A real-valued process $X = (X(t))_{t\in\mathbb{R}}$ is self-similar with exponent $H (H$-ss), if $X(a\cdot) =_d a^HX$ for all $a > 0$. Sample path properties of $H$-ss processes with stationary increments are investigated. The main result is that the sample paths have nowhere bounded variation if $0 < H \leq 1$, unless $X(t) \equiv tX(1)$ and $H = 1$, and apart from this can have locally bounded variation only for $H > 1$, in which case they are singular. However, nowhere bounded variation may occur also for $H > 1$. Examples exhibiting this combination of properties are constructed, as well as many others. Most are obtained by subordination of random measures to point processes in $\mathbb{R}^2$ that are Poincare, i.e., invariant in distribution for the transformations $(t, x) \mapsto (at + b, ax)$ of $\mathbb{R}^2$. In a final section it is shown that the self-similarity and stationary increment properties are preserved under composition of independent processes: $X_1 \circ X_2 = (X_1(X_2(t)))_{t\in \mathbb{R}}$. Some interesting examples are obtained this way.

A real-valued process $X = (X(t))_{t\in \mathbb{R}}$ is self-similar with exponent $H(H$-ss), if $X(a\cdot) =_d a^HX$ for all $a > 0$. The present paper studies $H$-ss processes $X_H$ with stationary increments that can be represented for $t > 0$ by $X_H(t) := \int |x|^H \operatorname{sgn} x \Pi((0, t\rbrack, dx) =: \int x \Pi^H((0, t\rbrack, dx)$, where $\Pi$ is a point process in $\mathbb{R}^2$ that is Poincare, i.e., invariant in distribution under the transformations $(t, x) \mapsto (\text{at} + b, ax)$ of $\mathbb{R}^2$. In particular, $X_H$ allows such a representation if it is a jump process, $\Pi^H$ being the graph of its jumps. Several examples of Poincare processes $\Pi$ are presented. These lead in many cases to new examples of $H$-ss processes $X_H$ with stationary increments. Furthermore, it is investigated for which $H$ the integral expression for $X_H$ converges, conditionally or absolutely. If $\Pi$ has finite intensity $\mathbb{E}\Pi$, then $(1, \infty)$ is wp1 the set of $H$ for which $X_H$ converges absolutely. If $\mathbb{E}\Pi$ is not finite, then the situation is more complicated, as is the case for conditional convergence. Several examples illustrate this. In the final section the integrator $\pi$ in the expression for $X_H$ is replaced by $\Pi - \mathbb{E}\Pi$, which gives conditional convergence for more $H$ in $(0, 1)$.

We consider an infinite particle system, the simple exclusion process, which was introduced in the 1970 paper "Interaction of Markov Processes," by Spitzer. In this system, particles attempt to move independently according to a Markov kernel on a countable set of sites, but any jump which would take a particle to an already occupied site is suppressed. In the case that the Markov kernel is symmetric, an inequality by Liggett gives a comparison, for expectations of positive definite functions, between the exclusion process and a system of independent particles. We apply a special case of this inequality to an auxiliary process, to prove another comparison inequality, and to derive a large deviation result for the symmetric exclusion system. In the special case of simple random walks on $Z$, this result can be transformed into a large deviation result for an infinite network of queues.

The invariant measures with finite first moment of the smoothing process are characterized in terms of the harmonic functions. It is also shown that under some restriction on the initial configuration, the difference between the mass at a given site and the mean of the masses of its neighbours converges in probability to zero.

We consider one dimensional stochastic Ising models with finite range interactions. For such processes we first prove that the semi-group of the process converges exponentially fast on the $L^2$ space of the Gibbs states. Under the additional hypothesis that the flip rates are attractive, we prove that the semigroup acting on the cylinder functions converges to equilibrium exponentially fast in the uniform norm.

In this note, we prove for a large class of random walks on $R^n$ that $\lim \inf_{n\rightarrow\infty}(1/n)\log P_x(L_n(\omega, \cdot) \in N) \geq - I(\mu)$ where $L_n(\omega, \cdot)$ is the occupation measure, $N$ is a weak neighborhood of $\mu$ and $I(\mu)$ is the usual Donsker-Varadhan functional. This generalizes a previous theorem of the author where the state space is assumed to be compact.

The results of W. Richter (Theory Probab. Appl. (1957) 2 206-219) on sums of independent, identically distributed random variables are generalized to arbitrary sequences of random variables $T_n$. Under simple conditions on the moment generating function of $T_n$, which imply that $T_n/n$ converges to zero, it is shown, for arbitrary sequences $\{m_n\}$, that $k_n(m_n)$, the probability density function of $T_n/n$ at $m_n$, is asymptotic to an expression involving the large deviation rate of $T_n/n$. Analogous results for lattice valued random variables are also given. Applications of these results to statistics appearing in nonparametric inference are presented. Other applications to asymptotic distributions in statistical mechanics are pursued in another paper.

A Cramer type large deviation theorem is proved under alternatives as well as under hypothesis for the generalized linear rank statistic which includes as special cases (unsigned) linear rank statistics, signed linear rank statistics, linear combination of functions of order statistics, and a rank combinatorial statistic.

This paper presents correction terms to the tangent approximation for the first-exit density of Brownian motion at distant boundaries. These lead to second-order approximations to the first-exit distribution. Asymptotic formulas for the mean and variance of the first-exit time are derived. Numerical comparisons show the accuracy of the approximations.

We present a "Tanaka-like" representation for $\alpha(x, B)$, the local time of intersection for Brownian motion in 2 and 3 dimensions, where $\alpha(x, B)$ is formally $\int_B \int \delta_x(\omega_t - \omega_s) ds dt$.

This paper considers random walks on a finite group $G$, in which the probability of going from $x$ to $yx, x, y \in G$, depends only on $y$. The main results concern the distribution of the number of steps it takes to reach a particular element of $G$ if one starts with the uniform distribution on $G$. These results answer some random sorting questions. They are attained by applications of group representation theory.

Let $\{Z_k, -\infty < k < \infty\}$ be iid where the $Z_k$'s have regularly varying tail probabilities. Under mild conditions on a real sequence $\{c_j, j \geq 0\}$ the stationary process $\{X_n: = \sum^\infty_{j=0} c_jZ_{n-j}, n \geq 1\}$ exists. A point process based on $\{X_n\}$ converges weakly and from this, a host of weak limit results for functionals of $\{X_n\}$ ensue. We study sums, extremes, excedences and first passages as well as behavior of sample covariance functions.

An absolutely continuous distribution $F$ is said to be in the domain of uniform local attraction of the absolutely continuous distribution $H$ if the density of the normalized maximum of an independent sample of size $n$ converges locally uniformly to the density of $H$ as $n \rightarrow \infty$. Under the sole restriction that $F$ is eventually increasing, the domains of uniform local attraction to the three types of extreme value distribution are shown to be characterized by the usual Von Mises' conditions. The equivalent form of conditions used here greatly simplifies and shortens proofs of known results. In particular, $L_p$ convergence and convergence of the $k$ upper sample extremes are investigated and extensions to known results derived.

In this paper we investigate the differentiability preserving properties of the semigroup $\{T_t: t \geq 0\}$ whose infinitesimal generator is a closed extension of the one-dimensional diffusion operator $L = a(x)d^2/dx^2 + b(x)d/dx$ acting on $C^2(I)$, where $I$ is a closed and bounded interval. Especially we treat the case in which the smoothness of the diffusion coefficient fails at the boundary. We get that $\{T_t: t \geq 0\}$ preserves the one and two-times differentiabilities but does not the three-times one of sufficiently many initial data.

The series expressions for the linear predictors of a stationary process have been known for a long time, but necessary and sufficient conditions for the mean square convergence of these series are still not available. It is shown that an equivalent problem is to find necessary and sufficient conditions for the invertibility of the infinite moving average representation of the process. Two known sufficient conditions are discussed, and a more general condition that includes both as special cases is given. The process that arises from fractional differencing of a random walk is discussed as an example.

In [4] Rosenthal proved the following generalization of Khintchine's inequality: \begin{equation*} \tag{B} \begin{cases} \max \{ \| \sigma^n_{i=1} X_i \|_2, (\sigma^n_{i=1} \| X_i \|^p_p)^{1/p}\} \\ \leq \| \sigma^n_{i=1} X_i \|_p \leq B_p\max \{ \| \sigma^n_{i=1} X_i \|_2, (\sigma^n_{i-i} \| X_i \|^p_p)^{1/p}\} \\ \text{for all independent symmetric random variables} X_1, X_2,\cdots, \text{with finite} pth \text{moment}, 2 < p < \infty.\end{cases}\end{equation*} Rosenthal's proof of (B) as well as later proofs of more general results by Burkholder [1] yielded only exponential of $p$ estimates for the growth rate of $B_p$ as $p \rightarrow \infty$. The main result of this paper is that the actual growth rate of $B_p$ as $p \rightarrow \infty$ is $p/\operatorname{Log} p$, as compared with a growth rate of $\sqrt p$ in Khintchine's inequality.

In order to model the diurnal variation of certain cell populations we study general one-type branching processes whose individuals reproduce in a manner that is influenced by time in a periodic fashion. A Malthusian growth rate is established as well as a periodically varying asymptotic composition, over ages e.g.

This paper deals with zero-sum discrete-time stationary models of stochastic games with Borel state and action spaces. A mathematical framework introduced here for such games refers to the minimax theorem of Ky Fan involving certain asymmetric assumptions on the primitive data. This approach ensures the existence and the universal measurability of the value functions and the existence for either or both players of optimal or $\varepsilon$-optimal universally measurable strategies in the finite horizon games as well as in certain infinite horizon games. The fundamental result of this paper is a minimax selection theorem extending a selection theorem of Brown and Purves. As applications of this basic result, we obtain some new theorems on absorbing, discounted, and positive stochastic games.

Different necessary and sufficient conditions for the existence of regular conditional probabilities are found for the cases of countably generated, countably separated, and complete probability spaces. Perfection is $n.$ and $s.$ for countably generated spaces, "almost pre-standardness" for the countably generated and countably separated cases, and discreteness for complete spaces. Several different forms of the regular conditional probability property must be distinguished.

It is shown that if $\{p_j\}$ is a discrete density function on the integers with support contained in $\{0, 1, \cdots, d\}$, and $p_0 > 0, p_1 > 0, p_{d - 1} > 0, p_d > 0$, then there is an $n_0$ such that the $n$-fold convolution $\{p_j\}^{\ast_n}$ is unimodal for all $n \geq n_0$. Examples show that this result is nearly best possible, but weaker results are proved under less restrictive assumptions.

A convexification argument is offered which implies that the strong law of large numbers for random compact sets in a Banach space is valid without a convexity assumption.

A new topology is introduced on the group of $\mu$-preserving automorphisms of a Lebesgue space $(X, \Sigma, \mu)$ so that $f_k \rightarrow f$ if $\mu(f^n_kA \cap B) \rightarrow \mu(f^nA \cap B)$ uniformly in $n$ for all $A, B$ in $\Sigma$. The subspace of mixing automorphisms is a Baire space in the relative topology. A conjecture (about the extent to which a mixing stationary process is determined by its two-dimension distributions) is stated, which is true implies that the two-fold mixing automorphisms are of first category in the mixing ones. So if the conjecture is true then by Baire's Theorem there is a mixing but not two-fold mixing automorphism.

The squared $n^k$-dimensional Euclidean distance $f_k$ between a given joint distribution of $k$ random variables with values in $1, \cdots, n$ and the joint distribution of independent variables with the same respective marginals has been suggested as a measure of dependence. The following facts are established for $M_k$, the maximum of $f_k$ over all joint distributions for fixed $k$: (1) $M_k$ is attained among the distributions with all $k$ variables equal to a variable $X$ that takes on just two values. (2) For $k \leq 6, M_k = 1/2 - (1/2)^k$ is attained when the distribution of $X$ is $\{1/2, 1/2\}$. (3) For $k \geq 7, M_k$ is not attained at $\{1/2, 1/2\}$ and strictly exceeds $1/2 - (1/2)^k$. (4) For $k \rightarrow \infty$, the distributions of $X$ where $M_k$ is attained approach $\{0, 1\}$, and $M_k \nearrow 1$.