A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.

Recent results on the Berry-Esseen bound for $U$-statistics assumed the following conditions: Suppose a $U$-statistic (of degree 2) is nondegenerate. Then the rate of convergence in the CLT is of the order $O(n^{-1/2})$ provided that $\mathbb{E}|\mathbb{E}\{h(X_1, X_2)|X_1\}|^3 < \infty, \mathbb{E}|h(X_1, X_2)|^{5/3} < \infty,$ where $h$ is a symmetric kernel corresponding to the $U$-statistic. It follows from our results that these moment conditions are final. In particular, the last moment condition cannot be replaced by a moment of order $5/3 - \epsilon$ for any $\epsilon > 0$. Similar results hold for von Mises statistics.

Let $\{X_i, 1 \leq i \leq n\}$ be independent and identically distributed random variables. For a symmetric function $h$ of $m$ arguments, with $\theta = Eh(X_1,\ldots, X_m)$, we propose estimators $\theta_n$ of $\theta$ that have the property that $\theta_n \rightarrow \theta$ almost surely (a.s.) and $\theta_n \geq \theta$ a.s. for all large $n$. This extends the results of Gilat and Hill, who proved this result for $\theta = Eh(X_1)$. The proofs here are based on an almost sure representation that we establish for $U$ statistics. As a consequence of this representation, we obtain the Marcinkiewicz-Zygmund strong law of large numbers for $U$ statistics and for a special class of $L$ statistics.

For i.i.d. data $X_1,\cdots, X_n$ and a kernel $h$, the associated $U$-statistic process is defined as $U_n (u, v) = \frac{1}{n(n-1)} \sum_{1\leq i\neq j\leq n} h(X_i, X_j)1_{\{X_i\leq u,X_j\leq \nu\}}.$ Variants of these processes occur, for example, in the representation of the product-limit estimator of a lifetime distribution for censored/truncated data or in trimmed $U$-statistics. We derive an almost sure representation of $U_n$ under weak moment assumptions on $h$. Proofs rely on a proper decomposition of the remainder term into strong two-parameter martingales.

We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity and so forth.

Let $X_{n,i}$ denote an infinitesimal array of independent random variables with convergent partial sums $Z_n = \sum^n_{i=1} X_{n,i} -a_n \rightarrow_\mathscr{D}\xi$. Throughout, we find conditions for the convergence of the portion $k_n$ of lower extremes $L_n(k_n) = \sum^{k_n}_{i=1}X_{i:n} - b_n$ given by order statistics $X_{i:n}$. Similarly, $W_n(r_n)$ denotes the sum of the $r_n$ upper extremes and $M_n = Z_n - L_n - W_n$ stands for the middle part of the sum. It is shown that $(L_n, M_n, W_n) \rightarrow_\mathscr{D} (\xi_1, \xi_2, \xi_3)$ jointly converges for various sequences $k_n, r_n \rightarrow \infty$, where the components of the limit law are independent such that $\xi_1 + \xi_2 + \xi_3 =_\mathscr{D} \xi$. The limit of the middle part $\xi_2$ is asymptotically normal and $\xi_1 (\xi_3)$ gives the negative (positive) spectral Poisson part of $\xi$. In the case of a compound Poisson limit distribution we obtain rates of convergence that can be used for applications to insurance mathematics.

We prove that there exists a constant $a(A) \in (0, \infty)$ such that $\lim \inf_{t \rightarrow \infty} (\log \log t/t)\sup_{0 \leq s \leq t}|\int^s_0\langle AW_u, dW_u\rangle | = a(A)$ with probability 1, where $A$ is a skew-symmetric $d \times d$ matrix, $A \neq 0$, and $\{W_t\}_{t\geq 0}$ is a $d$-dimensional Wiener process.

Let $X_1, X_2,\ldots$, be a sequence of independent and identically distributed random variables. We find sequences of norming and centering constants $\alpha_n$ and $\beta_n$ such that a universal Chung-type law of the iterated logarithm holds, namely, $\lim \inf_{n\rightarrow \infty} \max_{1\leq k \leq n}|S_k - k\beta_n|/\alpha_n < \infty$ almost surely, where $S_k$ denotes the sum of the first $k$ of $X_1, X_2,\ldots, k \geq 1$. If the underlying distribution function is in the Feller class, we show that this $\lim \inf$ is strictly positive with probability 1.

We prove that the $\lim \inf$ of suitably normalized sums of i.i.d. nonnegative and nondegenerate random variables can with probability 1 only be a constant between $-2^{1/2}$ and 0. Moreover, we show that each value within this range is attainable by an appropriate choice of the underlying common distribution function.

We give a necessary and sufficient condition for the Strassen functional law of the iterated logarithm for the Wiener process to hold when the topology is defined by a general class of norms on $\mathbb{C}(0, 1)$.

Let $S_n = X_1 + \cdots + X_n$ be partial sums of independent identically distributed random variables and let $a_n$ be an increasing sequence of positive constants tending to $\infty$. This paper concerns the almost sure lower limit of $\max_{1\leq j \leq n} S_j/a_n$. We prove that the lower limit is either 0 or $\infty$ under mild conditions and give integral tests to determine which is the case. Let $\tau = \inf\{n \geq 1: S_n > 0\}$ and $\tau_- = \inf\{n \geq 1: S_n \leq 0\}$. Several inequalities are given that determine up to scale constants various quantities involving truncated moments of the ladder variables $S_\tau$ and $\tau$ under three different conditions: $ES_\tau < \infty, E|S_{\tau-}| < \infty$ and $X$ symmetric. Moments of ladder variables are also discussed.

Lim inf results for Gaussian samples are obtained. In particular, there are applications to rates of convergence for the functional form of Chung's law of the iterated logarithm for Brownian motion.

Let $X = \{X(t); t \in T\}$ be a measurable symmetric $\alpha$-stable process, $0 < \alpha < 2$. In this paper necessary and sufficient conditions for $X$ to have almost all sample paths in an Orlicz space $\mathbb{L}_\psi(T, \mu)$ with a function $\psi$ satisfying the $\Delta_2$-condition are given.

We study Slepian inequalities for general non-Gaussian infinitely divisible random vectors. Conditions for such inequalities are expressed in terms of the corresponding Levy measures of these vectors. These conditions are shown to be nearly best possible, and for a large subfamily of infinitely divisible random vectors these conditions are necessary and sufficient for Slepian inequalities. As an application we consider symmetric $\alpha$-stable Ornstein-Uhlenbeck processes and a family of infinitely divisible random vectors introduced by Brown and Rinott.

Let $S_n = X_1 + \cdots + X_n, n \geq 1$, be a $d$-dimensional random walk and let $T_a = \inf\{n \geq n_a: ng(S_n/n) \geq a\}$, where $n_a = o(a)$. Let $\theta = g(EX_1), \hat{\theta}_n = g(S_n/n)$ and $\Delta_a = T_a\hat{\theta}_{T_a} - a$. Edgeworth-type expansions are developed for $P\{T_a = n, y_1 \leq \Delta_a \leq y_2\}$ and for the distribution functions of $T_a$ and of $\sqrt T_a(h(\hat{\theta}_{T_a}) - h(\theta))$, where $h$ is a real-valued function such that $h'(\theta) \neq 0$.

Consider two independent sequences $X_1,\ldots, X_n$ and $Y_1,\ldots, Y_n$. Suppose that $X_1,\ldots, X_n$ are i.i.d. $\mu X$ and $Y_1,\ldots, Y_n$ are i.i.d. $\mu_Y$, where $\mu_X$ and $\mu_Y$ are distributions on finite alphabets $\sum_X$ and $\sum_Y$, respectively. A score $F: \sum_X \times \sum_Y \rightarrow \mathbb{R}$ is assigned to each pair $(X_i, Y_j)$ and the maximal nonaligned segment score is $M_n = \max_{0\leq i, j \leq n - \Delta, \Delta \geq 0}\{\sum^\Delta_{l=1}F(X_{i+l}, Y_{j+l})\}$. Our result is that $M_n/\log n \rightarrow \gamma^\ast(\mu_X, \mu_Y)$ a.s. with $\gamma^\ast$ determined by a tractable variational formula. Moreover, the pair empirical measure of $(X_{i+l}, Y_{j+l})$ during the segment where $M_n$ is achieved converges to a probability measure $\nu^\ast$, which is accessible by the same formula. These results generalize to $X_i, Y_j$ taking values in any Polish space, to intrasequence scores under shifts, to long quality segments and to more than two sequences.

Consider two independent sequences $X_1,\ldots, X_n$ and $Y_1,\ldots, Y_n$. Suppose that $X_1,\ldots, X_n$ are i.i.d. $\mu_X$ and $Y_1,\ldots, Y_n$ are i.i.d. $\mu_Y$, where $\mu_X$ and $\mu_Y$ are distributions on finite alphabets $\sigma_X$ and $\sigma_Y$, respectively. A score $F: \sigma_X \times \sigma_Y\rightarrow \mathbb{R}$ is assigned to each pair $(X_i, Y_j)$ and the maximal nonaligned segment score is $M_n = \max_{0\leq i, j\leq n - \Delta, \Delta \geq 0} \{\sum^\Delta_{k=1} F(X_{i+k}, Y_{j+k})\}$. The limit distribution of $M_n$ is derived here when $\mu_X$ and $\mu_Y$ are not too far apart and $F$ is slightly constrained.

A reaction-diffusion model, constructed on a grid by linking nonlinear density-dependent birth and death processes through particle diffusion, is studied by proving two laws of large numbers and a fluctuation theorem under varying assumptions on the density parameter.

In this paper we study multidimensional stochastic reaction-diffusion equations (SRDE's) with white noise boundary data. More precisely, we consider a general SRDE with Robin data to be a white noise field. Because this boundary data is very irregular, we formulate a set of conditions that a random field must satisfy to solve the SRDE. We show that a unique solution exists, and we study the boundary-layer behavior of the solution. This boundary-layer analysis reveals some natural restrictions on the reaction term of the SRDE that ensure that the reaction term does not qualitatively affect the boundary layer. The boundary-layer analysis also leads to the definition of some functional Banach spaces into which are encoded the boundary-layer degeneracies and that would be the natural settings for other analyses of the SRDE of this paper (e.g., large deviations and central limit theorems, approximation theorems).

A large deviation system is established for the empirical processes of a mean-field interacting particle system with unbounded jump rates under assumptions that are satisfied by many interesting models including the first and the second Schlogl models. The action functional obtained has a form that is very useful for applications.

Let $S$ be a real-valued random walk that does not drift to $\infty$, so $P(S_k \geq 0$ for all $k) = 0$. We condition $S$ to exceed $n$ before hitting the negative half-line, respectively, to stay nonnegative up to time $n$. We study, under various hypotheses, the convergence of these conditional laws as $n \rightarrow \infty$. First, when $S$ oscillates, the two approximations converge to the same probability law. This feature may be lost when $S$ drifts to $-\infty$. Specifically, under suitable assumptions on the upper tail of the step distribution, the two approximations then converge to different probability laws.

We show that a Jackson network relaxes exponentially fast to its steady state by giving a lower bound on the Cheeger constant for the associated Markov process. We also give lower bounds on the mean time until some node of the network overflows.

In this article, we give some sufficient conditions for the weak convergence of Markov processes in terms of their extended generators. For the convergence of Markov processes with generators defined by Dynkin, necessary and sufficient conditions are obtained. As an application, we will discuss the convergence of diffusion processes with jumps.

We determine the Martin boundary for aperiodic simple random walk on a bi-tree, that is, the Cartesian product of two homogeneous trees. This is obtained by first deriving a "renewal theorem," giving an asymptotic estimate of the Green kernel as the space variable tends to infinity. The basic tool is a result of Lalley that gives a uniform estimate of transition probabilities of nearest neighbour random walks on trees.

In this paper, we prove a necessary and sufficient condition for tightness of products of i.i.d. finite-dimensional random real matrices. We give an example illustrating the use of our theorem and treat in more details the case of $2 \times 2$ real matrices.

We give estimates of the distances between the densities of measures having the same finite number of the same marginals. These estimates give a solution of the computer tomography paradox of Gutman, Kemperman, Reeds and Shepp, and open the possibility for construction of a new method of inversion of the Radon transformation.

Limit theorems for functions of stationary mean-zero Gaussian sequences of vectors satisfying long range dependence conditions are considered. Depending on the rate of decay of the coefficients, the limit law can be either Gaussian or the law of a multiple Ito-Wiener integral. We prove the bootstrap of these limit theorems in the case when the limit is normal. A sufficient bracketing condition for these limit theorems to happen uniformly over a class of functions is presented.