Matryoshka dolls, the traditional Russian nesting figurines, are known worldwide for each doll’s encapsulation of a sequence of smaller dolls. In this paper, we exploit the structure of a new sequence of nested matrices we call matryoshkan matrices in order to compute the moments of the one-dimensional polynomial processes, a large class of Markov processes. We characterize the salient properties of matryoshkan matrices that allow us to compute these moments in closed form at a specific time without computing the entire path of the process. This simplifies the computation of the polynomial process moments significantly. Through our method, we derive explicit expressions for both transient and steady-state moments of this class of Markov processes. We demonstrate the applicability of this method through explicit examples such as shot noise processes, growth–collapse processes, ephemerally self-exciting processes, and affine stochastic differential equations from the finance literature. We also show that we can derive explicit expressions for the self-exciting Hawkes process, for which finding closed-form moment expressions has been an open problem since their introduction in 1971. In general, our techniques can be used for any Markov process for which the infinitesimal generator of an arbitrary polynomial is itself a polynomial of equal or lower order.

We apply general moment identities for Poisson stochastic integrals with random integrands to the computation of the moments of Markovian growth–collapse processes. This extends existing formulas for mean and variance available in the literature to closed-form moment expressions of all orders. In comparison with other methods based on differential equations, our approach yields explicit summations in terms of the time parameter. We also treat the case of the associated embedded chain, and provide recursive codes in Maple and Mathematica for the computation of moments and cumulants of any order with arbitrary cut-off moment sequences and jump size functions.

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

In this paper we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of distributions, that we call projective, for which we give detailed properties. This allows us to obtain necessary conditions for a given construction to be projective. We illustrate our results by proposing some theoretical projective distributions, as elliptical distributions or a new class of distribution having given bivariate margins. In the case where the data does not necessarily correspond to a projective distribution, we also explain how to build proper distributions while checking that the distance to the prescribed projections is small enough.

In this paper we develop a constructive structure theory for a class of exponential functionals of Brownian motion which includes Asian option values. This is done in two stages of differing natures. As a first step, the functionals are represented as Laguerre reduction series obtained from main results of Schröder (2006), this paper's companion paper. These reduction series are new and given in terms of the negative moments of the integral of geometric Brownian motion, whose structure theory is developed in a second step. Providing a new angle on these processes, this is done by establishing connections with theta functions. Integral representations and computable formulae for the negative moments are thus derived and then shown to furnish highly efficient ways for computing the negative moments. Application of this paper's Laguerre reduction series in numerical examples suggests that one of the most efficient methods for the explicit valuation of Asian options is obtained. The paper also provides mathematical background results referred to in Schröder (2005c).

Suppose where S is a compact set (of say). Let Φ (mapping S into S) and Ψ be continuous on S and such that Ψ(xk) is monotone (non-decreasing, say) as k increases, and xk+1 = Φ(xk). Put Ψ∗ = lim (k → ∞)Ψ(xk); if {xk(i)}∞i=0 is a convergent subsequence of {xk}, with a limit-point α, then Ψ∗ = Ψ(α), and and

The strength and range of interpoint interactions in a spatial point process can be quantified by the function J = (1 - G)/(1 - F), where G is the nearest-neighbour distance distribution function and F the empty space function of the process. J(r) is identically equal to 1 for a Poisson process; values of J(r) smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a very large class of point processes, J(r) is constant for distances r greater than the range of spatial interaction. Hence both the range and type of interpoint interaction may be inferred from J without parametric model assumptions. We evaluate J(r) explicitly for a variety of point processes. The J function of the superposition of independent point processes is a weighted mean of the J functions of the individual processes.

In order to study the transience of Hendricks libraries, we introduce and study a special class of Markov chains, the Tsetlin d-piles, generalizing Tsetlin libraries and briefly defined as follows: a 1-pile is a Tsetlin library and a d-pile is a Tsetlin library where each book is replaced by a (d − 1)-pile. We give a stationary measure of these chains and establish the necessary and sufficient conditions for positive recurrence and transience. Finally, the study of d-piles allows us to determine a sufficient condition for transience of quite a large class of Hendricks libraries.

This paper deals with the stationary bilinear model with Hermite degree 2 in discrete time which is built up by the first- and second-order Hermite polynomial of a Gaussian white noise process. The exact spectrum and bispectrum is constructed in terms of the transfer functions of the model.

The multiplexing of variable bit rate traffic streams in a packet network gives rise to two types of queueing. On a small time-scale, the rates at which the sources send is more or less constant, but there is queueing due to simultaneous packet arrivals (packet-level effect). On a somewhat larger time-scale, queueing is the result of a relatively high number of sources sending at a rate that is higher than their average rate (burst-level effect). This paper explores these effects. In particular, we give asymptotics of the overflow probability in the combined packet/burst scale model. It is shown that there is a specific size of the buffer (i.e. the ‘critical buffer size’) below which packet-scale effects are dominant, and above which burst-scale effects essentially determine the performance—strikingly, there is a sharp demarcation: theso-called ‘phase transition’. The results are asymptotic in the number of sources n. We scale buffer space B and link rate C by n, to nb and nc, respectively; then we let n grow large. Applying large deviations theory we show that in this regime the overflow probability decays exponentially in the number of sources n. For small buffers the corresponding decay rate can be calculated explicitly, for large buffers we derive an asymptote (linear in b). The results for small and large buffers give rise to an approximation for the decay rate (with general b), as well as for the critical buffer size. A numerical example (multiplexing of voice streams) confirms the accuracy of these approximations.

A weighted graph G is a pair (V, ℰ) containing vertex set V and edge set ℰ, where each edge e ∈ ℰ is associated with a weight We. A subgraph of G is a forest if it has no cycles. All forests on the graph G form a probability space, where the probability of each forest is proportional to the product of the weights of its edges. This paper aims to simulate forests exactly from the target distribution. Methods based on coupling from the past (CFTP) and rejection sampling are presented. Comparisons of these methods are given theoretically and via simulation.

We study the harmonic moments of Galton-Watson processes that are possibly inhomogeneous and have positive values. Good estimates of these are needed to compute unbiased estimators for noncanonical branching Markov processes, which occur, for instance, in the modelling of the polymerase chain reaction. By convexity, the ratio of the harmonic mean to the mean is at most 1. We prove that, for every square-integrable branching mechanism, this ratio lies between 1-A/k and 1-A/k for every initial population of size k>A. The positive constants A and Aͤ are such that A≥Aͤ, are explicit, and depend only on the generation-by-generation branching mechanisms. In particular, we do not use the distribution of the limit of the classical martingale associated with the Galton-Watson process. Thus, emphasis is put on nonasymptotic bounds and on the dependence of the harmonic mean upon the size of the initial population. In the Bernoulli case, which is relevant for the modelling of the polymerase chain reaction, we prove essentially optimal bounds that are valid for every initial population size k≥1. Finally, in the general case and for sufficiently large initial populations, similar techniques yield sharp estimates of the harmonic moments of higher degree.

We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

Large samples from a light-tailed distribution often have a well-defined shape. This paper examines the implications of the assumption that there is a limit shape. We show that the limit shape determines the upper quantiles for a large class of random variables. These variables may be described loosely as continuous homogeneous functionals of the underlying random vector. They play an important role in evaluating risk in a multivariate setting. The paper also looks at various coefficients of tail dependence and at the distribution of the scaled sample points for large samples. The paper assumes convergence in probability rather than almost sure convergence. This results in an elegant theory. In particular, there is a simple characterization of domains of attraction.

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.