We are interested in the geometric properties of real-valued Gaussian random fields defined on manifolds. Our manifolds, $M$, are of class $C^3$ and the random fields $f$ are smooth. Our interest in these fields focuses on their excursion sets, $f^{-1}[u, +\infty)$, and their geometric properties. Specifically, we derive the expected Euler characteristic $\Ee[\chi(f^{-1}[u, +\infty))]$ of an excursion set of a smooth Gaussian random field. Part of the motivation for this comes from the fact that $\Ee[\chi(f^{-1}[u,+\infty))]$ relates global properties of $M$ to a geometry related to the covariance structure of $f$. Of further interest is the relation between the expected Euler characteristic of an excursion set above a level $u$ and $\Pp[ \sup_{p \in M} f(p) \geq u ]$. Our proofs rely on results from random fields on $\Rr^n$ as well as differential and Riemannian geometry.

We consider two stochastic partial differential equations \[ du_{\varepsilon}(t)= (L_ru_{\varepsilon}(t)+f_{r}(t)) \,dV_{\varepsilon t}^r+(M_{k}u_{\varepsilon}(t)+g_k(t))\, \circ dY_t^k, \qquad\hspace*{-5pt} \varepsilon=0,1, \] driven by the same multidimensional martingale $Y=(Y^k)$ and by different increasing processes $V_{0}^r$, $V_1^r$, $r=1,2,\ldots,d_1$, where $L_r$ and $M^k$ are second-and first-order partial differential operators and $\circ$ stands for the Stratonovich differential. We estimate the moments of the supremum in $t$ of the Sobolev norms of $u_1(t)-u_0(t)$ in terms of the supremum of the differences\break $|V^r_{0t}-V^{r}_{1t}|$. Hence, we obtain moment estimates for the error of a multistage splitting-up method for stochastic PDEs, in particular, for the equation of the unnormalized conditional density in nonlinear filtering.

We study the rate at which the difference $X^n_t=X_t-X_{[nt]/n}$ between a process $X$ and its time-discretization converges. When $X$ is a continuous semimartingale it is known that, under appropriate assumptions, the rate is $\sqrt{n}$, so we focus here on the discontinuous case. Then $\alpha_nX^n$ explodes for any sequence $\alpha_n$ going to infinity, so we consider "integrated errors'' of the form $Y^n_t=\int_0^tX^n_s\,ds$ or $Z^{n,p}_t=\int_0^t|X^n_s|^p\,ds$ for $p\in(0,\infty)$: we essentially prove that the variables $\sup_{s\leq t}|nY^n_s|$ and $\sup_{s\leq t}nZ^{n,p}_s$ are tight for any finite $t$ when $X$ is an arbitrary semimartingale, provided either $p\geq2$ or\break $p\in(0,2)$ and $X$ has no continuous martingale part and the sum $\sum_{s\leq t}|\Delta X_s|^p$ converges a.s. for all $t<\infty$, and in addition $X$ is the sum of its jumps when $p<1$. Under suitable additional assumptions, we even prove that the discretized processes $nY^n_{[nt]/n}$ and $nZ^{n,p}_{[nt]/n}$\vadjust{\vspace{1pt}} converge in law to nontrivial processes which are explicitly given.

As a by-product, we also obtain a generalization of Itö's formula for functions that are not twice continuously differentiable and which may be of interest by itself.

We study the convergence rate to normal limit law for the space requirement of random $m$-ary search trees. While it is known that the random variable is asymptotically normally distributed for $3\le m\le 26$ and that the limit law does not exist for $m>26$, we show that the convergence rate is $O(n^{-1/2})$ for $3\le m\le 19$ and is $O(n^{-3(3/2-\alpha)})$, where $4/3<\alpha<3/2$ is a parameter depending on $m$ for $20\le m\le 26$. Our approach is based on a refinement to the method of moments and applicable to other recursive random variables; we briefly mention the applications to quicksort proper and the generalized quicksort of Hennequin, where more phase changes are given. These results provide natural, concrete examples for which the Berry--Esseen bounds are not necessarily proportional to the reciprocal of the standard deviation. Local limit theorems are also derived.

In this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables. We present results providing sharp analogues of the Burkholder--Rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r.v.'s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms. Among others, we obtain the following sharp inequalities: $E(\sum_{k=1}^n X_k)^t\le 2 \max (\sum_{k=1}^n EX_k^t, (\sum_{k=1}^n a_k)^t)$ for all nonnegative r.v.'s $X_1, \ldots, X_n$ with $E(X_k\mid X_1, \ldots, X_{k-1})\le a_k$, $EX_k^t<\infty$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$; $E(\sum_{k=1}^n X_k)^t\le E\theta^t(1) \max (\sum_{k=1}^n b_k, (\sum_{k=1}^n a_k^s)^{t/s})$ for all nonnegative r.v.'s $X_1, \ldots, X_n$ with $E(X_k^s\mid X_1, \ldots, X_{k-1})\le a_k^s$, $E(X_k^t\mid X_1, \ldots, X_{k-1})\le b_k$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$, $0#x003C;s\le t-1$ or $t\ge 2$, $0#x003C;s\le 1$, where $\theta(1)$ is a Poisson random variable with parameter 1. As applications, new decoupling inequalities for sums of multilinear forms are presented and sharp Khintchine--Marcinkiewicz--Zygmund inequalities for generalized moving averages are obtained. The results can also be used in the study of a wide class of nonlinear statistics connected to problems of long-range dependence and in an econometric setup, in particular, in stabilization policy problems and in the study of properties of moving average and autocorrelation processes. The results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved.

Let $X,\, X_1,\, X_2,\ldots$ be i.i.d. nondegenerate random variables, $S_n=\sum_{j=1}^nX_j$ and $V_n^2=\sum_{j=1}^nX_j^2$. We investigate the asymptotic \vspace*{1pt} behavior in distribution of the maximum of self-normalized sums, $\max_{1\le k\le n}S_k/V_k$, and the law of the iterated logarithm for self-normalized sums, $S_n/V_n$, when $X$ belongs to the domain of attraction of the normal law. In this context, we establish a Darling--Erdős-type theorem as well as an Erdős--Feller--Kolmogorov--Petrovski-type test for self-normalized sums.

In Crisan, Gaines and Lyons [SIAM J. Appl. Probab. 58 (1998) 313--342] we describe a branching particle algorithm that produces a particle approximation to the solution of the Zakai equation and find an upper bound for the rate of convergence of the mean square error. In this paper, the exact rate of convergence of the mean square error is deduced. Also, several variations of the branching algorithm with better rates of convergence are introduced.

The notion of an $L_{1}$-norm density estimator process indexed by a class of kernels is introduced. Then a functional central limit theorem and a Glivenko--Cantelli theorem are established for this process. While assembling the necessary machinery to prove these results, a body of Poissonization techniques and restricted chaining methods is developed, which is useful for studying weak convergence of general processes indexed by a class of functions. None of the theorems imposes any condition at all on the underlying Lebesgue density $f$. Also, somewhat unexpectedly, the distribution of the limiting Gaussian process does not depend on $f$.

We characterize Yang--Mills connections in vector bundles in terms of covariant derivatives of stochastic parallel transport along variations of Brownian bridges on the base manifold. In particular, we prove that a connection in a vector bundle $E$ is Yang--Mills if and only if the covariant derivative of parallel transport along Brownian bridges (in the direction of their drift) is a local martingale, when transported back to the starting point. We present a Taylor expansion up to order $3$ for stochastic parallel transport in $E$ along small rescaled Brownian bridges and prove that the connection in $E$ is Yang--Mills if and only if all drift terms in the expansion (up to order 3) vanish or, equivalently, if and only if the average rotation of parallel transport along small bridges and loops is of order $4$.

We consider the stochastic differential equation \[ dX_t=dW_t+dA_t, \] where $W_t$ is $d$-dimensional Brownian motion with $d\geq 2$ and the $i$th component of $A_t$ is a process of bounded variation that stands in the same relationship to a measure $\pi^i$ as $\int_0^t f(X_s)\, ds$ does to the measure $f(x)\, dx$. We prove weak existence and uniqueness for the above stochastic differential equation when the measures $\pi^i$ are members of the Kato class $\K_{d-1}$. As a typical example, we obtain a Brownian motion that has upward drift when in certain fractal-like sets and show that such a process is unique in law.

Physicists have studied the stochastic assignment problem using ideas from statistical mechanics. For a version of this problem, we give, at high enough temperature, a complete proof of the existence of the structure they predict.

$m$-dependent stationary infinitely divisible sequences are characterized as a class of generalized finite moving average sequences via the structure of the associated Lévy measure. This characterization is used to find necessary and sufficient conditions for the weak convergence of centered and normalized partial sums of $m$-dependent stationary infinitely divisible sequences. Partial sum convergence for stationary infinitely divisible sequences that can be approximated by $m$-dependent ones is then studied.

A new global isomorphism theorem is obtained that expresses the local times of transient regular diffusions under $P^{x,y}$, in terms of related Gaussian processes. This theorem immediately gives an explicit description of the local times of diffusions in terms of $0$th order squared Bessel processes similar to that of Eisenbaum and Ray's classical description in terms of certain randomized fourth order squared Bessel processes. The proofs given are very simple. They depend on a new version of Kac's lemma for $h$-transformed Markov processes and employ little more than standard linear algebra. The global isomorphism theorem leads to an elementary proof of the Markov property of the local times of diffusions and to other recent results about the local times of general strongly symmetric Markov processes. The new version of Kac's lemma gives simple, short proofs of Dynkin's isomorphism theorem and an unconditioned isomorphism theorem due to Eisenbaum.

This is a contribution to the theory of sums of independent random variables at the level of optimal explicit inequalities: we compute the optimal constants in Hornich's lower bounds for the mean absolute deviations of sample means. This is done by reducing the original problem to the elementary one of determining the minimally concentrated binomial distributions $B_{n,p}$ with fixed sample size parameter $n$.

Extending results by P. 1981--1983 we give strong approximation results for sample maxima by simple transformations of extremal processes and discuss the quality of the approximations. The limiting process becomes stationary after a transformation of the argument.

We investigate martingale inequalities in noncommutative $L^p$-spaces associated with a von Neumann algebra equipped with a faithful normal state. We prove the noncommutative analogue of the classical Burkholder inequality on the conditioned (or little) square function and extend the noncommutative Burkholder--Gundy inequalities from Comm. Math. Phys. 189 (1997) 667--698 to this nontracial setting. We include several related results.

In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index $\alpha \in (1, 2]$. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one ancestor and conditioned on the total progeny, converges in a functional sense, to a new process: the normalized excursion of the continuous height process associated with the $\alpha $-stable CSBP. We deduce from this convergence an analogous limit theorem for the contour process. In the Brownian case $\alpha =2$, the limiting process is the normalized Brownian excursion that codes the continuum random tree: the result is due to Aldous who used a different method.

We study the paths of minimal cost for first-passage percolation in two dimensions and obtain an exponential bound on the tail probability of the ratio of the lengths of the shortest and longest of these. This inequality permits us to answer a long-standing question of Hammersley and Welsh on the shift differentiability of the time constant. Specifically, we show that for subcritical Bernoulli percolation the time constant is not shift differentiable when $p$ is close to one-half.

Let the Gaussian process $X_m(t)$ be the $m$-fold integrated Brownian motion for positive integer $m$. The Laplace transform of the quadratic functional of $X_m(t)$ is found by using an appropriate self-adjoint integral operator. The result is then used to show the power of a general connection between small ball probabilities for the Gaussian process. The connection is discovered by introducing an independent random shift. The interplay between our results and the principal eigenvalues for nonuniform elliptic generators on an unbounded domain is also discussed.

Consider the first exit time $\tau_D$ of a $(d+1)$-dimensional Brownian motion from an unbounded open domain $D=\set{ (x, y) \in \R^{d+1} \dvtx y > f(x), x \in \R^d }$ starting at\vspace{0.5pt} $(x_0, f(x_0)+1)\in \R^{d+1}$ for some $x_0 \in \R^d$, where the function $f(x)$ on $\R^d$ is convex and $f(x) \to \infty$ as the Euclidean norm $|x| \to \infty$. Very general estimates for the asymptotics of $\log \pr{\tau_D >t}$ are given by using Gaussian techniques. In particular, for $f(x)=\exp\{ |x|^p\}$, $p >0$, \[ \lim_{t\to\infty} t^{-1} (\log t)^{2/p} \log \pr{\tau_D>t}=-j_\nu^2/2, \] where $\nu=(d-2)/2$ and $j_\nu$ is the smallest positive zero of the Bessel function $J_{\nu}$.

The primary goal of this paper is to study the range of the random field $X(t) = \sum_{j=1}^N X_j(t_j)$, where $X_1,\ldots, X_N$\vspace*{-1pt} are independent Lévy processes in $\R^d$.

To cite a typical result of this paper, let us suppose that $\Psi_i$ denotes the Lévy exponent of $X_i$ for each $i=1,\ldots,N$. Then, under certain mild conditions, we show that a necessary and sufficient condition for $X(\R^N_+)$ to have positive $d$-dimensional Lebesgue measure is the integrability of the function $\R^d \ni \xi \mapsto \prod_{j=1}^N \Re \{ 1+ \Psi_j(\xi)\}^{-1}$. This extends a celebrated result of Kesten and of Bretagnolle in the one-parameter setting. Furthermore, we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion. This extends a theorem of Hawkes to the present random fields setting and completes the analysis of local times for additive Lévy processes initiated in a companion by paper Khoshnevisan, Xiao and Zhong.