In this paper, we shall develop certain inequalities to bound the difference between distributions in terms of their Stieltjes transforms. Using these inequalities, convergence rates of expected spectral distributions of large dimensional Wigner and sample covariance matrices are established. The paper is organized into two parts. This is the first part, which is devoted to establishing the basic inequalities and a convergence rate for Wigner matrices.

In the first part of the paper, we develop certain inequalities to bound the difference between distributions in terms of their Stieltjes transforms and established a convergence rate of expected spectral distributions of large Wigner matrices. The second part is devoted to establishing convergence rates for the sample covariance matrices, for the cases where the ratio of the dimension to the degrees of freedom is bounded away from 1 or close to 1, respectively.

A Gaussian upper bound for the iterated kernels of Markov chains is obtained under some natural conditions. This result applies in particular to simple random walks on any locally compact unimodular group $G$ which is compactly generated. Moreover, if $G$ has polynomial volume growth, the Gaussian upper bound can be complemented with a similar lower bound. Various applications are presented. In the process, we offer a new proof of Varopoulos' results relating the uniform decay of convolution powers to the volume growth of $G$.

This paper considers random processes of the form $X_{n + 1} = a_nX_n + b_n (\operatorname{mod} p)$, where $X_0 = 0$ and the sequences $a_n$ and $b_n$ are independent with $a_n$ identically distributed for $n = 0, 1, 2, \ldots$ and $b_n$ identically distributed for $n = 0, 1, 2, \ldots$. Chung, Diaconis and Graham studied such processes where $a_n = 2$ always; this paper considers more general distributions for $a_n$ and $b_n$. The question is how long does it take these processes to get close to the uniform distribution? If $a_n$ is a distribution on $\mathbf{Z}^+$ which does not vary with $p$ and $b_n$ is a distribution on $\mathbf{Z}$ which also does not vary with $p$, an upper bound on this time is $O((\log p)^2)$ with appropriate restrictions on $p$ unless $a_n = 1$ always, $b_n = 0$ always or $a_n$ and $b_n$ can each take on only one value. This paper uses a recursive relation involving the discrete Fourier transform to find the bound. Under more restrictive conditions for $a_n$ and $b_n$, this paper finds that a generalization of the technique of Chung, Diaconis and Graham shows that $O(\log p \log \log p)$ steps suffice.

Let $(X, \mathscr{B})$ be a compact metric space with $\mathscr{B}$ the $\sigma$-field of Borel sets. Suppose this is the state space of a discrete parameter Markov process. Let $C$ be a closed convex set of probability measures on $X$. Known results on the asymptotic behavior of the probability that the empirical distributions $\hat{P}_n$ belong to $C$ and new results on the Markov process distribution of $\omega_0, \ldots, \omega_{n - 1}$ under the condition $\hat{P}_n \in C$ are obtained simultaneously through a large deviations estimate. In particular, the Markov process distribution under the condition $\hat{P}_n \in C$ is shown to have an asymptotic quasi-Markov property, generalizing a concept of Csiszar.

Let $(X_i)_{i \in \mathbb{Z}^d_+}$ be an array of independent identically distributed zero-mean random vectors with values in $\mathbb{R}^k$. When $E(|X_1|^r) < + \infty$, for some $r > 2$, we obtain the strong approximation of the partial sum process $(\sum_{i \in \nu S}X_i: S \in \mathscr{J})$ by a Gaussian partial sum process $(\sum_{i \in \nu S}Y_i: S \in \mathscr{J})$, uniformly over all sets in a certain Vapnik-Chervonenkis class $\mathscr{J}$ of subsets of $\lbrack 0, 1\rbrack^d$. The most striking result is that both an array $(X_i)_{i \in \mathbb{Z}^d_+}$ of i.i.d. random vectors and an array $(Y_i)_{i \in \mathbb{Z}^d_+}$ of independent $N(0, \operatorname{Var} X_1)$-distributed random vectors may be constructed in such a way that, up to a power of $\log \nu$, $\sup_S \in \mathscr{J} |\sum_{i \in \nu S} (X_i - Y_i)| = O(\nu^{(d - 1)/2} \vee \nu^{d/r}) \mathrm{a.s.},$ for any Vapnik-Chervonenkis class $\mathscr{J}$ fulfilling the uniform Minkowsky condition. From a 1985 paper of Beck, it is straightforward to prove that such a result cannot be improved, when $\mathscr{J}$ is the class of Euclidean balls.

Let $S_n$ denote the $n$th normalized partial sum of a sequence of mean zero, weakly dependent random vectors. This paper gives asymptotic expansions for $Ef(S_n)$ under weaker moment conditions than those of Gotze and Hipp (1983). It is also shown that an expansion for $Ef(S_n)$ with an error term $o(n^{-(s - 2)/2})$ is valid without any Cramer-type condition, if $f$ has partial derivatives of order $(s - 1)$ only. This settles a conjecture of Gotze and Hipp in their 1983 paper.

The paper contains a "smoothed" one-step triangular array asymptotic expansion for discrete-time martingales. An important element of the proof is a second-order description of Skorokhod embedding of discrete martingales in continuous ones. An application to Markov processes is given, along with a bootstrapping example.

Let $\{X_k\}_{k \in \mathbb{N}}$ be a nonstationary sequence of random variables. Sufficient conditions are found for the existence of an independent sequence $\{\tilde{X}_k\}_{k \in \mathbb{N}}$ such that $\sup_{x \in \mathbb{R}^1}|P(M_n \leq x) - P(\tilde{M}_n \leq x)| \rightarrow 0$ as $n \rightarrow \infty$, where $M_n$ and $\tilde{M}_n$ are $n$th partial maxima for $\{X_k\}$ and $\{\tilde{X}_k\}$, respectively.

Let $X_i, i \geq 1$, be independent random variables with a common distribution in the domain of attraction of a strictly stable law, and for each $n \geq 1$ let $X_{1, n} \leq \cdots \leq X_{n, n}$ denote the order statistics of $X_1, \ldots, X_n$. In 1986, S. Csorgo, Horvath and Mason showed that for each sequence $k_n, n \geq 1$, of nonnegative integers with $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$, the trimmed sums $S_n(k_n) = X_{k_n + 1, n} + \cdots + X_{n - k_n, n}$ converge in distribution to the standard normal distribution, when properly centered and normalized, despite the fact that the entire sums $X_1 + \cdots + X_n$ have a strictly stable limit, when properly centered and normalized. The asymptotic almost sure behavior of $S_n(k_n)$ strongly depends on the rate at which $k_n$ converges to $\infty$. The sequences $k_n \sim c \log \log n$ as $n \rightarrow \infty$ for $0 < c < \infty$ constitute a borderline case between a classical law of the iterated logarithm and a radically different behavior. This borderline case is investigated in detail for nonnegative summands $X_i$.

We consider two-person zero-sum stochastic games with limit superior payoff function and Borel measurable state and action spaces. The games are shown to have a value and the value function is calculated by transfinite iteration of an operator and proved to be upper analytic. The paper extends results of our earlier article [17] in which the same class of games was considered for countable state spaces and finite action sets.

Let $\mathbf{B}$ be a separable Banach space and let $\mu$ be a centered Poisson probability measure on $\mathbf{B}$ with Levy measure $M$. Assume that $M$ admits a polar decomposition in terms of a finite measure $\sigma$ on the unit sphere of $\mathbf{B}$ and a Levy measure $\rho$ on $(0, \infty)$. The main result of this paper provides a complete description of the structure of $\mathscr{J}_\mu$, the support of $\mu$. Specifically, it is shown that: (i) if $\int_{(0, 1\lbrack} s\rho(ds) = \infty$, then $\mathscr{J}_\mu$ is a linear space and is equal to the closure of the semigroup generated by $\mathscr{J}_M$ (the support of $M$) and the negative of the barycenter of $\sigma$; and (ii) if $\int_{(0, 1\rbrack} s\rho(ds) < \infty$ and zero is in the support of $\rho$, then $\mathscr{J}_\mu$ is a convex cone and is equal to the closure of the semigroup generated by $\mathscr{J}_M$. The result (i) yields an affirmative answer to the question, open for some time, of whether the support of a stable probability measure of index $1 \leq \alpha < 2$ on $B$ is a translate of a linear space. Analogs of these results, for both Poisson and stable probability measures defined on general locally convex spaces, are also provided.

The infinite secretary problem, in which an infinite number of rankable items arrive at times which are i.i.d., uniform on (0, 1), is modified to allow for a fixed period of recall of length $\alpha, 0 \leq \alpha \leq 1$. The goal is to find the maximum probability of best choice, $v = v(\alpha)$, as well as an optimal stopping time $\tau^\ast = \tau^\ast(\alpha)$. A differential-delay equation is derived, the solution of which yields $v(\alpha)$ and $\tau^\ast(\alpha)$, the latter given in terms of a constant $t^\ast \lbrack = t^\ast(\alpha)\rbrack$. For $\alpha \geq 1/2$, the complete solution to the problem is obtained. For $0 < \alpha < 1/2, v(\alpha)$ cannot be put in closed form, so upper and lower bounds for $v(\alpha)$ and $t^\ast(\alpha)$ are obtained and are investigated for $\alpha$ near 0 and near 1/2, where the solutions are known. We also find asymptotic expansions of $v(\alpha)$ and $t^\ast(\alpha)$ about $\alpha = 0$ and $\alpha = 1/2$. Finally, the solution to the finite, $n$-item length-$m$ recall problem introduced by Smith and Deely is shown to converge to the solution of the infinite problem when $m/n \rightarrow \alpha$.

We consider the problem of when the bootstrap sample mean, appropriately normalized and centered, converges in distribution along almost every sample path. We allow the normalizing sequence to be an arbitrary sequence of positive random variables. It is proved that the only possible normalizing sequence is essentially $(\sum^n_{i = 1}X^2_i)^{1/2}$. Furthermore, if the bootstrap sample mean converges along almost every sample path, then either the variance is finite or else the distribution of $X$ is extremely heavy tailed. In the latter case, the distribution of the bootstrap sample mean is completely determined by how many times the maximum order statistic from the original random sample is repeated in the bootstrap sample. The necessary condition on how heavy the tails must be is $(\sum^n_{i = 1}|X_i|^p)^{1/p}/(\sum^n_{i = 1}X^2_i)^{1/2} \rightarrow 1$ almost surely for all $p \in (0, \infty\rbrack$. Furthermore, we show that in this case the limit of the bootstrap sample mean normalized by $(\sum^n_{i = 1}X^2_i)^{1/2}$ is Poisson with mean 1.

Bootstrap percolation is a model in which an element of $\mathbf{Z}^2$ becomes occupied in one time unit if two appropriately chosen neighbors are occupied. Schonmann [4] proved that starting from a Bernoulli product measure of positive density, the distribution of the time needed to occupy the origin decays exponentially. We show that for $\alpha > 1$, the exponent can be taken as $\delta p^{2\alpha}$ for some $\delta > 0$, thus showing that the associated characteristic exponent is at most two. Another characteristic exponent associated to this model is shown to be equal to one.

A random walk with obstacles in $\mathbf{R}^d, d \geq 2$, is considered. A probability measure is put on a space of obstacles, giving a random walk with random obstacles. A central limit theorem is then proven for this process when the obstacles are distributed by a Gibbs state with sufficiently low activity. The same problem is treated for a tagged particle of an infinite hard core particle system.

A Cameron-Martin-Girsanov-Maruyama type formula for symmetric diffusions on infinite dimensional state space is proved. In particular, relaxations of the usual assumptions which still imply absolute continuity (but possibly no longer equivalence) of the path space measures are discussed. In addition a converse result is proved, that is, we show that absolute continuity of the path space measures enables us to identify the underlying Dirichlet form.

The empirical measure, a generalization of occupation times, of a super-Brownian motion is studied. In our case the empirical measure tends almost surely to Lebesgue measure as time $t \rightarrow \infty$. Asymptotic probabilities of deviation from this central behavior by various orders (large, not very large and normal deviations) are estimated. Extension to similar superprocesses, that is, Dawson-Watanabe processes, is discussed. Our analytic approach also produces new results for semilinear PDE's.

Subadditive functionals on the space of sample paths include suprema, integrals of paths, oscillation on sets and many others. In this paper we find an optimal condition which ensures that the distribution of a subadditive functional of sample paths of an infinitely divisible process belongs to the subexponential class of distributions. Further, we give exact tail behavior for the distributions of such functionals, thus improving many recent results obtained for particular forms of subadditive functionals and for particular infinitely divisible processes.

We continue the study of the asymptotic behavior of Markov processes $(X^\varepsilon(t), \nu^\varepsilon(t))$ corresponding to systems of elliptic PDE with a small parameter $\varepsilon > 0$. In the present paper we consider the case where the process $(X^\varepsilon(t), \nu^\varepsilon(t))$ can leave a given domain $D$ only due to large deviations from the degenerate process $(X^0(t), \nu^0(t))$. In this way we study the limit behavior of solutions of corresponding Dirichlet problems.

We prove a local version of Strassen's law of the iterated logarithm. Instead of shrinking larger and larger pieces of a Brownian path and letting time go to infinity, we look at a sequence of functions we get by blowing up smaller and smaller pieces and we investigate the asymptotic behaviour of this sequence as time goes to zero. It turns out that this sequence of functions is a relatively compact subset of $C\lbrack 0, 1\rbrack$ with probability 1, and the set of its limit points is the same as in Strassen's theorem.

Let $f$ be a square-integrable function on the unit square. Assume that the singular numbers $(a_i)_{i \in \mathbb{N}}$ of the Hilbert-Schmidt operator associated with $f$ admit some $0 < \alpha < \frac{1}{3}$ such that $\sum^\infty_{i = 1}|a_i|^\alpha < \infty$. We present a purely stochastic method to investigate the occupation densities of the Skorohod integral process $U$ induced by $f$. It allows us to show that $U$ possesses continuous square-integrable occupation densities and obviously generalizes beyond the second Wiener chaos.

The Brownian and stable density processes are distribution valued processes that arise both via limiting operations on infinite collections of Brownian motions and stable Levy processes and as the solutions of certain stochastic partial differential equations. Their (self-)intersection local times (ILT's) of various orders can be defined in a manner somewhat akin to that used to define the self-intersection local times of simple $\mathscr{R}^d$-valued processes; that is, via a limiting operation involving approximate delta functions. We obtain a full characterisation of this limiting procedure, determining precisely in which cases we have convergence and deriving the appropriate renormalisation for obtaining weak convergence when only this is available. We also obtain results of a fluctuation nature, that describe the rate of convergence in the former case. Our results cover all dimensions and all levels of self-intersection.

If $L$ generates a transient diffusion, then the corresponding exterior Dirichlet problem (EP) has in general many bounded solutions. We consider perturbations of $L$ by a first-order term and assume that EP can be solved uniquely for each perturbed operator. Then as the perturbation tends to 0, the sequence of perturbed solutions may converge to a solution of the original EP. Using a skew-product representation of diffusions, we give an integral criterion for the uniqueness of this limit and show that it takes place iff the Kuramochi boundary of $L$ at $\infty$ is a singleton. In the case when uniqueness fails, we provide a description of a subclass of limiting solutions in terms of boundary conditions for the original process in the natural scale.

Under suitable conditions on a stopping time $T$ and zero mean i.i.d. random variables $\{X_n, n \geq 1\}$, a Wald-type equation $ES_{k, T} = 0$ is obtained where $S_{k, n}$ is the sum of products of $k$ of the $X$'s with indices from 1 to $n$. This, in turn, is utilized to obtain information about the moments of $T_k = \inf\{n \geq k: S_{k, n} \geq 0\}$ and $W_c = \inf\{n \geq 2: S^2_{1, n} > c\sum^n_{j = 1}X^2_j\}, c > 0$.

The limiting distribution of weighted $U$-statistics of degree 2 is found for a wide class of weights, including uniform weights. Nonnormal limits can occur for both degenerate and nondegenerate kernels. A compact expression is given for the cumulants of the distribution. Incomplete and randomly weighted $U$-statistics are also analyzed.

Empirical processes of $U$-statistic structure were introduced by Serfling and studied in detail by Silverman, who proved weak convergence of weighted versions in the i.i.d. case. Our main theorem shows that this result can be generalized in two directions: First, the i.i.d. assumption can be omitted, and second, our proofs holds for a richer class of weight functions. In addition, we obtain almost sure convergence of weighted $U$-processes in the i.i.d. case which improves the results of Helmers, Janssen and Serfling, Aerts, Janssen and Mason and (in the special situation of the real line) Nolan and Pollard.