Simulated annealing algorithms are time inhomogeneous controlled Markov chains used to search for the minima of energy functions defined on finite state spaces. The control parameters, the so-called cooling schedule, control the probability that the energy should increase during one step of the algorithm. Most of the studies on simulated annealing have dealt with limit theorems, such as characterizing convergence conditions on the cooling schedule, or giving an equivalent of the law of the process for one fixed cooling schedule. In this paper we derive finite time estimates. These estimates are uniform in the cooling schedule and in the energy function. With new technical tools, we gain a new insight into the algorithm. We give a sharp upper bound for the probability that the energy is close to its minimum value. Hence we characterize the optimal convergence rate. This involves a new constant, the "difficulty" of the energy landscape. We calculate two cooling schedules for which our bound is almost reached. In one case it is reached up to a multiplicative constant for one energy function. In the other case it is reached in the sense of logarithmic equivalence uniformly in the energy function. These two schedules are both triangular: There is one different schedule for each finite simulation time. For each fixed finite time the second schedule has the currently used but previously mathematically unjustified exponential form. Finally, the title is "Rough large deviation estimates" because we have computed sharper ones (i.e., with sharp multiplicative constants) in two other papers.

Say that a family $\{P_\theta^n: \theta \in \Theta\}$ of sequences of probability measures is exponentially continuous if whenever $\theta_n \rightarrow \theta$, the sequence $\{P_{\theta_n}^n\}$ satisfies a large deviation principle with rate function $\lambda_\theta$. If $\Theta$ is compact and $\{P_\theta^n\}$ is exponentially continuous, then the mixture $P^n(A) =: \int_\Theta P_\theta^n(A)d\mu(\theta)$ satisfies a large deviation principle with rate function $\lambda(x) =: \inf\{\lambda_\theta(x): \theta \in S(\mu)\}$, where $S(\mu)$ is the support of the mixing measure $\mu$. If $X_1,X_2,\ldots$ is a sequence of i.i.d. random vectors, $\{\bar{X}_n\}$ the corresponding sequence of sample means and $P_\theta^n =: P_\theta\circ\bar{X}^{-1}_n$, then $\{P_\theta^n\}$ is exponentially continuous if the classical rate function $\lambda_\theta(\nu)$ is jointly lower semicontinuous and a uniform integrability condition introduced by de Acosta is satisfied. These results are applied in Section 4 to derive a large deviation theory for exchangeable random variables; the resulting rate functions are typically nonconvex. If the parameter space $\Theta$ is not compact, then examples can be constructed where a full large deviation principle is not satisfied because the upper bound fails for a noncompact set.

We answer two questions: "What is the probability that a randomly chosen distribution function on $\{0,1,\ldots, n\}$ is a mixture of binomial distributions?" and "What is the probability that an $n$-exchangeable sequence is the initial segment of an infinite exchangeable sequence?" Curiously, the answers are the same.

Given a finite alphabet, there is an inductive method for constructing a stationary measure on doubly infinite words from this alphabet. This construction can be randomized; the main focus here is on a particular uniform randomization which intuitively corresponds to the idea of choosing a generic stationary process. It is shown that with probability 1, the random stationary process has zero entropy and gives positive probability to every periodic infinite word.

Of interest in DNA analysis is the length $L(x^n_1)$ of the longest sequence that appears twice in a sequence $x^n_1$ of length $n$. Karlin and Ghandour and Arratia and Waterman have shown that if the sequence is a sample path from an i.i.d. or Markov process, then $L(x^n_1) = O(\log n)$. In this paper, examples of ergodic processes are constructed for which the asymptotic growth rate is infinitely often as large as $\lambda(n)$, where $\lambda(n)$ is subject only to the condition that it be $o(n)$.

Let $X_n, n \geq 1$, be a sequence of independent random variables, and let $F_N$ be the distribution function of the partial sums $\sum^N_{n = 1}X_n$. Motivated by a conjecture of Erdos in probabilistic number theory, we investigate conditions under which the convergence of $F_N(x)$ at two points $x = x_1,x_2$ with different limit values already implies the weak convergence of the distributions $F_N$. We show that this is the case if $\sum^\infty_{n = 1}\rho(X_n,c_n) = \infty$ whenever $\sum^\infty_{n = 1}c_n$ diverges, where $\rho(X,c)$ denotes the Levy distance between $X$ and the constant random variable $c$. In particular, this condition is satisfied if $\lim\inf_{n \rightarrow\infty}P(X_n = 0) > 0$.

A one-sided refinement of the strong law of large numbers is found for which the partial weighted sums not only converge almost surely to the expected value, but also the convergence is such that eventually the partial sums all exceed the expected value. The new weights are distribution-free, depending only on the relative ranks of the observations. A similar refinement of the Glivenko-Cantelli theorem is obtained, in which a new empirical distribution function not only has the usual uniformly almost-sure convergence property of the classical empirical distribution function, but also has the property that all its quantiles converge almost surely. A tool in the proofs is a strong law of large numbers for order statistics.

Let $X_i$ be i.i.d. random variables, $0 \leq X_i \leq 1$ and $c \geq 0$, and let $Y_i = X_i - ic$. It is shown that for all $n$, all $c$ and all such $X_i, E(\max_{i \geq 1} Y_i) - \sup_t EY_t < e^{-1}$, where $t$ is a stopping rule and $e^{-1}$ is shown to be the best bound for which the inequality holds. Specific bounds are also obtained for fixed $n$ or fixed $c$. These results are very similar to those obtained by Jones for a similar problem, where $0 \leq X_i \leq 1$ are independent but not necessarily identically distributed. All results are valid and unchanged also when $Y_i$ is replaced by $Y^\ast_i = \max_{1 \leq j \leq i} X_j - ic$.

Let $X_1,X_2,\ldots$ be i.i.d. random variables with common mean $\mu \geq 0$ and associated random walk $S_0 = 0, S_n = X_1 + \cdots + X_n, n \geq 1$. For a regularly varying function $\phi(t) = t^\alpha L(t), \alpha > -1$ with slowly varying $L(t)$, we consider the generalized renewal function $U_\phi(t) = \sum_{n \geq 0} \phi(n)P(S_n \leq t),\quad t \in \mathbb{R},$ by relating it to the family $\tau = \tau(t) = \inf\{n \geq 1: S_n > t\} t \geq 0$. One of the major results is that $U_\phi(t) < \infty$ for all $t \in \mathbb{R}, \operatorname{iff} \phi(t)^{-1}U_\phi(t) \sim 1/(\alpha + 1)\mu^{\alpha + 1}$ as $t \rightarrow \infty, \operatorname{iff} E(X^-_1)^2\phi(X^-_1) < \infty$, provided $\phi$ is ultimately increasing $(\Rightarrow \alpha \geq 0)$. A related result is proved for $U_\phi(t + h) - U_\phi(t)$ and $U^+_\phi(t) = \sum_{n \geq 0}\phi(n)P(M_n \leq t)$, where $M_n = \max_{0 \leq j \leq n} S_j$. Our results form extensions of earlier ones by Heyde, Kalma, Gut and others, who either considered more specific functions $\phi$ or used stronger moment assumptions. The proofs are based on a regeneration technique from renewal theory and two martingale inequalities by Burkholder, Davis and Gundy.

Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(t), 0 \leq t \leq 1\}$ be the empirical and quantile processes generated by the first $n$ observations from an i.i.d. sequence of uniformly distributed random variables on (0,1). Let $0 < a_n < 1$ be a sequence of constants such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. We investigate the strong limiting behavior as $n \rightarrow \infty$ of the increment functions $\{\alpha_n(t + a_ns) - \alpha_n(t), 0 \leq s \leq 1\}$ and $\{\beta_n(t + a_ns) - \beta_n(t), 0 \leq s \leq 1\},$ where $0 \leq t \leq 1 - a_n$. Under suitable regularity assumptions imposed upon $a_n$, we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.

A branching diffusion process is studied when its diffusivity decreases to 0 at the rate of $\varepsilon \ll 1$ and its branching/transmutation intensity increases at the rate of $\varepsilon^{-1}$. We derive the action functionals which describe some large deviations of the processes as $\varepsilon$ tends to 0. The branching diffusion processes are closely related to systems of semilinear parabolic differential equations.

Let $R_t$ be the position of the rightmost particle at time $t$ in a time-homogeneous one-dimensional branching diffusion process. Let $\gamma(\alpha,t)$ be the $\alpha$th quantile of $R_t$ under $P^0$, where $P^x$ denotes the probability measure of the branching diffusion process starting with a single particle at position $x$. We show that $\gamma(\alpha,t)$ is a limiting quantile of $R_t$ under $P^x$ in the sense that $\lim_{t \rightarrow\infty}P^x\{R_t \leq \gamma(\alpha,t)\}$ exists for all $\alpha \in (0,1)$ and all $x \in \mathbb{R}$. If the underlying diffusion is recurrent, we show that, after an appropriate rescaling of space, the $P^x$ distribution of $R_t - t$ converges weakly to a nontrivial limiting distribution $w_x$.

In this paper, we study a renormalized self-intersection local time for superprocesses over stable processes and classical diffusions. When the renormalization breaks down, we obtain limit theorems.

This paper is concerned with accurate asymptotic estimates for exit time probabilities associated with nearly deterministic Markov diffusions. The exit time probabilities are expressed as asymptotic series of WKB type in a small parameter, which measures the strength of the random Brownian motion inputs. This series is valid in certain regions in which the minimum action function $u(x,s)$ is a smooth function of state $x$ and time $s$. The function $u$ is a solution to the corresponding Hamilton-Jacobi PDE of first order.

The small noise exit problem of Wentzell and Freidlin is of particular interest for regions whose boundary consists of trajectories of the underlying (unperturbed) dynamical system. This is called the case of characteristic boundary. One fruitful approach to attacking this problem involves conditioning the probability measure so that exit to the boundary occurs more quickly. In a previous paper, this approach was applied to some simplified examples, revealing some previously unanticipated phenomena for the characteristic boundary exit problem. In this paper we develop certain aspects of this approach more generally. In particular we present stochastic differential equations which give an asymptotically correct description of this conditioned process by using a carefully chosen system of coordinates near the boundary.

A sequence of measures on a topological space is perturbed by a sequence of elements of a Lie group acting on that space. Criteria are given for the singularity and equivalence of the corresponding product measures. These criteria extend the results of Shepp and Steele. In particular, Fisher information comes into the scene and its role is further clarified.

Let $\{B,H,P_0\}$ be an abstract Wiener space and for every real $\rho$, let $T_\rho\omega = \omega + \rho F(\omega)$ be a transformation from $B$ to $B$. It is well known that under certain assumptions the measures induced by $T_\rho$ or $T_\rho^{-1}$ are mutually absolutely continuous with respect to $P_0$ and the density function is represented by the Ramer formula. In this formula, the Carleman-Fredholm determinant $\det_2(I_H + \rho\nabla F)$ appears as a factor. We characterize the class of $\nabla F$ for which a.s.-$P_0, \det_2(I_H + \rho\nabla F) = 1$ for all $\rho$ in an open subset of $\mathbb{R}$, in which case the form of Ramer's expression reduces to the familiar Cameron-Martin-Maruyama-Girsanov form. The proof is based on a characterization of quasinilpotent Hilbert-Schmidt operators.

In this note, we study the regularity of $\mathbb{R^N}$-valued random functions $X - (X_n, n \in \mathbb{N})$ on $\mathbb{R}$ such that $X_n(t) = a_nx_n(b_nt),\quad t \in \mathbb{R}, n \in \mathbb{N},$ where $\mathbf{a} = (a_n) \subset \mathbb{R}^+, \mathbf{b} = (b_n) \subset \mathbb{R}^+$ and $(x_n, n \in \mathbb{N})$ is an i.i.d. sequence of gaussian centered stationary real random functions on $\mathbb{R}$. If the common covariance of the $x_n$'s verifies some very weak regularity assumptions, then their paths are continuous in $l_p, p \in \lbrack 1,\infty\lbrack$ if and only if they are in this space and some integral depending uniquely on $p$ and on $\mathbf{a}$ and $\mathbf{b}$ is convergent. These results extend and refine some previous results concerning only the case $p \in \lbrack 2, \infty\lbrack$.

We define a semi-min-stable (SMS) process $Y(t)$ in $\lbrack 0,\infty)$ to be one which is stable under the simultaneous operations of taking the minima of $n$ independent copies of $Y(t)$ (pointwise over time $t$) and rescaling space and time. We show that the only possible rescaling of time is by a fixed power of $n$ and that SMS processes are essentially the only possible weak limits for large $m$ of a process obtained by taking the minimum, pointwise over $t$, of $m$ independent copies of a given process and then rescaling space and time. We describe the representation of a SMS process as the minimum of a Poisson process on a function space. We obtain a partial characterization of sample continuous SMS processes, similar to that of de Haan in the case of max-stable processes.

If $X$ is a spectrally positive Levy process, $\bar{X}^c$ the continuous part of its maximum process, and $J$ the sum of the jumps of $X$ across its previous maximum, then $X - 2\bar{X}^c - J$ has the same law as $X$ conditioned to stay negative. This extends a result due to Pitman, who links the real Brownian motion and the three-dimensional Bessel process. Several other relations between the Brownian motion and the Bessel process are extended in this setting.

We derive simple explicit formulas for the Fourier-Laplace transforms of the Hilbert transform and related functionals of the local time of a symmetric Levy process. These formulas generalize results of Biane and Yor for Brownian motion. The method of proof provides an explanation (of sorts) for the presence of the hyperbolic functions in such formulas.

The joint density of the total winding and the radius of a planar Brownian motion is calculated, by solving the associated backward equation. An explicit expression for the distribution of the hitting time of an angular barrier is shown; in particular, the behavior of the tail of the distribution is determined.

The paper gives bounds for the accuracy of Poisson approximation to the distribution of the number of points in a point process. There are two principal bounds, one in terms of reduced Palm probabilities for general point processes, and one involving compensators for point processes on the line. The latter bound is frequently sharper than the previously used compensator bounds when the expected number of points is large, and examples show that little improvement is possible without changing the form of the bound.

This paper investigates the dynamics of the one-dimensional three-color cyclic cellular automaton. The author has previously shown that this process fluctuates, meaning that each lattice site changes color infinitely often, so that there is no "final state" for the system. The focus of the current work is on the clustering properties of this system. This paper demonstrates that the one-dimensional three-color cyclic cellular automaton clusters, and the mean cluster size, as a function of time $t$, is asymptotic to $ct^{1/2}$, where $c$ is an explicitly calculable constant. The method of proof also allows us to compute asymptotic estimates of the mean interparticle distance for a one-dimensional system of particles which undergo deterministic motion and which annihilate upon collision. No clustering results are known about the four-color process, but evidence is presented to suggest that the mean cluster size of such systems grows at a rate different from $t^{1/2}$.

Consider a collection of real-valued random variables indexed by the integers. It is well known that such a process can be stationary, that is, translation invariant, and ergodic and yet have very strong associations: The one-sided tail field may determine the sample; the measure may fail to be mixing in any sense; the weak law of large numbers may fail on some infinite subset of the integers. The main result of this paper is that this cannot happen if the integers are replaced by an infinite homogeneous tree and the translations are replaced by all graph automorphisms. In fact, any automorphism-invariant process indexed by the tree is a mixture of extremal processes whose one-sided tail fields are trivial, from which the mixing properties follow.

The total variation distance between the process which counts cycles of size $1,2,\ldots, b$ of a random permutation of $n$ objects and a process $(Z_1,Z_2,\ldots, Z_b)$ of independent Poisson random variables with $\mathbb{E}Z_i = 1/i$ converges to 0 if and only if $b/n \rightarrow 0$. This Poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations. These limit theorems include the Erdos-Turan theorem for the asymptotic normality of the log of the order of a random permutation, and the DeLaurentis-Pittel functional central limit theorem for the cycle sizes. We give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of $n/b \rightarrow \infty$. A similar result holds for derangements and, more generally, for permutations conditioned to have given numbers of cycles of various sizes. Comparison results are included to show that in approximating the cycle structure by an independent Poisson process the main discrepancy arises from independence rather than from Poisson marginals.

The two-dimensional grid $Z^2$ and any graph of smaller growth rate is recurrent. We show that any graph satisfying an isoperimetric inequality only slightly stronger than that of $Z^2$ is transient. More precisely, if $f(k)$ denotes the smallest number of vertices in the boundary of a connected subgraph with $k$ vertices, then the graph is transient if the infinite sum $\sum f(k)^{-2}$ converges. This can be applied to parabolicity versus hyperbolicity of surfaces.