We investigate local MCMC algorithms, namely the random-walk Metropolis and the Langevin algorithms, and identify the optimal choice of the local step-size as a function of the dimension n of the state space, asymptotically as n→∞. We consider target distributions defined as a change of measure from a product law. Such structures arise, for instance, in inverse problems or Bayesian contexts when a product prior is combined with the likelihood. We state analytical results on the asymptotic behavior of the algorithms under general conditions on the change of measure. Our theory is motivated by applications on conditioned diffusion processes and inverse problems related to the 2D Navier–Stokes equation.

We prove that an averaging principle holds for a general class of stochastic reaction–diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite-dimensional systems.

Importance sampling has been reported to produce algorithms with excellent empirical performance in counting problems. However, the theoretical support for its efficiency in these applications has been very limited. In this paper, we propose a methodology that can be used to design efficient importance sampling algorithms for counting and test their efficiency rigorously. We apply our techniques after transforming the problem into a rare-event simulation problem—thereby connecting complexity analysis of counting problems with efficiency in the context of rare-event simulation. As an illustration of our approach, we consider the problem of counting the number of binary tables with fixed column and row sums, c_j’s and r_i’s, respectively, and total marginal sums d=∑_jc_j. Assuming that max _jc_j=o(d^1/2), ∑c_j²=O(d) and the r_j’s are bounded, we show that a suitable importance sampling algorithm, proposed by Chen et al. [J. Amer. Statist. Assoc. 100 (2005) 109–120], requires O(d³ɛ⁻²δ⁻¹) operations to produce an estimate that has ɛ-relative error with probability 1−δ. In addition, if max _jc_j=o(d^1/4−δ₀) for some δ₀>0, the same coverage can be guaranteed with O(d³ɛ⁻²log(δ⁻¹)) operations.

Assuming that the stock price Z=(Z_t)_0≤t≤T follows a geometric Brownian motion with drift μ∈ℝ and volatility σ>0, and letting M_t=max _0≤s≤tZ_s for t∈[0, T], we consider the optimal prediction problems $$V_{1}=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_{T}}{Z_{\tau}}\biggr) \quad \mathrm{and} \quad V_{2}=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_{T}}\biggr),$$ where the infimum and supremum are taken over all stopping times τ of Z. We show that the following strategy is optimal in the first problem: if μ≤0 stop immediately; if μ∈(0, σ²) stop as soon as M_t/Z_t hits a specified function of time; and if μ≥σ² wait until the final time T. By contrast we show that the following strategy is optimal in the second problem: if μ≤σ²/2 stop immediately, and if μ>σ²/2 wait until the final time T. Both solutions support and reinforce the widely held financial view that “one should sell bad stocks and keep good ones.” The method of proof makes use of parabolic free-boundary problems and local time–space calculus techniques. The resulting inequalities are unusual and interesting in their own right as they involve the future and as such have a predictive element.

Consider a random walk whose (light-tailed) increments have positive mean. Lower and upper bounds are provided for the expected maximal value of the random walk until it experiences a given drawdown d. These bounds, related to the Calmar ratio in finance, are of the form (exp{αd}−1)/α and (K exp{αd}−1)/α for some K>1, in terms of the adjustment coefficient α (E[exp{−αX}]=1) of the insurance risk literature. Its inverse 1/alpha has been recently derived by Aumann and Serrano as an index of riskiness of the random variable X.

This article also complements the Lundberg exponential stochastic upper bound and the Crámer–Lundberg approximation for the expected minimum of the random walk, with an exponential stochastic lower bound. The tail probability bounds are of the form C exp{−αx} and exp{−αx}, respectively, for some 1/K<C<1.

Our treatment of the problem involves Skorokhod embeddings of random walks in martingales, especially via the Azéma–Yor and Dubins stopping times, adapted from standard Brownian motion to exponential martingales.

Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present a general framework based on semigroup expansions for the construction of higher-order discretization schemes and analyze its rate of convergence. We also apply it to approximate general Lévy driven stochastic differential equations.

We study the classical problem of noisy constrained capacity in the case of the binary symmetric channel (BSC), namely, the capacity of a BSC whose inputs are sequences chosen from a constrained set. Motivated by a result of Ordentlich and Weissman [In Proceedings of IEEE Information Theory Workshop (2004) 117–122], we derive an asymptotic formula (when the noise parameter is small) for the entropy rate of a hidden Markov chain, observed when a Markov chain passes through a BSC. Using this result, we establish an asymptotic formula for the capacity of a BSC with input process supported on an irreducible finite type constraint, as the noise parameter tends to zero.

We investigate the effective resistance R_n and conductance C_n between the root and leaves of a binary tree of height n. In this electrical network, the resistance of each edge e at distance d from the root is defined by r_e=2^dX_e where the X_e are i.i.d. positive random variables bounded away from zero and infinity. It is shown that ER_n=nEX_e−(Var (X_e)/EX_e)ln n+O(1) and Var (R_n)=O(1). Moreover, we establish sub-Gaussian tail bounds for R_n. We also discuss some possible extensions to supercritical Galton–Watson trees.

We establish the asymptotic normality of the G-measure of the symmetric difference between the level set and a plug-in-type estimator of it formed by replacing the density in the definition of the level set by a kernel density estimator. Our proof will highlight the efficacy of Poissonization methods in the treatment of large sample theory problems of this kind.

In this work, we examine spectral properties of Markov transition operators corresponding to Gaussian perturbations of discrete time dynamical systems on the circle. We develop a method for calculating asymptotic expressions for eigenvalues (in the zero noise limit) and show that changes to the number or period of stable orbits for the deterministic system correspond to changes in the number of limiting modulus 1 eigenvalues of the transition operator for the perturbed process. We call this phenomenon a λ-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.

A hidden Markov model is called observable if distinct initial laws give rise to distinct laws of the observation process. Observability implies stability of the nonlinear filter when the signal process is tight, but this need not be the case when the signal process is unstable. This paper introduces a stronger notion of uniform observability which guarantees stability of the nonlinear filter in the absence of stability assumptions on the signal. By developing certain uniform approximation properties of convolution operators, we subsequently demonstrate that the uniform observability condition is satisfied for various classes of filtering models with white-noise type observations. This includes the case of observable linear Gaussian filtering models, so that standard results on stability of the Kalman–Bucy filter are obtained as a special case.

We consider Kac’s random walk on n-dimensional rotation matrices, where each step is a random rotation in the plane generated by two randomly picked coordinates. We show that this process converges to the Haar measure on SO(n) in the L² transportation cost (Wasserstein) metric in O(n²ln n) steps. We also prove that our bound is at most a O(ln n) factor away from optimal. Previous bounds, due to Diaconis/Saloff-Coste and Pak/Sidenko, had extra powers of n and held only for L¹ transportation cost.

Our proof method includes a general result of independent interest, akin to the path coupling method of Bubley and Dyer. Suppose that P is a Markov chain on a Polish length space (M, d) and that for all x, y∈M with d(x, y)≪1 there is a coupling (X, Y) of one step of P from x and y (resp.) that contracts distances by a (ξ+o(1)) factor on average. Then the map μ↦μP is ξ-contracting in the transportation cost metric.

We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate proportional to the number of catalyst individuals alive. The reactant forms a process in random medium.

We describe asymptotically the genealogy of catalytic branching populations coded as the induced forest of ℝ-trees using the many individuals—rapid branching continuum limit. The limiting continuum genealogical forests are then studied in detail from both the quenched and annealed points of view. The result is obtained by constructing a contour process and analyzing the appropriately rescaled version and its limit. The genealogy of the limiting forest is described by a point process. We compare geometric properties and statistics of the reactant limit forest with those of the “classical” forest.