Consider an N×n random matrix Y_n=(Yⁿ_ij) with entries given by $$Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X_{ij}^{n},$$ the Xⁿ_ij being centered, independent and identically distributed random variables with unit variance and (σ_ij(n); 1≤i≤N, 1≤j≤n) being an array of numbers we shall refer to as a variance profile. In this article, we study the fluctuations of the random variable

log det(Y_nY^*_n+ρI_N),

where Y^* is the Hermitian adjoint of Y and ρ>0 is an additional parameter. We prove that, when centered and properly rescaled, this random variable satisfies a central limit theorem (CLT) and has a Gaussian limit whose parameters are identified whenever N goes to infinity and N/n→c∈(0, ∞). A complete description of the scaling parameter is given; in particular, it is shown that an additional term appears in this parameter in the case where the fourth moment of the X_ij’s differs from the fourth moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.

We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience and existence of the sth moment of the return time to the empty state for this model. This paper generalizes the model, when only two stations accept arriving jobs, which was considered in [Ann. Appl. Probab. 17 (2007) 1447–1473]. Our results are stated in terms of Lyapunov exponents for random matrices. From the recurrence criteria it can be seen that the polling model with parameter regeneration can exhibit the unusual phenomenon of null recurrence over a thick region of parameter space.

In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/∞. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space $\mathcal{S}^{\prime}$ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a $\mathcal{S}^{\prime}$-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes.

We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long-range behavior of the spacings between the Brownian motions arranged in increasing order. For finitely many Brownian motions interacting in this manner, we characterize drifts for which the family of laws of the vector of spacings is tight and show its convergence to a unique stationary joint distribution given by independent exponential distributions with varying means. We also study one particular countably infinite system, where only the minimum Brownian particle gets a constant upward drift, and prove that independent and identically distributed exponential spacings remain stationary under the dynamics of such a process. Some related conjectures in this direction are also discussed.

In modeling parasitic diseases, it is natural to distinguish hosts according to the number of parasites that they carry, leading to a countably infinite type space. Proving the analogue of the deterministic equations, used in models with finitely many types as a “law of large numbers” approximation to the underlying stochastic model, has previously either been done case by case, using some special structure, or else not attempted. In this paper we prove a general theorem of this sort, and complement it with a rate of convergence in the ℓ₁-norm.

We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks.

We study a random growth model on ℝ^d introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim to find conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results for greedy paths from the lattice setting to the continuous setting.

We prove that the solution of the Kac analogue of Boltzmann’s equation can be viewed as a probability distribution of a sum of a random number of random variables. This fact allows us to study convergence to equilibrium by means of a few classical statements pertaining to the central limit theorem. In particular, a new proof of the convergence to the Maxwellian distribution is provided, with a rate information both under the sole hypothesis that the initial energy is finite and under the additional condition that the initial distribution has finite moment of order 2+δ for some δ in (0, 1]. Moreover, it is proved that finiteness of initial energy is necessary in order that the solution of Kac’s equation can converge weakly. While this statement may seem to be intuitively clear, to our knowledge there is no proof of it as yet.

Let X₁, …, X_n be a random sample from a p-dimensional population distribution. Assume that c₁n^α≤p≤c₂n^α for some positive constants c₁, c₂ and α. In this paper we introduce a new statistic for testing independence of the p-variates of the population and prove that the limiting distribution is the extreme distribution of type I with a rate of convergence $O((\log n)^{5/2}/\sqrt{n})$. This is much faster than O(1/log n), a typical convergence rate for this type of extreme distribution. A simulation study and application to stochastic optimization are discussed.

A singular stochastic control problem with state constraints in two-dimensions is studied. We show that the value function is C¹ and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem, we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133–2171] on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not C¹. This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is C² in the interior of the no-action region. We then use a verification argument applied to a suitable C² approximation of the value function to establish optimality of the conjectured control.

We study a contact process running in a random environment in ℤ^d where sites flip, independently of each other, between blocking and nonblocking states, and the contact process is restricted to live in the space given by nonblocked sites. We give a partial description of the phase diagram of the process, showing in particular that, depending on the flip rates of the environment, survival of the contact process may or may not be possible for large values of the birth rate. We prove block conditions for the process that parallel the ones for the ordinary contact process and use these to conclude that the critical process dies out and that the complete convergence theorem holds in the supercritical case.

Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramér light-tail assumption on the claim size distribution.

We study the parabolic Anderson problem, that is, the heat equation ∂_tu=Δu+ξu on (0, ∞)×ℤ^d with independent identically distributed random potential {ξ(z) : z∈ℤ^d} and localized initial condition u(0, x)=1₀(x). Our interest is in the long-term behavior of the random total mass U(t)=∑_zu(t, z) of the unique nonnegative solution in the case that the distribution of ξ(0) is heavy tailed. For this, we study two paradigm cases of distributions with infinite moment generating functions: the case of polynomial or Pareto tails, and the case of stretched exponential or Weibull tails. In both cases we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory.

In the paper we study dynamics of the arbitrage prices of credit default swaps within a hazard process model of credit risk. We derive these dynamics without postulating that the immersion property is satisfied between some relevant filtrations. These results are then applied so to study the problem of replication of general defaultable claims, including some basket claims, by means of dynamic trading of credit default swaps.