In this paper we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a partial differential equation, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for a more accurate comparison among competing importance sampling schemes.

We study the asymptotics of the survival probability for the critical and decomposable branching processes in a random environment and prove Yaglom-type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in a constant environment.

For any strictly positive martingale S = eX for which X has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log-strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model, Merton (1976), one infinite activity exponential Lévy model (variance gamma), and one stochastic volatility model, Heston (1993). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

We take a fresh look at the classical problem of runs in a sequence of independent and identically distributed coin tosses and derive a general identity/recursion which can be used to compute (joint) distributions of functionals of run types. This generalizes and unifies already existing approaches. We give several examples, derive asymptotics, and pose some further questions.

A random k-out mapping (digraph) on [n] is generated by choosing k random images of each vertex one at a time, subject to a 'preferential attachment' rule: the current vertex selects an image i with probability proportional to a given parameter α = α(n) plus the number of times i has already been selected. Intuitively, the larger α becomes, the closer the resulting k-out mapping is to the uniformly random k-out mapping. We prove that α = Θ(n1/2) is the threshold for α growing 'fast enough' to make the random digraph approach the uniformly random digraph in terms of the total variation distance. We also determine an exact limit for this distance for the α = βn1/2 case.

We study a multistage epidemic model which generalizes the SIR model and where infected individuals go through K ≥ 1 stages of the epidemic before being removed. An infected individual in stage k ∈ {1, …, K} may infect a susceptible individual, who directly goes to stage k of the epidemic; or it may go to the next stage k + 1 of the epidemic. For this model, we identify the critical regime in which we establish diffusion approximations. Surprisingly, the limiting diffusion exhibits an unusual form of state space collapse which we analyze in detail.

Retransmission-based failure recovery represents a primary approach in existing communication networks that guarantees data delivery in the presence of channel failures. Recent work has shown that, when data sizes have infinite support, retransmissions can cause long (-tailed) delays even if all traffic and network characteristics are light-tailed. In this paper we investigate the practically important case of bounded data units 0 ≤ Lb ≤ b under the condition that the hazard functions of the distributions of data sizes and channel statistics are proportional. To this end, we provide an explicit and uniform characterization of the entire body of the retransmission distribution ℙ[Nb > n] in both n and b. Our main discovery is that this distribution can be represented as the product of a power law and gamma distribution. This rigorous approximation clearly demonstrates the coupling of a power law distribution, dominating the main body, and the gamma distribution, determining the exponential tail. Our results are validated via simulation experiments and can be useful for designing retransmission-based systems with the required performance characteristics. From a broader perspective, this study applies to any other system, e.g. computing, where restart mechanisms are employed after a job processing failure.

For triangular arrays $\{X_{n,k}:1\leqslant k\leqslant n,n\geqslant 1\}$ of upper extended negatively dependent random variables weakly mean dominated by a random variable $X$ and sequences $\{b_{n}\}$ of positive constants, conditions are given to guarantee an almost sure finite upper bound to $\sum _{k=1}^{n}(X_{n,k}-\mathbb{E}X_{n,k})/\!\sqrt{b_{n}\,\text{Log}\,n}$, where $\text{Log}\,n:=\max \{1,\log n\}$, thus getting control over the limiting rate in terms of the prescribed sequence $\{b_{n}\}$ and permitting us to weaken or strengthen the assumptions on the random variables.

In [1], the authors consider a random walk (Zn,1, . . ., Zn,K+1) ∈ ${\mathbb{Z}}$K+1 with the constraint that each coordinate of the walk is at distance one from the following coordinate. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with that obtained in [1].

We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

Consider a real-valued discrete-time stationary and ergodic stochastic process, called the noise process. For each dimension n, we can choose a stationary point process in ℝn and a translation invariant tessellation of ℝn. Each point is randomly displaced, with a displacement vector being a section of length n of the noise process, independent from point to point. The aim is to find a point process and a tessellation that minimizes the probability of decoding error, defined as the probability that the displaced version of the typical point does not belong to the cell of this point. We consider the Shannon regime, in which the dimension n tends to ∞, while the logarithm of the intensity of the point processes, normalized by dimension, tends to a constant. We first show that this problem exhibits a sharp threshold: if the sum of the asymptotic normalized logarithmic intensity and of the differential entropy rate of the noise process is positive, then the probability of error tends to 1 with n for all point processes and all tessellations. If it is negative then there exist point processes and tessellations for which this probability tends to 0. The error exponent function, which denotes how quickly the probability of error goes to 0 in n, is then derived using large deviations theory. If the entropy spectrum of the noise satisfies a large deviations principle, then, below the threshold, the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is obtained for two classes of point processes: the Poisson process and a Matérn hard-core point process. New lower bounds on error exponents are derived from this for Shannon's additive noise channel in the high signal-to-noise ratio limit that hold for all stationary and ergodic noises with the above properties and that match the best known bounds in the white Gaussian noise case.

For a Markov two-dimensional death-process of a special class we consider the use of Fourier methods to obtain an exact solution of the Kolmogorov equations for the exponential (double) generating function of the transition probabilities. Using special functions, we obtain an integral representation for the generating function of the transition probabilities. We state the expression of the expectation and variance of the stochastic process and establish a limit theorem.

Let X1, X2, …, Xn be independent and uniformly distributed random variables in the unit square [0, 1]2, and let L(X1, …, Xn) be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton and Hammersley proved the existence of a universal constant β such that limn→∞n−1/2L(X1, …, Xn) = β almost surely. The best bounds for β are still those originally established by Beardwood, Halton and Hammersley, namely 0.625 ≤ β ≤ 0.922. We slightly improve both upper and lower bounds.

A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routeing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a leaf activity priority (LAP) policy, introduced by Stolyar and Yudovina (2012). An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to ∞ in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by r−1/2) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U[0, 1]d random particle. We show that the limiting configuration contains N − 1 coincident particles at a random location ξN ∈ [0, 1]d. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN, showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.

Let W = {Wn: n ∈ N} be a sequence of random vectors in Rd, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ∈ N: Wnuq) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, (1 / vn)logP(Wn / an ≥ uq) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components of the original random structure. In this paper we build on previous work, and derive results in the multivariate case and in situations where regular variation is not restricted to one particular direction or quadrant.

This paper provides tools for the study of the Dirichlet random walk in Rd. We compute explicitly, for a number of cases, the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere of Rd.