Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k 1, k 2) denote the number of times that k 1 failures are followed by k 2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k 1, k 2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for N k3 (n; k 1, k 2), the number of times that k 1 failures followed by k 2 successes occur k 3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).

We derive results that show the impact of aggregation in a queueing network. Our model consists of a two-stage queueing system where the first (upstream) queue serves many flows, of which a certain subset arrive at the second (downstream) queue. The downstream queue experiences arbitrary interfering traffic. In this setup, we prove that, as the number of flows being aggregated in the upstream queue increases, the overflow probability of the downstream queue converges uniformly in the buffer level to the overflow probability of a single queueing system obtained by simply removing the upstream queue in the original two-stage queueing system. We also provide the speed of convergence and show that it is at least exponentially fast. We then extend our results to non-i.i.d. traffic arrivals.

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x α F *[i](x) + o(x α ) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x −β/(1−β) G*[i](x) + o (x −β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = x δ/(δ−1)H *[i](x) + o(x δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F *, G * and H * are multiplicatively periodic functions.

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

In Bhatt and Roy's minimal directed spanning tree construction for n random points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

We introduce a class of discrete-time stochastic processes generated by interacting systems of reinforced urns. We show that such processes are asymptotically partially exchangeable and we prove a strong law of large numbers. Examples and the analysis of particular cases show that interacting reinforced-urn systems are very flexible representations for modelling countable collections of dependent and asymptotically exchangeable sequences of random variables.

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.

Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as being discrete and is identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is proved under minimal conditions. As a result, we obtain a formula for effective population size, generalising the well-known harmonic mean expression for effective size.

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

We derive a large-deviation principle for super-Brownian motion with immigration, where the immigration is governed by the Lebesgue measure. We show that the speed function is t 1/2 for d = 1, t/logt for d = 2 and t for d ≥ 3, which is different from that of the occupation-time process counterpart (without immigration) and the model of random immigration.

We consider minimum relative entropy calibration of a given prior distribution to a finite set of moment constraints. We show that the calibration algorithm is stable (in the Prokhorov metric) under a perturbation of the prior and the calibrated distributions converge in variation to the measure from which the moments have been taken as more constraints are added. These facts are used to explain the limiting properties of the minimum relative entropy Monte Carlo calibration algorithm.

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabei et al. for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for an arbitrary level of randomness.

This paper introduces a method of generating real harmonizable multifractional Lévy motions (RHMLMs). The simulation of these fields is closely related to that of infinitely divisible laws or Lévy processes. In the case where the control measure of the RHMLM is finite, generalized shot-noise series are used. An estimation of the error is also given. Otherwise, the RHMLM X h is split into two independent RHMLMs, X ε,1 and X ε,2. More precisely, X ε,2 is an RHMLM whose control measure is finite. It can then be rewritten as a generalized shot-noise series. The asymptotic behaviour of X ε,1as ε → 0+ is further elaborated. Sufficient conditions to approximate X ε,1by a multifractional Brownian motion are given. The error rate in terms of Berry-Esseen bounds is then discussed. Finally, some examples of simulation are given.

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ > 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N(t), where N(·) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

We prove some central limit theorems for a two-level super-Brownian motion with random immigration, which lead to limiting Gaussian random fields. The covariances of those Gaussian fields are explicitly characterized.

We investigate some effects that the ‘light' trimming of a sum Sn = X 1 + X 2 + · ·· + Xn of independent and identically distributed random variables has on behaviour of iterated logarithm type. Light trimming is defined as removing a constant number of summands from Sn. We consider two versions: (r) Sn, which is obtained by deleting the r largest Xi from Sn , and , which is obtained by deleting the r variables Xi which are largest in absolute value from Sn. We summarise some relevant results from Rogozin (1968), Heyde (1969), and later writers concerning the untrimmed sum, and add some new results concerning trimmed sums. Among other things we show that a general form of the law of the iterated logarithm holds for but not (completely) for .

This paper investigates the probabilistic behaviour of the eigenvalue of the empirical transition matrix of a Markov chain which is of largest modulus other than 1, loosely called the second-largest eigenvalue. A central limit theorem is obtained for nonmultiple eigenvalues of the empirical transition matrix. When the Markov chain is actually a sequence of independent observations the distribution of the second-largest eigenvalue is determined and a test for independence is developed. The independence case is considered in more detail when the Markov chain has only two states, and some applications are given.