Scaling of proposals for Metropolis algorithms is an important practical problem in Markov chain Monte Carlo implementation. Analyses of the random walk Metropolis for high-dimensional targets with specific functional forms have shown that in many cases the optimal scaling is achieved when the acceptance rate is approximately 0.234, but that there are exceptions. We present a general set of sufficient conditions which are invariant to orthonormal transformation of the coordinate axes and which ensure that the limiting optimal acceptance rate is 0.234. The criteria are shown to hold for the joint distribution of successive elements of a stationary pth-order multivariate Markov process.

We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/N/N queue in a many-server heavy traffic regime.

The problem of detecting an abrupt change in the distribution of an arbitrary, sequentially observed, continuous-path stochastic process is considered and the optimality of the CUSUM test is established with respect to a modified version of Lorden's criterion. We apply this result to the case that a random drift emerges in a fractional Brownian motion and we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H+1/2.

Sharp tail bounds for the sum of d random variables with given marginal distributions and arbitrary dependence structure have been known since Makarov (1981) and Rüschendorf (1982) for d=2 and, in some examples, for d≥3. Based on a duality result, dual bounds have been introduced in Embrechts and Puccetti (2006b). In the homogeneous case,\break $F₁=···=F_n, with monotone density, sharp tail bounds were recently found in Wang and Wang (2011). In this paper we establish the sharpness of the dual bounds in the homogeneous case under general conditions which include, in particular, the case of monotone densities and concave densities. We derive the corresponding optimal couplings and also give an effective method to calculate the sharp bounds.

In this paper we study limit theorems for the Feller game which is constructed from one-dimensional simple symmetric random walks, and corresponds to the St. Petersburg game. Motivated by a generalization of the St. Petersburg game which was investigated by Gut (2010), we generalize the Feller game by introducing the parameter α. We investigate limit distributions of the generalized Feller game corresponding to the results of Gut. Firstly, we give the weak law of large numbers for α=1. Moreover, for 0<α≤1, we have convergence in distribution to a stable law with index α. Finally, some limit theorems for a polynomial size and a geometric size deviation are given.

We study the exact asymptotics for the distribution of the first time, τ_x, a Lévy process X_t crosses a fixed negative level -x. We prove that ℙ{τ_x >t} ~V(x) ℙ{X_t≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τ_x>t} explicitly in both light- and heavy-tailed cases.

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state (i₀,j₀), we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive a certain integral representation for the probability of this event, and an asymptotic expression for the case when i₀ becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model, and the asymmetric exclusion process.

Let {X(t):t∈ℝ} be the integrated on–off process with regularly varying on-periods, and let {Y(t):t∈ℝ} be a centered Lévy process with regularly varying positive jumps (independent of X(·)). We study the exact asymptotics of ℙ(sup_t≥0{X(t)+Y(t)-ct}>u) as u→∞, with special attention to the case r=c, where r is the increase rate of the on–off process during the on-periods.

We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λ_n,n≥0, and death coefficients μ_n,n≥0. If we define A=∑_n=1^∞ 1/λ_nπ_n and S=∑_n=1^∞ (1/λ_nπ_n)∑_i=n+1^∞ π_i, where {π_n,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λ_C>0 and there is precisely one quasistationary distribution, namely, {a_j(λ_C)}, where λ_C is the decay parameter of {X(t),t≥0} in C={1,2,...} and a_j(x)≡μ₁^-1π_jxQ_j(x), j=1,2,... . In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={m_i, i=1,2,...}, we have lim_t\→∞ℙ_M(X(t)=j∣ T>t)= a_j(λ_C), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.

In this paper we quantify the efficiency of parallelism in systems that are prone to failures and exhibit power law processing delays. We characterize the performance of two prototype schemes of parallelism, redundant and split, in terms of both the power law exponent and exact asymptotics of the delay distribution tail. We also develop the optimal splitting scheme which ensures that split always outperforms redundant.

Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.

Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

In this paper, new conditions for the stability of V-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with independent and identically distributed noises (with a nonstandard probability density function) is obtained.

We study the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtain explicit formulae for the Laplace transforms associated with the two-sided exit problem. The expected present value of tax payments until default, the two-sided exit probabilities, and, hence, the nondefault probability with the default threshold equal to the lower bound are solved as immediate corollaries. A sufficient and necessary condition for the tax identity in ruin theory is discovered.

We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let N_t denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (N_t, T, X^(T)), where X^(T) is the jumping contour process of the tree truncated at time T, is equal to that of (M, -I_M, Y'_M) conditional on {M≠0}. Here M+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; I_M is the infimum of the path Y_M, which in turn is defined as Y killed at its last visit of 0 before e; and Y'_M is the Vervaat transform of Y_M. This identity yields an explanation for the geometric distribution of N_T (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {N_T=n}, and also on {N_T=n,T<a}, the ages and residual lifetimes of the n alive individuals at time T are i.i.d. and independent of n. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital.

We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ>0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ=1, the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n→∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived.

Gene conversion is a genetic mechanism by which one gene is `copied and pasted' onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within a d-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with a d-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.

Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.

In this note we find a new result concerning the asymptotic expected number of passages of a finite or infinite interval (x,x+h) as x→∞ for a random walk with increments having a positive expected value. If the increments are distributed like X then the limit for 0<h<∞ turns out to have the form Emin(|X|,h)/EX, which unexpectedly is independent of h for the special case where |X|≤b<∞ almost surely and h>b. When h=∞, the limit is Emax(X,0)/EX. For the case of a simple random walk, a more pedestrian derivation of the limit is given.

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.