Importance sampling is a variance reduction technique for efficient estimation of rare-event probabilities by Monte Carlo. In standard importance sampling schemes, the system is simulated using an a priori fixed change of measure suggested by a large deviation lower bound analysis. Recent work, however, has suggested that such schemes do not work well in many situations. In this paper we consider dynamic importance sampling in the setting of uniformly recurrent Markov chains. By “dynamic” we mean that in the course of a single simulation, the change of measure can depend on the outcome of the simulation up till that time. Based on a control-theoretic approach to large deviations, the existence of asymptotically optimal dynamic schemes is demonstrated in great generality. The implementation of the dynamic schemes is carried out with the help of a limiting Bellman equation. Numerical examples are presented to contrast the dynamic and standard schemes.

Physical notions of stochastic resonance for potential diffusions in periodically changing double-well potentials such as the spectral power amplification have proved to be defective. They are not robust for the passage to their effective dynamics: continuous-time finite-state Markov chains describing the rough features of transitions between different domains of attraction of metastable points. In the framework of one-dimensional diffusions moving in periodically changing double-well potentials we design a new notion of stochastic resonance which refines Freidlin’s concept of quasi-periodic motion. It is based on exact exponential rates for the transition probabilities between the domains of attraction which are robust with respect to the reduced Markov chains. The quality of periodic tuning is measured by the probability for transition during fixed time windows depending on a time scale parameter. Maximizing it in this parameter produces the stochastic resonance points.

Mixtures of Gaussian (or normal) distributions arise in a variety of application areas. Many heuristics have been proposed for the task of finding the component Gaussians given samples from the mixture, such as the EM algorithm, a local-search heuristic from Dempster, Laird and Rubin [J. Roy. Statist. Soc. Ser. B 39 (1977) 1–38]. These do not provably run in polynomial time.

We present the first algorithm that provably learns the component Gaussians in time that is polynomial in the dimension. The Gaussians may have arbitrary shape, but they must satisfy a “separation condition” which places a lower bound on the distance between the centers of any two component Gaussians. The mathematical results at the heart of our proof are “distance concentration” results—proved using isoperimetric inequalities—which establish bounds on the probability distribution of the distance between a pair of points generated according to the mixture.

We also formalize the more general problem of max-likelihood fit of a Gaussian mixture to unstructured data.

Let S⊂(0,1). Given a known function f:S→(0,1), we consider the problem of using independent tosses of a coin with probability of heads p (where p∈S is unknown) to simulate a coin with probability of heads f(p). We prove that if S is a closed interval and f is real analytic on S, then f has a fast simulation on S (the number of p-coin tosses needed has exponential tails). Conversely, if a function f has a fast simulation on an open set, then it is real analytic on that set.

The present paper deals with the characterization of no-arbitrage properties of a continuous semimartingale. The first main result, Theorem 2.1, extends the no-arbitrage criterion by Levental and Skorohod [Ann. Appl. Probab. 5 (1995) 906–925] from diffusion processes to arbitrary continuous semimartingales. The second main result, Theorem 2.4, is a characterization of a weaker notion of no-arbitrage in terms of the existence of supermartingale densities. The pertaining weaker notion of no-arbitrage is equivalent to the absence of immediate arbitrage opportunities, a concept introduced by Delbaen and Schachermayer [Ann. Appl. Probab. 5 (1995) 926–945].

Both results are stated in terms of conditions for any semimartingales starting at arbitrary stopping times σ. The necessity parts of both results are known for the stopping time σ=0 from Delbaen and Schachermayer [Ann. Appl. Probab. 5 (1995) 926–945]. The contribution of the present paper is the proofs of the corresponding sufficiency parts.

The theme of this paper is the derivation of analytic formulae for certain large combinatorial structures. The formulae are obtained via fluid limits of pure jump-type Markov processes, established under simple conditions on the Laplace transforms of their Lévy kernels. Furthermore, a related Gaussian approximation allows us to describe the randomness which may persist in the limit when certain parameters take critical values. Our method is quite general, but is applied here to vertex identifiability in random hypergraphs. A vertex $v$ is identifiable in $n$ steps if there is a hyperedge containing $v$ all of whose other vertices are identifiable in fewer steps. We say that a hyperedge is identifiable if every one of its vertices is identifiable. Our analytic formulae describe the asymptotics of the number of identifiable vertices and the number of identifiable hyperedges for a Poisson $(β)$ random hypergraph $Λ$ on a set $V$ of $N$ vertices, in the limit as $N→∞$. Here $β$ is a formal power series with nonnegative coefficients $β_0,β_1,…,$ and $(Λ(A))_{A⊆V}$ are independent Poisson random variables such that $Λ(A)$, the number of hyperedges on $A$, has mean $Nβ_j/\pmatrix{{N}\cr{j}}$ whenever $|A|=j$.

We study the asymptotics related to the following matching criteria for two independent realizations of point processes X∼X and Y∼Y. Given l>0, X∩[0,l) serves as a template. For each t>0, the matching score between the template and Y∩[t,t+l) is a weighted sum of the Euclidean distances from y−t to the template over all y∈Y∩[t,t+l). The template matching criteria are used in neuroscience to detect neural activity with certain patterns. We first consider W_l(θ), the waiting time until the matching score is above a given threshold θ. We show that whether the score is scalar- or vector-valued, (1/l)logW_l(θ) converges almost surely to a constant whose explicit form is available, when X is a stationary ergodic process and Y is a homogeneous Poisson point process. Second, as l → ∞, a strong approximation for −log[Pr{W_l(θ)=0}] by its rate function is established, and in the case where X is sufficiently mixing, the rates, after being centered and normalized by $\sqrt{l}$, satisfy a central limit theorem and almost sure invariance principle. The explicit form of the variance of the normal distribution is given for the case where X is a homogeneous Poisson process as well.

This work concerns controlled Markov chains with finite state and action spaces. The transition law satisfies the simultaneous Doeblin condition, and the performance of a control policy is measured by the (long-run) risk-sensitive average cost criterion associated to a positive, but otherwise arbitrary, risk sensitivity coefficient. Within this context, the optimal risk-sensitive average cost is characterized via a minimization problem in a finite-dimensional Euclidean space.

We establish Gaussian limits for general measures induced by binomial and Poisson point processes in d-dimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general results are used to deduce central limit theorems for measures induced by random graphs (nearest neighbor, Voronoi and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth–growth models) and statistics of germ–grain models.

Let I be a compact d-dimensional manifold, let X:I→ℛ be a Gaussian process with regular paths and let F_I(u), u∈ℛ, be the probability distribution function of sup _t∈IX(t).

We prove that under certain regularity and nondegeneracy conditions, F_I is a C¹-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u→+∞. This is a partial extension of previous results by the authors in the case d=1.

Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t)=x, where Z:I→ℛ^d is a random field and x is a fixed point in ℛ^d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.

We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by Poisson and mixed Poisson distributions; these lead to asymptotic normality results. We are particularly interested in the influence of the key value on the distribution of the node depth. Methodologically our message is that the explicit representation may provide additional insight if compared to the standard approach that is based on the recursive structure of the trees. Further, in order to exhibit the influence of the key on the distributional asymptotics, a suitable choice of distance of probability distributions is important. Our results are also applicable in connection with the number of recursions needed in Hoare’s [Comm. ACM 4 (1961) 321–322] selection algorithm FIND.

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on ℤ^d or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x,y∈ℤ^d, there is a strictly positive probability that {z∈ℤ^d;d(y,z)<d(x,z)} and {z∈ℤ^d;d(y,z)>d(x,z)} are both infinite sets. We also show that there is a strictly positive probability that the graph of time-minimizing path from the origin in first-passage percolation has at least two topological ends. This generalizes results obtained by Häggström and Pemantle for independent exponential times on the square lattice.

We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks we consider satisfy the so-called complete resource pooling condition and therefore have one-dimensional approximating Brownian control problems. We propose a simple discrete review policy for controlling such networks. Assuming 2+ɛ moments on the interarrival times and processing times, we provide a conceptually simple proof of asymptotic optimality of the proposed policy.

We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function L_n(z)=|W_n|⁻¹logEexp{z|Ξ∩W_n|} of the empirical volume fraction |Ξ∩W_n|/|W_n|, where |⋅| denotes the d-dimensional Lebesgue measure. Here Ξ=⋃_i≥1(Ξ_i+X_i) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process Π_λ=∑_i≥1δ_Xi with intensity λ>0 and a sequence of independent copies Ξ₁,Ξ₂,… of a random compact set Ξ₀. For an increasing family of compact convex sets {W_n, n≥1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit lim _n→∞L_n(z) on some disk in the complex plane whenever Eexp{a|Ξ₀|}<∞ for some a>0. Moreover, closely connected with this result, we obtain exponential inequalities and the exact asymptotics for the large deviation probabilities of the empirical volume fraction in the sense of Cramér and Chernoff.

Large deviation results are given for a class of perturbed nonhomogeneous Markov chains on finite state space which formally includes some stochastic optimization algorithms. Specifically, let {P_n} be a sequence of transition matrices on a finite state space which converge to a limit transition matrix P. Let {X_n} be the associated nonhomogeneous Markov chain where P_n controls movement from time n−1 to n. The main statements are a large deviation principle and bounds for additive functionals of the nonhomogeneous process under some regularity conditions. In particular, when P is reducible, three regimes that depend on the decay of certain “connection” P_n probabilities are identified. Roughly, if the decay is too slow, too fast or in an intermediate range, the large deviation behavior is trivial, the same as the time-homogeneous chain run with P or nontrivial and involving the decay rates. Examples of anomalous behaviors are also given when the approach P_n→P is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics.

The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of “disorder” when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Lévy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jumps. The method of proof is based on reducing the Bayesian problem to an integro-differential free-boundary problem where, in some cases, the smooth-fit principle breaks down and is replaced by the principle of continuous fit.

We ķ investigate the tail asymptotics of the supremum of X(t)+Y(t)−ct, where X={X(t),t≥0} and Y={Y(t),t≥0} are two independent stochastic processes. We assume that the process Y has subexponential characteristics and that the process X is more regular in a certain sense than Y. A key issue examined in earlier studies is under what conditions the process X contributes to large values of the supremum only through its average behavior. The present paper studies various scenarios where the latter is not the case, and the process X shows some form of “atypical” behavior as well. In particular, we consider a fluid model fed by a Gaussian process X and an (integrated) On-Off process Y. We show that, depending on the model parameters, the Gaussian process may contribute to the tail asymptotics by its moderate deviations, large deviations, or oscillatory behavior.