The approximation of partial sums of $\phi$-mixing random variables with values in a Banach space $B$ by a $B$-valued Brownian motion is obtained. This result yields the compact as well as the functional law of the iterated logarithm for these sums. As an application we strengthen a uniform law of the iterated logarithm for classes of functions recently obtained by Kaufman and Philipp (1978). As byproducts we obtain necessary and sufficient conditions for an almost sure invariance principle for independent identically distributed $B$-valued random variables and an almost sure invariance principle for sums of $d$-dimensional random vectors satisfying a strong mixing condition.

Asymptotic upper and lower bounds are obtained for the uniform measure of the rate of convergence in the central limit theorem using a variety of norming constants. For many distributions the upper and lower bounds are of the same order of magnitude. As easy corollaries we deduce extensive generalizations of the classical characterizations of the rate of convergence in terms of series and order of magnitude conditions.

We show the existence and uniqueness of the solution of stochastic equations of reflection on $\mathbb{R}^+_n$ associated to semimartingales with right continuous and left limited paths. This result is used to prove the existence and uniqueness of a Markov process associated with a boundary condition of Neumann type with degenerate infinitesimal generator.

A joint generalization of Harris-Spitzer's and Szatzschneider's one-dimensional collision models is given and for the path of a given particle an approximation is obtained.

Given a process $\{X_i\}$, any permutation $\sigma: \lbrack 1, n\rbrack \rightarrow \lbrack 1, n\rbrack$ determines an order statistical event $A(\sigma) = \{X_{\sigma(1)} < X_{\sigma(2)} < \cdots < X_{\sigma(n)}\}$. How many events $A(\sigma)$ are needed to form a union whose probability exceeds $1 - \epsilon$? This question is answered in the case of stationary ergodic processes with finite entropy.

The existing sliding-block entropy compression coding with a fidelity criterion theorem is generalized to abstract source and source reproduction alphabets, stationary and ergodic sources and nonmetric distortion measures. This establishes the sliding-block entropy compression coding theorem in the same generality as the corresponding block entropy compression coding theorem.

The subject of this work is a study of four properties of an isotropic Gaussian process on an infinite dimensional sphere in Hilbert space. The process is deterministic in the sense that its values on an arbitrary nonempty open subset of the sphere determine its values throughout the sphere. An harmonic property is defined, and is characterized in terms of the covariance function of the process. If $f$ is a function of a real variable which is square-integrable with respect to the Gaussian density, then the average of $f$ over the values of the process on an $n$-dimensional subsphere converges with probability 1 for $n \rightarrow \infty$. Under further conditions on the process the average of $f$ has, with appropriate normalization, a limiting Gaussian distribution for $n \rightarrow \infty$.

Kazamaki has shown that if $(M^n)_{n\geqq 1}, M$ are BMO martingales with continuous paths and $\lim M^n = M$ in BMO, then $\mathscr{E}(M^n)$ converges in $\mathscr{H}^1$ to $\mathscr{E}(M)$, where $\mathscr{E}(M)$ denotes the stochastic exponential of $M$. While Kazamaki's result does not extend to the right continuous case, it does extend "locally." It is shown here that if $M^n, M$ are semimartingales and $M^n$ converges locally in $\mathscr{H}^\omega$ (a semimartingale BMO-type norm) to $M$ then $X^n$ converges locally in $\mathscr{H}^p (1 \leqq p < \infty)$ to $X$, where $X^n, X$ are respectively solutions of stochastic integral equations with Lipschitz-type coefficients and differentials $dM^n, dM$. (The coefficients are also allowed to vary.) This is a stronger stability than usually holds for solutions of stochastic integral equations, reflecting the strength of the $\mathscr{H}^\omega$ norm.

We develop a general theory of duality for Markov processes satisfying Meyer's hypothesis (L) and possessing an excessive reference measure. We make use of a compactification introduced by Walsh which allows a right process and its moderate dual to have strong Markov versions on an enlarged state space. The representation theory for potentials of additive functionals due to Revuz and Sharpe can be extended to this setting. Using this theory, we show that the conatural additive functionals introduced by Garcia-Alvarez are natural additive functionals in the new topology. A general version of Motoo's theorem is given, and the Getoor-Sharpe approach to capacities is extended to this situation. Finally, we show that if the original process satisfies Hunt's hypothesis (H), then a version of the bounded maximum principle holds on the compactified space.

In this paper we make comparisons involving stopping times $\tau$ of a process $X$ and the maximal function $X^\ast_\tau$ of that process, where $X$ is either Brownian motion or random walk. In particular, we give conditions implying that $P(X^\ast_\tau > \lambda) \approx P(\tau^{1/2} > \lambda)$ in the sense of a two-sided inequality holding. We show that if, for all large $\lambda$ there exist constants $\beta > 1$ and $\gamma > 0$ satisfying $$0 < P(\tau^{1/2} > \lambda) \leq \gamma P(\tau^{1/2} > \beta\lambda),$$ and if $X$ is a one-dimensional Brownian motion, then $P(X^\ast_\tau > \lambda) \approx P(\tau^{1/2} > \lambda)$. An analogous result is given for $n$-dimensional Brownian motion $(n \geq 3)$. We also consider a similar result for one-sided maximal functions of local martingales. Finally, we look at a random walk $X$, where $X_n = x_1 + x_2 + \cdots + x_n$, and give two different sets of conditions on $\tau$ and the $x_i$'s under which the result $P(\tau^{1/2} > \lambda) \approx P(X^\ast_\tau > \lambda)$ is true.

Necessary and sufficient conditions are presented for the measurability of $p$th order or symmetric stable stochastic processes and for the integrability or absolute continuity of sample paths of symmetric stable processes. Also obtained are sufficient conditions for absolute continuity of $p$th order processes.

The method of time reversal of a Markov process from a cooptional time, introduced by Nagasawa, is used to show that certain occupation time and last exit time problems for one linear diffusion are equivalent to first passage time problems for certain other diffusions. Another proof of Nagasawa's theorem is given, based on the measures of Revuz.

In this paper we obtain characterizations of large classes of nonparametric life distributions, such as the increasing (decreasing) failure rate, increasing (decreasing failure rate average, new better (worse) than used, etc., classes. The methods used differ from the usual functional equation methods used for the far more common characterizations of parametric families of life distributions.

We apply the concept of stopping times of martingales to problems in classical probability theory regarding the occurrence of sequence patterns in repeated experiments. For every finite collection of sequences of possible outcomes, we compute the expected waiting time till one of them is observed in a run of experiments. Also we compute the probability for each sequence to be the first to appear. The main result, with a transparent proof, is a generalization of some well-known facts on Bernoulli process including formulas of Feller and the "leading number" algorithm of Conway.

Let $Q$ be a fixed probability on the Borel $\sigma$-field in $R^n$ and $H$ be an energy function continuous in $R^n$. A set $N$ is related to $H$ by $N = \{x \mid\inf_yH(y) = H(x)\}$. Laplace's method, which is interpreted as weak convergence of probabilities, is used to introduce a probability $P$ on $N$. The general properties of $P$ are studied. When $N$ is a union of smooth compact manifolds and $H$ satisfies some smooth conditions, $P$ can be written in terms of the intrinsic measures on the highest dimensional mainfolds in $N$.

Etant donne une martingale a deux indices reguliere (en un sens convenable), on majore la "norme" $L_p (p \in\rbrack 0, \infty\lbrack)$ de $S$ (racine de la variaton quadratique) par la "norme" $L_p$ de $u^\ast$ (variable maximale). Pour cela on demontre une inegalite faisant intervenir les fonctions de repartition de $S$ et $u^\ast$. La methode permet aussi de montrer le theoreme local correspondant, a savoir que $S$ est presque surement fini la ou $u^\ast$ est fini.

There is no Borel function $f$, defined for all infinite sequences of 0's and 1's, such that for every sequence $X$ of 0-1 random variables that converges in probability to a constant $c$, we have $f(x) = c$ a.s.

Separated $B$-spaces introduced by E. B. Dynkin are just standard Borel and thus here we have an intrinsic definition of standard Borel spaces.

We give an example of a Gaussian measure on the $\sigma$-algebra of $l^\infty$ generated by all the continuous linear functionals on $l^\infty$, which is not a Radon measure.

The note contains a construction of (possibly explosive) diffusions whose coefficients satisfy Stroock-Varadhan-type regularity conditions, but are unbounded. The appropriate state space here is the one-point compactification of $R^k$, and the sample paths of the diffusions are shown to be a.s. continuous on this state space.