Consider a random directed graph on n vertices with independent and identically distributed outdegrees with distribution F having mean μ, and destinations of arcs selected uniformly at random. We show that if μ > 1 then for large n there is very likely to be a unique giant strong component with proportionate size given as the product of two branching process survival probabilities, one with offspring distribution F and the other with Poisson offspring distribution with mean μ. If μ ≤ 1 there is very likely to be no giant strong component. We also extend this to allow for F varying with n.

A counterexample is given to demonstrate that a previously proposed characterization of optimal inspection policy for series systems is not correct in the discounted case. A set of valid necessary conditions is provided which leads to the characterization of optimality for a special case. The limiting case of these necessary conditions is also examined.

We consider storage models where the input rate and the demand are modulated by a Markov jump process. One particular example from teletraffic theory is a fluid model of a multiplexer loaded by exponential on-off sources. Although the storage level process has been widely studied, little attention has been paid to the output rate process. We will show that, under certain assumptions, there exists another Markov jump process that modulates the output rate. The modulating process is explicitly constructed. It turns out to be a modification of a GI/G/1 queueing process

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

We present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact simulation in this setting is challenging because of the presence of correlated local-time-like terms in the definition of RBM. We apply recently developed so-called ε-strong simulation techniques (also known as tolerance-enforced simulation) which allow us to provide a piecewise linear approximation to RBM with ε (deterministic) error in uniform norm. A novel conditional acceptance–rejection step is then used to eliminate the error. In particular, we condition on a suitably designed information structure so that a feasible proposal distribution can be applied.

An analogue of the deterministic pandemic threshold theorem for a measles epidemic is established for a viral host-vector epidemic. The severity of the epidemic is shown to be the unique positive solution of a pair of equations.

Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

Let (Zn) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn) consider the right eigenvector u = (u1, · ··, up) > 0, and set . It is shown that lim inf, lim sup whenever Z0 = i, where .

Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on independent and identically distributed vertex weights and the Euclidean distance $|x-y|$ . Depending on the various parameters involved, we get a rich phase diagram. We study graph distance and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.

For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

Consider a sequence of Markov-dependent trials where each trial produces a letter of a finite alphabet. Given a collection of patterns, we look at this sequence until one of these patterns appears as a run. We show how the method of gambling teams can be employed to compute the probability that a given pattern is the first pattern to occur.

We study the convergence of centered and normalized sums of independent and identically distributed random elements of the space D of càdlàg functions endowed with Skorokhod's J1 topology, to stable distributions in D. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.

Admission control can be employed to avoid congestion in queueing networks subject to overload. In distributed networks, the admission decisions are often based on imperfect measurements on the network state. In this paper, we study how the lack of complete state information affects the system performance, by considering a simple network model for distributed admission control. The stability region of the network is characterized and it is shown how feedback signaling makes the system very sensitive to its parameters.

We consider a diffusion process X(t) with a one-sided Brownian potential starting from the origin. The limiting behavior of the process as time goes to infinity is studied. For each t > 0, the sample space describing the random potential is divided into two parts, Ãt and B̃t, both having probability ½, in such a way that our diffusion process X(t) exhibits quite different limiting behavior depending on whether it is conditioned on Ãt or on B̃t (t → ∞). The asymptotic behavior of the maximum process of X(t) is also investigated. Our results improve those of Kawazu, Suzuki, and Tanaka (2001).

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus Td. In the discrete torus N-1TdN, we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is ej, then the exchange rate is defined as N-1 times the absolute value of the inner product between ej and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.

The purpose of this paper is to study an optimal stopping problem with constraints for a Markov chain with general state space by using the convex analytic approach. The costs are assumed to be nonnegative. Our model is not assumed to be transient or absorbing and the stopping time does not necessarily have a finite expectation. As a consequence, the occupation measure is not necessarily finite, which poses some difficulties in the analysis of the associated linear program. Under a very weak hypothesis, it is shown that the linear problem admits an optimal solution, guaranteeing the existence of an optimal stopping strategy for the optimal stopping problem with constraints.

The limiting behavior of the probability of extinction of critical generation-dependent branching processes is determined. Lower bounds are determined for this probability. A means of classifying these branching processes is suggested.

We provide upper and lower bounds for the mean $\mathscr{M}(H)$ of $\sup_{t\geq 0} \{B_H(t) - t\}$ , with $B_H(\!\cdot\!)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$ . We find bounds in (semi-) closed form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$ , where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$ , the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$ , $H\in(0,\frac{1}{2}]$ , which is tight around $H=\frac{1}{2}$ .

The classical exponential smoothing procedure to be applied to a sequence of observations is modified into some adaptive variants and also generalized to the recursive scheme

This means that given the history of observations and smoothed values up to time n, the updated smoothed value Wn +1 is given by a function u of Wn and Xn, which satisfies only certain structural properties. We study these procedures within the framework of random systems with complete connections and exploit results from Markov process theory to study the sequence . A broad class of ‘smoothing functions' u, which induce sequences with nice statistical properties, is presented. A method to estimate the limit of the expected smoothed values is also developed.,

We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion Ex[exp(-∫0τr(Xs)ds)f(Xτ)] over all stopping times τ, where the reward function f can take only a finite number of values and has a ‘staircase’ form. This problem is partly motivated by applications to financial asset pricing. Our results are of an explicit analytic nature and completely characterise the optimal stopping time. Also, it turns out that the problem's value function is not C1, which is due to the fact that the reward function f is not continuous.