Let $X(t)$ be a measurable stochastic process on a countably generated space $(E, \mathscr{E})$, and let $G(t) = \cap_{\delta>o} \mathscr{F}^\circ (t, t + \delta)$ be its germ field. By transferring the probabilities to a representation space, we define and analyze the class of such processes which are Markovian relative to $G(t)$ and autonomous, in the sense that they have a stationary transition mechanism. These processes are reduced to Ray processes on an abstract space with a certain weak topology. Five kinds of examples are indicated.

It is shown that infinite divisibility of random variables, such as first passage times in a stochastic process, is often connected with the existence of an imbedded terminating renewal process. The idea is used to prove that for a continuous time Markov chain with two, three or four states all first passage times are infinitely divisible but for more than four states there are first passage times which are not infinitely divisible.

It is shown that a discrete time voter model in equilibrium on $\mathbb{Z}_3$ approaches the 0-mass free field of 3-dimensional Euclidean field theory under appropriate renormalization. This result is of interest because the strong correlation between distant sites gives rise to the renormalization exponent $- \frac{5}{2}$ instead of the usual $- \frac{3}{2}.$ Dawson, Ivanoff, and Spitzer have examined models on $\mathbb{R}_3$ which exhibit precisely the same limit. Because the process we consider lives on a lattice, our method of proof is necessarily quite different from theirs. In particular, we make use of a "duality" between voter models and coalescing random walks which has been exploited effectively by Holley and Liggett.

Let $(X, \mathscr{A}, P)$ be a probability space and $\mathscr{B}$ a sub-$\sigma$-algebra of $\mathscr{A}$. Some results on regular conditional probabilities given $\mathscr{B}$ are proved. Using these, when $\mathscr{A}$ is separable and $\mathscr{B}$ is a countably generated sub-$\sigma$-algebra of $\mathscr{A}$ such that there is a regular conditional probability given $\mathscr{B}$, necessary and sufficient conditions for the existence of an independent complement for $\mathscr{B}$ are given.

It is shown that all the possible cases can arise in the mixture problem with respect to perfectness of probability measures. A characterization of perfectness is obtained through properties of a countably generated sub-$\sigma$-algebra given which there is a regular conditional probability. Perfectness of a perfect mixture of perfect measures is characterized.

We show that the asymptotic behavior of the normalized maxima of a stationary sequence satisfying a weak distributional mixing and bivariate condition is completely determined by the marginal distribution of the process. Sufficient conditions are given in order for the maxima and minima to be asymptotically independent. An example of a 1-dependent sequence where the maxima and minima are not asymptotically independent is also provided.

Whether stationary families of strategies are uniformly adequate for a leavable, analytically measurable, nonnegative gambling problem whose optimal return function is everywhere finite is a question which remains open. It is, however, given an affirmative answer if, for example, the fortune space is Euclidean and all nontrivial, available gambles are absolutely continuous with respect to Lebesgue measure.

A zero-one law is derived for a class of Gaussian fields $\{X(t) : t \in R^d\}$ including the generalized multiparameter Brownian motion. Under very general conditions, the joint distribution of the suprema of several Gaussian processes defined over compact metric spaces is shown to be absolutely continuous with a bounded density. Sufficient conditions are given for the existence of proper scaling limits of $\{X(t)\}$. The results are then combined to study local oscillations and local maxima.

The class of infinitely divisible characteristic functions which have unimodal Levy spectral functions is determined. It is shown that membership in this class is related to solutions of the equations $\phi(u) = \phi^r(ru)\phi_r(u)$, where $r \in (0, 1)$ and $\phi$ and $\phi_r$ are characteristic functions. We point out how elements of this class can serve as limit laws as well as some connections between this class and the class of self-decomposable characteristic functions.

It is shown that if, for a piecewise $C^2$ mapping of the unit interval into itself where the absolute value of the derivative is greater than 1, an invariant measure is weak-mixing, then a central limit theorem holds for a class of real Holder functions.

The class of continuous functions $f(t, x)$ for which $f(t, X(t))$ are Markov processes is explicitly determined, where $X(t)$ is a Brownian motion on the real line. This extends a result by Walsh.

A local limit theorem is given for large deviations of sums of independent, nonidentically distributed, integer valued random variables.

Let $B^{(N,d)}$ be Levy's $N$-parameter Brownian motion in $d$-space. It is shown that almost surely $B^{(N,d)}$ doubles the Hausdorff dimension of every Borel set in the parameter space when $d \geqslant 2N$. The dimension of the range of $B$ is also determined in this case.

Let $p_t, t \geqslant 0$ be the probability transition semigroup of a continuous one-dimensional diffusion. We examine continuous Markov processes $\xi_s$, defined for $-\infty < s < \infty$, which are governed by $p_t$. It is shown that the class of such processes, modulo convex combinations and translations, can consist of at most three elements. In addition, it is shown that the first passage times for these processes are related to a previously known existence condition.

We give an example to show that, given a nonhomogeneous Markov process $X$ on $E$, one cannot, in general, produce a right continuous strong Markov version of $X$ on a compactification of $E$. In particular, the space-time regularization fails to produce one compact Hausdorff state space for the nonhomogeneous process.

Lower bounds are derived for the multivariate Mills' ratio by expressing it as an expectation of a convex function and using Jensen's inequality.