthermore, we prove the central limit theorem related to this distribution when both the mutation rate θ and the selection rate σ become large in a specified manner. As a consequence, we find that the properly scaled homozygosities have asymptotical normal behavior. In particular, there is a certain phase transition with the limit depending on the relative strength of σ and θ.

In Lyons, Pemantle and Peres (1995), a martingale change of measure method was developed in order to give an alternative proof of the Kesten–Stigum L log L theorem for single-type branching processes. Later, this method was extended to prove the L log L theorem for multiple- and general multiple-type branching processes in Biggins and Kyprianou (2004), Kurtz et al. (1997), and Lyons (1997). In this paper we extend this method to a class of superdiffusions and establish a Kesten–Stigum L log L type theorem for superdiffusions. One of our main tools is a spine decomposition of superdiffusions, which is a modification of the one in Englander and Kyprianou (2004).

An almost sure limit theorem for the maxima of multivariate stationary Gaussian sequences is proved under some mild conditions.

Consider an urn model whose replacement matrix is triangular, has all nonnegative entries, and the row sums are all equal to 1. We obtain strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use these strong laws to study further behavior of certain three-color urn models.

We show that, contrary to common wisdom, the cumulative input process in a fluid queue with cluster Poisson arrivals can converge, in the slow growth regime, to a fractional Brownian motion, and not to a Lévy stable motion. This emphasizes the lack of robustness of Lévy stable motions as ‘birds-eye’ descriptions of the traffic in communication networks.

In this paper we refine a Poisson limit theorem of Gnedenko and Kolmogorov (1954): we determine the error order of a Poisson approximation for sums of asymptotically negligible integer-valued random variables that converge in distribution to the Poisson law. As an application of our results, we investigate the case of the coupon collector's problem when the distribution of the collector's waiting time is asymptotically Poisson.

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

The speed of extinction for some generalized Jiřina processes {Xn} is discussed. We first discuss the geometric speed. Under some mild conditions, the results reveal that the sequence {cn}, where c does not equal the pseudo-drift parameter at x = 0, cannot estimate the speed of extinction accurately. Then the general case is studied. We determine a group of sufficient conditions such that Xn/cn, with a suitable constant cn, converges almost surely as n → ∞ to a proper, nondegenerate random variable. The main tools used in this paper are exponent martingales and stochastic growth models.

Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati, and others.

Let n points be chosen independently and uniformly in the unit cube [0,1]d, and suppose that each point is supplied with a mark, the marks being independent and identically distributed random variables independent of the location of the points. To each cube R contained in [0,1]d we associate its score defined as the sum of marks of all points contained in R. The scan statistic is defined as the maximum of taken over all cubes R contained in [0,1]d. We show that if the marks are nonlattice random variables with finite exponential moments, having negative mean and assuming positive values with nonzero probability, then the appropriately normalized distribution of the scan statistic converges as n → ∞ to the Gumbel distribution. We also prove a corresponding result for the scan statistic of a Lévy noise with negative mean. The more elementary cases of zero and positive mean are also considered.

The main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.

Let be a collection of n uniform, independent, and identically distributed points on the Cantor ternary set. We consider the asymptotics for the expected total edge length of the directed and undirected nearest-neighbor graph on We prove convergence to a constant of the rescaled expected total edge length of this random graph. The rescaling factor is a function of the fractal dimension and has a log-periodic, nonconstant behavior.

Mallows and Shepp (2008) developed the following necklace processes. Start with a necklace consisting of one white bead and one black bead, and insert, one at a time, under a deterministic rule, a white bead or a black bead between a randomly chosen adjacent pair. They studied the statistical properties of the number of white beads by investigating the nature of the moments and the expected number of gaps of given length between white beads. In this note we study the number of white beads via Pólya urns and give a classification of necklace processes for some general rules. Additionally, we discuss the number of runs, i.e. the number of consecutive same color beads, instead of the number of gaps.

We study a process where balls are repeatedly thrown into n boxes independently according to some probability distribution p. We start with n balls, and at each step, all balls landing in the same box are fused into a single ball; the process terminates when there is only one ball left (coalescence). Let c := ∑jpj2, the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c−1, under two constraints on p that exclude a thin set of distributions p. One of the constraints is c = o(ln−2n). This ln−2n is shown to be a threshold value: for c = ω(ln−2n), there exists p with c(p) = c such that the expected coalescence time far exceeds c−1. Connections to coalescent processes in population biology and theoretical computer science are discussed.

We study sample covariance matrices of the form W = (1 / n)CCT, where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or kn → ∞ with kn = o(n / log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C. This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.