The cross entropy is a well-known adaptive importance sampling method which requires estimating an optimal importance sampling distribution within a parametric class. In this paper we analyze an alternative version of the cross entropy, where the importance sampling distribution is selected instead within a general semiparametric class of distributions. We show that the semiparametric cross entropy method delivers efficient estimators in a wide variety of rare-event problems. We illustrate the favourable performance of the method with numerical experiments.

We call `bits' a sequence of devices indexed by positive integers, where every device can be in two states: 0 (idle) and 1 (active). Start from the `ground state' of the system when all bits are in 0-state. In our first binary flipping (BF) model the evolution of the system behaves as follows. At each time step choose one bit from a given distribution P on the positive integers independently of anything else, then flip the state of this bit to the opposite state. In our second damaged bits (DB) model a `damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both the BF and DB models to show recurrent or transient behaviour, depending on the properties of the distribution P. We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a central limit theorem for the number of active bits for both models.

A large number n of sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional to n. In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections, and weak convergence, we investigate asymptotic behaviour of optimal costs as n increases to ∞. The introduced methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.

We consider continuous review inventory systems with general doubly stochastic Poisson demand. In this specific case the demand rate, experienced by the system, varies as a function of the age of the oldest unit in the system. It is known that the stationary distributions of the ages in such models often have a strikingly simple form. In particular, they exhibit a typical feature of a Poisson process: given the age of the oldest unit the remaining ages are uniform. The model we treat here generalizes some known inventory models dealing with partial backorders, perishable items, and emergency replenishment. We derive the limiting joint density of the ages of the units in the system by solving partial differential equations. We also answer the question of the uniqueness of the stationary distributions which was not addressed in the related literature.

We consider compound geometric approximation for a nonnegative, integer-valued random variable W. The bound we give is straightforward but relies on having a lower bound on the failure rate of W. Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth–death processes and Poisson processes.

In this paper we estimate quantile sensitivities for dependent sequences via infinitesimal perturbation analysis, and prove asymptotic unbiasedness, weak consistency, and a central limit theorem for the estimators under some mild conditions. Two common cases, the regenerative setting and ϕ-mixing, are analyzed further, and a new batched estimator is constructed based on regenerative cycles for regenerative processes. Two numerical examples, the G/G/1 queue and the Ornstein–Uhlenbeck process, are given to show the effectiveness of the estimator.

The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log_1∕pn and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

We consider a Markov chain (M_n)_n≥0 on the set ℕ₀ of nonnegative integers which is eventually decreasing, i.e. ℙ{M_n+1<M_n | M_n≥a}=1 for some a∈ℕ and all n≥0. We are interested in the asymptotic behavior of the law of the stopping time T=T(a)≔inf{k∈ℕ₀: M_k<a} under ℙ_n≔ℙ (· | M₀=n) as n→∞. Assuming that the decrements of (M_n)_n≥0 given M₀=n possess a kind of stationarity for large n, we derive sufficient conditions for the convergence in the minimal L^p-distance of ℙ_n(T−a_n)∕bn∈·) to some nondegenerate, proper law and give an explicit form of the constants a_n and b_n.

The susceptible→exposed→infectious→susceptible (SEIS) model is well known in mathematical epidemiology as a model of infection in which there is a latent period between the moment of infection and the onset of infectiousness. The compartment model is well studied, but the corresponding particle system has so far received no attention. For the particle system model in one spatial dimension, we give upper and lower bounds on the critical values, prove convergence of critical values in the limit of small and large latent time, and identify a limiting process to which the SEIS model converges in the limit of large latent time.

Consider a d-type (d<∞) Galton–Watson branching process, conditioned on the event that there are at least k≥2 individuals in the nth generation, pick k individuals at random from the nth generation and trace their lines of descent backward in time till they meet. In this paper, the limit behaviors of the distributions of the generation number of the most recent common ancestor of any k chosen individuals and of the whole population are studied for both critical and subcritical cases. Also, we investigate the limit distribution of the joint distribution of the generation number and their types.

In this paper we study a reflected AR(1) process, i.e. a process (Z_n)_n obeying the recursion Z_n+1= max{aZ_n+X_n,0}, with (X_n)_n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z_n (in terms of transforms) in case X_n can be written as Y_n−B_n, with (B_n)_n being a sequence of independent random variables which are all Exp(λ) distributed, and (Y_n)_n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B_n)_n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate the distribution of the minimum number of hops that are needed to reach some point of the base station process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the base station process as its intensity tends to 0. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the base station process in a given number of hops.

Let r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯_d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯_d,t has less than t^δ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t^𝛿, which verifies a conjecture of Cooper and Frieze (2015).

We obtain a complete description of anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian, and ‘intermediate’ infinitely divisible random fields. The asymptotic form of the covariance function of the random grain model is obtained. Application to superimposed network traffic is included.

We consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy-tailed steps, the limiting behavior of extreme observations until a given time t tends to be rather involved. We describe the asymptotics and extend several partial results which appeared in this setting. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous-time random walks.

We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color, given the past, is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.

We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n≥1, choose a ball from the urn uniformly at random. With probability ½<p<1, return the ball to the urn along with another ball of the same colour. With probability 1−p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri (2015), (2016) and Thörnblad (2015). We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1−p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p=1).

We consider translation-invariant, finite-range, supercritical contact processes. We show the existence of unbounded space-time cones within which the descendancy of the process from full occupancy may with positive probability be identical to that of the process from the single site at its apex. The proof comprises an argument that leans upon refinements of a successful coupling among these two processes, and is valid in d-dimensions.

We consider the coupon collection problem, where each coupon is one of the types 1,…,s with probabilities given by a vector 𝒑. For specified numbers r₁,…,r_s, we are interested in finding 𝒑 that minimizes the expected time to obtain at least r_i type-i coupons for all i=1,…,s. For example, for s=2, r₁=1, and r₂=r, we show that p₁=(logr−log(logr))∕r is close to optimal.

The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius r_ess(P_{|𝓁²(𝜋)}) of P_{|𝓁²(𝜋)} derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG₂) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG₂) is shown to hold under the condition α₀≔∑_m=−N^Nlim sup_i→+∞(P(i,i+m)P^*(i+m,i) ^1∕2<1. Moreover, r_ess(P_{|𝓁²(𝜋)}≤α₀. Effective bounds on the convergence rate can be provided from a truncation procedure.

We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius r_ess(P_{|𝓁²(𝜋)}) of P_{|𝓁²(𝜋)} derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG₂) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG₂) is shown to hold under the condition α₀≔∑_m=−N^Nlim sup_i→+∞(P(i,i+m)P^*(i+m,i)^1∕2<1. Moreover, r_ess(P_{|𝓁²(𝜋)}≤α₀. Effective bounds on the convergence rate can be provided from a truncation procedure.