The Metropolis-Hastings algorithm is a method of constructing a reversible Markov transition kernel with a specified invariant distribution. This note describes necessary and sufficient conditions on the candidate generation kernel and the acceptance probability function for the resulting transition kernel and invariant distribution to satisfy the detailed balance conditions. A simple general formulation is used that covers a range of special cases treated separately in the literature. In addition, results on a useful partial ordering of finite state space reversible transition kernels are extended to general state spaces and used to compare the performance of two approaches to using mixtures in Metropolis-Hastings kernels.

We study convergence rates of $\mathbb{R}$-valued algorithms, especially in the case of multiple targets and simulated annealing. We precise, for example, the convergence rate of simulated annealing algorithms, whose weak convergence to a distribution concentrated on the potential's minima had been established by Gelfand and Mitter or by Hwang and Sheu.

This paper establishes a process-level large deviations principle for Markov processes in the Euclidean space with a discontinuity in the transition mechanism along a hyperplane. The transition mechanism of the process is assumed to be continuous on one closed half-space and also continuous on the complementary open half-space. Similar results were recently obtained for discrete time processes by Dupuis and Ellis and by Nagot. Our proof relies on the work of Blinovskii and Dobrushin, which in turn is based on an earlier work of Dupuis and Ellis.

Three policies, namely optimal repacking, least load routing, and Bernoulli splitting, are considered for dynamic resource allocation in load sharing networks with standard Erlang type statistics. Large deviations principles are established for the three policies in a simple network of three consumer types and two resource locations and are used to identify the network overflow exponents. The overflow exponents for networks with arbitrary topologies are identified for optimal repacking and Bernoulli splitting policies and conjectured for the least load routing policy.

If a FIFO queue is fed by several input streams that jointly satisfy a sample path large deviation principle (LDP) with "linear geodesics," then the cumulative departures (up to a large time) also satisfy the LDP with a rate function which depends in a relatively simple way on the rate function corresponding to the inputs: this was demonstrated in a recent paper by the second author. It suggests the possibility of an iterative scheme which would allow one to determine the large deviation behavior of more complicated networks. To do this, however, one would require that the linear geodesic property be preserved: in this paper we demonstrate that in general it is not preserved. This is true even in the case of a single input stream.

We study the class of tree-growing search strategies introduced by Lent and Mahmoud, searches for which data are stored in a deterministic sequence of tree structures (e.g., linear search in forward order). Specifically, we study the conditions under which the number of comparisons needed to sort a sequence of randomly ordered numbers is asymptotically normal. Our main result is a sufficient condition for normality in terms of the growth rate of tree height alone; this condition is easily computed and is satisfied by all standard deterministic search strategies. We also give some examples of normal search strategies with surprisingly small variance, in particular, much smaller than is possible for the class of consistent strategies that are the focus of the work by Lent and Mahmoud.

For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution $\pi$ on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate normalizing constant. The same difficulty arises in computer science problems where one seeks to sample randomly from a large finite distributive lattice whose precise size cannot be ascertained in any reasonable amount of time.

The Markov chain Monte Carlo (MCMC) approximate sampling approach to such a problem is to construct and run "for a long time" a Markov chain with long-run distribution $\pi$. But determining how long is long enough to get a good approximation can be both analytically and empirically difficult.

Recently, Propp and Wilson have devised an ingenious and efficient algorithm to use the same Markov chains to produce perfect (i.e., exact) samples from $\pi$. However, the running time of their algorithm is an unbounded random variable whose order of magnitude is typically unknown a priori and which is not independent of the state sampled, so a naive user with limited patience who aborts a long run of the algorithm will introduce bias.

We present a new algorithm which (1) again uses the same Markov chains to produce perfect samples from $\pi$, but is based on a different idea (namely, acceptance/rejection sampling); and (2) eliminates user-impatience bias. Like the Propp-Wilson algorithm, the new algorithm applies to a general class of suitably monotone chains, and also (with modification) to "anti-monotone" chains. When the chain is reversible, naive implementation of the algorithm uses fewer transitions but more space than Propp-Wilson. When fine-tuned and applied with the aid of a typical pseudorandom number generator to an attractive spin system on n sites using a random site updating Gibbs sampler whose mixing time $\tau$ is polynomial in n, the algorithm runs in time of the same order (bound) as Propp-Wilson [expectation $O(\tau \log n)$] and uses only logarithmically more space [expectation $O(n \log n)$, vs.$O9n)$ for Propp-Wilson].

We derive a general local limit theorem for probabilities of large deviations for a sequence of random variables by means of the saddlepoint method on Laplace-type integrals. This result is applicable to parameters in a number of combinatorial structures and the distribution of additive arithmetical functions.

The present paper continues the work of two previous papers on the variational behavior, over a large number of generations, of a Wright-Fisher process modelling an even larger reproducing population. It was shown that a Wright-Fisher process subject to random drift and one-way mutation which undergoes a large deviation follows with near certainty a path which can be a trigonometric, exponential, hyperbolic or parabolic function. Here it is shown that a process subject to random drift and gamete selection follows in similar circumstances a path which is, apart from critical cases, a Jacobian elliptic function.

Consider the performance of an options writer who misspecifies the dynamics of the price process of the underlying asset by overestimating asset price volatility. When does he overprice the option? If he follows the hedging strategy suggested by his model, when does the terminal value of his strategy dominate the option payout?

We show that both these events happen if the option payoff is a convex function of the price of the underlying at maturity. The proofs involve the simple, powerful and intuitive techniques of coupling.

We establish some error estimates for the binomial approximation of American put prices in the Black-Scholes model. Namely, we prove that if P is the American put price and $P_n$ its n-step binomial approximation, there exist positive constants c and C such that $-c/n^{2/3} \leq P_n - P \leq C/n^{3/4}$. With an additional assumption on the interest rate and the volatility, a better upper bound is derived.

We study patterns of the phase diagram of ferromagnetic Ising models on graphs under an external magnetic field. We provide an example of a tree with only two types of vertices on which for a range of values of the external field there is a unique Gibbs distribution at low enough and at high enough temperatures, while at intermediate temperatures there is phase coexistence (in other words, a reentrance transition takes place).

We derive a shape theorem type result for the almost sure exponential decay of the Green's function of $-\Delta + V$, where the potentials $V(x), x \epsilon \mathbb{Z}^d$ are i.i.d. nonnegative random variables. This result implies a large deviation principle governing the position of a d-dimensional random walk moving in the same potential.

Here the almost sure convergence of one-dimensional Kohonen's algorithm in its general form, namely, the 2k-neighbor setting with a nonuniform stimuli distribution, is proved. We show that the asymptotic behavior of the algorithm is governed by a cooperative system of differential equations which is irreducible. The system of differential equations possesses an asymptotically stable equilibrium, a compact subset of whose domain of attraction will be visited by the state variable $X^n$ infinitely often. The assumptions on the stimuli distribution and the neighborhood functions are weakened, too.

We consider a Markov chain $X_n^{\varepsilon}$ obtained by adding small noise to a discrete time dynamical system and study the chain's quasi-stationary distribution (qsd). The dynamics are given by iterating a function $f:I \to I$ for some interval I when f has finitely many fixed points, some stable and some unstable. We show that under some conditions the quasi-stationary distribution of the chain concentrates around the stable fixed points when $\varepsilon \to 0$. As a corollary, we obtain the result for the case when f has a single attracting cycle and perhaps repelling cycles and fixed points. In this case, the quasi-stationary distribution concentrates on the attracting cycle. The result applies to the model of population dependent branching processes with periodic conditional mean function.