Motivated by the problem of the evolution of DNA sequences, Kauffman and Levin introduced a model in which fitnesses were assigned to strings of 0's and 1's of length N based on the values observed in a sliding window of length $K+1$. When $K\ge 1$, the landscape is quite complicated with many local maxima. Its properties have been extensively investigated by simulation but until our work and the independent investigations of Evans and Steinsaltz little was known rigorously about its properties except in the case $K=N-1$. Here, we prove results about the number of local maxima, their heights and the height of the global maximum. Our main tool is the theory of (substochastic) Harris chains.

We consider the approximation of the distribution of the sum of independent but not necessarily identically distributed random variables by a compound Poisson distribution and also by a finite signed measure of higher accuracy. Using Kerstan's method, some new bounds for the total variation distance are presented. Recently, several authors had difficulties applying Stein's method to the problem given. For instance, Barbour, Chen and Loh used this method in the case of random variables on the nonnegative integers. Under additional assumptions, they obtained some bounds for the total variation distance containing an undesirable log term. In the present paper, we shall show that Kerstan's approach works without such restrictions and yields bounds without log terms.

Given a locally bounded real function g, we examine the existence of a 4-covariation $[g(B^H), B^H, B^H, B^H]$, where $B^H$ is a fractional Brownian motion with a Hurst index $H \ge \tfrac{1}{4}$. We provide two essential applications. First, we relate the 4-covariation to one expression involving the derivative of local time, in the case $H = \tfrac{1}{4}$, generalizing an identity of Bouleau--Yor type, well known for the classical Brownian motion. A second application is an Itô formula of Stratonovich type for $f(B^H)$. The main difficulty comes from the fact $B^H$ has only a finite 4-variation.

We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of "asymptotic elasticity'' of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption--terminal wealth problems in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of $\lone$ to its topological bidual $\linfd$, a space of finitely additive measures. As an application, we treat a constrained Itô process market model, as well as a "totally incomplete'' model.

The odds theorem gives a unified answer to a class of stopping problems on sequences of independent indicator functions. The success probability of the optimal rule is known to be larger than $Re^{-R}$, where R defined in the theorem satisfies $R\ge 1$ in the more interesting case. The following findings strengthen this result by showing that $1/e$ is then a lower bound. Knowing that this is the best possible uniform lower bound motivates this addendum.

We consider a vortex structure based on a three-dimensional fractional Brownian motion with Hurst parameter $H>\frac{1}{2}.$ We show that the energy $\mathbb{H}$\vspace*{-1pt} has moments of any order under suitable conditions. When $H\in (\frac{1}{2},\frac{1}{3})$ we prove that the intersection energy $\mathbb{H}_{xy}$ can be decomposed into four terms, one of them being a weighted self-intersection local time of the fractional Brownian motion in $\mathbb{R}^{3}$.

We prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure. The method is based on a fixed point result in suitable Besov spaces.

This article examines the rate of escape for a random walk on $G\wr \Z$ and proves laws of the iterated logarithm for both the inner and outer radius of escape. The class of G for which these results hold includes finite, G as well as groups of the form $H\wr \Z$, so the construction can be iterated. Laws of the iterated logarithm are also found for random walk on Baumslag--Solitar groups and a discrete version of the Sol geometry.

This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations $dy_t=d\omega_t -\nabla V(y_t)\, dt$, $y_0=0$. When $d=1$ and V is not periodic but obtained as a superposition of an infinite number of periodic potentials with geometrically increasing periods [$V(x) = \sum_{k=0}^\infty U_k(x/R_k)$, where $U_k$ are smooth functions of period 1, $U_k(0)=0$, and $R_k$ grows exponentially fast with k] we can show that $y_t$ has an anomalous slow behavior and we obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Pointwise estimates are based on a new analytical inequality for subharmonic functions. When $d\geq 1$ and V is periodic, quantitative estimates are obtained on the heat kernel of $y_t$, showing the rate at which homogenization takes place. The latter result proves Davies' conjecture and is based on a quantitative estimate for the Laplace transform of martingales that can be used to obtain similar results for periodic elliptic generators.

Let $0 < a < b < \infty$, and for each edge e of $\Z^d$ let $\omega_e=a$ or $\omega_e=b$, each with probability $1/2$, independently. This induces a random metric $\dist_\omega$ on the vertices of $\Z^d$, called first passage percolation. We prove that for $d>1$, the distance $\dist_\omega(0,v)$ from the origin to a vertex $v$, $|v|>2$, has variance bounded by $C|v|/\log|v|$, where $C=C(a,b,d)$ is a constant which may only depend on a, b and d. Some related variants are also discussed.

We show that there is a finitary isomorphism from a finite state independent and identically distributed (i.i.d.) process to the $T,T^{-1}$ process associated to one-dimensional random walk with positive drift. This contrasts with the situation for simple symmetric random walk in any dimension, where it cannot be a finitary factor of any i.i.d. process, including in $d\ge 5$, where it becomes weak Bernoulli.

One of the key problems related to the Bak--Sneppen evolution model on the circle is computing the limit distribution of the fitness at a fixed observation vertex in the stationary regime as the size of the system tends to infinity. Some simulations have suggested that this limit distribution is uniform on $(f,1)$ for some $f\sim2/3$. In this article, we prove that the mean of the fitness in the stationary regime is bounded away from 1, uniformly in the size of the system, thereby establishing the nontriviality of the limit behavior. The Bak--Sneppen dynamics can easily be defined on any finite connected graph. We also present a generalization of the phase-transition result in the context of an increasing sequence of such graphs. This generalization covers the multidimentional Bak--Sneppen model as well as the Bak--Sneppen model on a tree. Our proofs are based on a "self-similar'' graphical representation of the avalanches.

In this article, we prove a large deviation principle for the empirical drift of a one-dimensional Brownian motion with self-repellence called the Edwards model. Our results extend earlier work in which a law of large numbers and a central limit theorem were derived. In the Edwards model, a path of length T receives a penalty $e^{-\beta H_T}$, where $ H_T$ is the self-intersection local time of the path and $\beta\in(0,\infty)$ is a parameter called the strength of self-repellence. We identify the rate function in the large deviation principle for the endpoint of the path as $\beta^{2/3} I(\beta^{-1/3}\cdot)$, with $I(\cdot)$ given in terms of the principal eigenvalues of a one-parameter family of Sturm--Liouville operators. We show that there exist numbers $0<b^{**}<b^*<\infty$ such that (1) I is linearly decreasing on $[0,b^{**}]$, (2) I is real-analytic and strictly convex on $(b^{**},\infty)$, (3) I is continuously differentiable at $b^{**}$ and\break (4) I has a unique zero at $b^*$. (The latter fact identifies $b^*$ as the asymptotic drift of the endpoint.) The critical drift $b^{**}$ is associated with a crossover in the optimal strategy of the path: for $b\geq b^{**}$ the path assumes local drift b during the full time T, while for $0\leq b<b^{**}$ it assumes local drift $b^{**}$ during time $\frac{b^{**}+b}{2b^{**}}T$ and local drift $-b^{**}$ during the remaining time $\frac{b^{**}-b}{2b^{**}}T$. Thus, in the second regime the path makes an overshoot of size $\frac{b^{**}-b}{2}T$ so as to reduce its intersection local time.

The auto-regressive model on $\RR ^{d}$ defined by the recurrence equation $ Y^{y}_{n}=a_{n}Y^{y}_{n-1}+B_{n} $, where $ \{ (a_{n},B_{n})\} _{n} $\vspace*{-0.5pt} is a sequence of i.i.d. random variables in $ \RR ^{*}_{+}\times \RR ^{d} $, has, in the critical case $ \esp {\log a_{1}}=0 $,\vspace*{-0.5pt} a local contraction property, that is, when $ Y^{y}_{n} $ is in a compact set the distance $ | Y^{y}_{n}-Y^{x}_{n}| $ converges almost surely to 0. We determine the speed of this convergence and we use this asymptotic estimate to deal with some higher-dimensional situations. In particular, we prove the recurrence and the local contraction property with speed for an autoregressive model whose linear part is given by triangular matrices with first Lyapounov exponent equal to 0. We extend the previous results to a Markov chain on a nilpotent Lie group induced by a random walk on a solvable Lie group of $ \mathcal{NA} $ type.

We introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional. As an application we give a new easy proof of Talagrand's concentration inequality for empirical processes, where besides symmetrization we use only Talagrand's concentration inequality on the discrete cube $\{0,1\}^n.$ As another application of this technique we prove new Vapnik--Chervonenkis type inequalities. For example, for VC-classes of functions we prove a classical inequality of Vapnik and Chervonenkis only with normalization by the sum of variance and sample variance.

We consider linear nth order stochastic differential equations on $[0,1]$, with linear boundary conditions supported by a finite subset of $[0,1]$. We study some features of the solution to these problems, and especially its conditional independence properties of Markovian type.

Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite- and infinite-dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite-dimensional random dynamical systems generated by stochastic partial differential equations. We first introduce a random graph transform and a fixed point theorem for nonautonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.

Let K be a smooth convex body. The convex hull of independent random points in K is a random polytope. Estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained. The essential step is the use of the Efron--Stein jackknife inequality for the variance of symmetric statistics. Consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope. A conjecture of Bárány concerning random and best-approximation of convex bodies is confirmed. Analogous results for random polytopes with vertices on the boundary of the convex body are given.

Let $X_1, X_2, \ldots $ be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramér-type large deviation result for the standardized partial sums. In this paper, we show that a Cramér-type large deviation theorem holds for self-normalized sums only under a finite $(2+\delta)$th moment, $0< \delta \leq 1$. In particular, we show $P(S_n /V_n \geq x)=\break (1-\Phi(x)) (1+O(1) (1+x)^{2+\delta} /d_{n,\delta}^{2+\delta})$ for $0 \leq x \leq d_{n,\delta}$,\vspace{1pt} where $d_{n,\delta} = (\sum_{i=1}^n EX_i^2)^{1/2}/(\sum_{i=1}^n E|X_i|^{2+\delta})^{1/(2+\delta)}$ and $V_n= (\sum_{i=1}^n X_i^2)^{1/2}$. Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.

A celebrated theorem of Burke's asserts that the Poisson process is a fixed point for a stable exponential single server queue; that is, when the arrival process is Poisson, the equilibrium departure process is Poisson of the same rate. This paper considers the following question: Do fixed points exist for queues which dispense i.i.d. services of finite mean, but otherwise of arbitrary distribution (i.e., the so-called $\cdot/\mathit{GI}/1/\infty/\mathit{FCFS}$ queues)? We show that if the service time $S$ is nonconstant and satisfies \mbox{$\int\!\! P\{S \geq u\}^{1/2}\,du <\infty$}, then there is an unbounded set $\fF\subset(E[S],\infty)$ such that for each $\alpha\in\fF$ there exists a unique ergodic fixed point with mean inter-arrival time equal to $\alpha$. We conjecture that in fact $\fF=(E[S],\infty)$.

We consider an infinite tandem of first-come-first-served queues. The service times have unit mean, and are independent and identically distributed across queues and customers. Let $\bI$ be a stationary and ergodic interarrival sequence with marginals of mean $\tau>1$, and suppose it is independent of all service times. The process $\bI$ is said to be a fixed point for the first, and hence for each, queue if the corresponding interdeparture sequence is distributed as $\bI$. Assuming that such a fixed point exists, we show that it is the distributional limit of passing an arbitrary stationary and ergodic interarrival process of mean $\tau$ through the infinite queueing tandem.

Motivated by multivariate random recurrence equations we prove a new analogue of the Key Renewal Theorem for functionals of a Markov chain with compact state space in the spirit of Kesten [Ann. Probab. 2 (1974) 355--386]. Compactness of the state space and a certain continuity condition allows us to simplify Kesten's proof considerably.