The purpose of this paper is to explain why it is fruitful to think of Wiener space as an infinite--dimensional sphere of radius $\infty\frac{1}{2}$. The idea goes back to Levy and Wiener and has recently been employed to advantage by Hida; Hida, Kubo, Nomoto and Yosizawa; Kono; Orihara and Umemura; their results will be reported upon below.

Mokobodzki has shown that any kernel which satisfies the complete maximum principle and the absolute continuity hypothesis has the same excessive functions as a compact kernel. We start with a nice semigroup which satisfies the absolute continuity hypothesis, build for it a coresolvent with the same property, and apply Mokobodzki's result to the coresolvent in order to get a Martin compactification for the original process. Since the coresolvent may not separate points on the state space, we are obliged to throw away a polar Borel set before the Martin procedure applies. We prove a representation theorem for excessive functions, which implies in particular that any measure on the original state space, which does not charge semipolar sets, is equivalent to (i.e. has the same null sets as) a continuous additive functional. This implies the "fine support theorem": any finely perfect set is the fine support of a continuous additive functional, a theorem which has been proved by Mokobodzki and Azema using other methods.

The paper deals with Markov processes which have both random starting and terminal times. Such processes were suggested by G. A. Hunt, were constructed by L. L. Helms (under the name Markov processes with creation and annihilation) and were treated also by M. Nagasawa and the author. The paper contains a new existence proof by a way of constructing such a process from its given associated semigroup of kernels $\tilde{P}_t, t \geqq 0$, and its (Markov) transition function. This construction is more general than that given by L. L. Helms (in terms of the Markov transition function and the creation measure) and is also more convenient as far as perturbation theory of Markov processes is concerned. Indeed more general relations between this theory and creation of mass processes are established. Finally an application to solving the Cauchy problem in partial differential equations is indicated.

Let $\{x_i(t): i = \cdots -1, 0, 1, \cdots\}$ be a collection of one-dimensional symmetric stable processes of order $\gamma \in (0, 1\rbrack$ with the property that the starting positions $\cdots < x_{-1}(0) < x_0(0) = 0 < x_1(0) < \cdots$ form a Poisson system with rate one. By generalizing the order preserving property of elastic collision, these can be used to define a set of collision processes $\{y_i(t)\}$. It is shown in this paper that for large values of $A$, the finite dimensional distributions of $y_0(At)/A^{1/2\gamma}$ approach the Gaussian distribution with mean zero and covariance $r(t, s) = c(t^{1/\gamma} + s^{1/\gamma} - |t - s|^{1/\gamma})$.

Existence and uniqueness is proved for Spitzer's zero range interaction process. It is then shown that a certain class of measures on the configuration space is invariant for this process.

Several methods for embedding discrete martingales in Brownian motion by means of stopping times have been presented. Various conditions on the increments of the martingales are sufficient to insure that the trajectories of the embedded process and the Brownian motion are close. This paper will characterize all discrete stochastic processes, which can be constructed on some probability space supporting a Brownian motion, in such a way that the constructed process and the Brownian motion are close in probability, under suitable normalization. These are exactly the processes $\{S_j | j = 0, 1, \cdots\}$ such that, for any $\varepsilon > 0$ and $M = 0, 1, \cdots$ the conditional probability that the $(M + 1)$st change in size of at least 1 completed by the trajectory $0, S_1/n, S_2/n, \cdots$ is in $\lbrack 1, 1 + \varepsilon \rbrack$ (or $\lbrack -(1 + \varepsilon), - 1 \rbrack)$, given the process up to the $M$th such change, converges in probability to $\frac{1}{2}$ as $n \rightarrow \infty$. It is not required that any moments exist, or even that $E(S_{j+1} |S_1, \cdots, S_j) = S_j, \text{a.s.}$ In proving the main result, a new technique for constructing discrete processes from Brownian motion is presented.

Let $X_1, X_2, \cdots$ be a sequence of random variables satisfying $E(X_{n + 1}\mid X_n, X_{n - 1}, \cdots, X_1) = a_1 X_n + a_2 X_{n - 1} + \cdots + X_{n - k - 1}, n \geqq k$, where $a_1 + a_2 + \cdots + a_k = 1$. Under certain general conditions, mainly that $\sup_nE|X_n| < \infty$, it is shown that $X_n - Y_n \rightarrow\operatorname{a.s.} 0$, where $\{Y_n\}$ is a solution of the homogeneous equation $y_n = a_1y_{n - 1} + a_2y_{n - 2} + \cdots + a_ky_{n - k}$. Several applications of possible theoretical interest are described. Also, the results suggest some extensions of classical results in the theory of random walks which are outlined.

For processes with independent increments and multi-dimensional time parameters, analogues of Strassen's version of the law of the iterated logarithm are proven using standard large-deviation and truncation techniques. Applications to empirical processes are included.

Suppose $x_1, \cdots, x_n$ are the order-statistics of a random sample from a distribution $F$. We assume that the expectations $\xi_{i:n} = E(x_i)$ are known, and derive sharp bounds on $F(x)$ for all $x$. These results are obtained by transforming the problem into a classical one involving ordinary power moments.

It is shown in this paper that the paths of a real separable Gaussian process on an interval have a number of interesting properties with probability zero on one. These properties include boundedness, bounded variation, continuity, uniform continuity, absolute continuity, a Lipschitz property, differentiability, absence of oscillatory discontinuities.

Some functions of stopping times are necessarily stopping times, but others need not be. For example, the sum $\tau_1 + \tau_2$ of two stopping times is, while for stochastic processes in continuous time, the product $\tau_1 \cdot \tau_2$ need not be. Determined here for each positive integer $n$ are those functions $\phi$ for which $\phi(\tau)$ is a stopping time for all $n$-triples of stopping times $\tau = \tau_1, \cdots, \tau_n$.

Let $S_n, n = 1, 2, \cdots$, be a sequence of sums of independent, identically distributed random variables $X_i$ such that $P\{X_i > y\}$ is a regularly varying function of $y$ at infinity. Let $N$ be a stopping time for $S_n$ with finite mean. A necessary and sufficient condition is given that $$\lim_{y\rightarrow \infty} P\{S_N > y\}/P\{X_1 > y\} = EN.$$ Examples further illustrate the role of this condition.

A supermartingale maximal inequality is derived. A maximal inequality is derived for arbitrary random variables $\{S_n, n \geqq 1\}$ (let $S_0 = 0$) satisfying $E\exp\lbrack u(S_{m + n} - S_m) \rbrack \leqq \exp(Knu^2)$ for all real $u$, all integers $m \geqq 0$ and $n \geqq 1$, and some constant $K$. These two maximal inequalities are used to derive upper half laws of the iterated logarithm for supermartingales, multiplicative random variables, and random variables not satisfying particular dependence assumptions.

This paper considers ergodic behavior of those non-stationary Markov processes which can be represented by a sequence of stochastic kernels, $\{P_n(x, y)\}$, defined on a $\sigma$-finite measure space $(S, \mathscr{F}, \mu)$. In particular, the convergence of the superpositions, $P_1P_2P_3 \cdots P_n$, of these kernels is related to the convergence of their corresponding left eigenfunctions, $\psi_n$, where $\psi_n(y) = \int \psi_n(x)P_n(x, y)\mu(dx)$ and $\int \psi_n(y)\mu(dy) = 1$. It is then shown how these results can easily be extended to the general case where densities are not assumed.

The Fourier transform of a stationary random measure on an arbitrary locally compact Abelian group is defined and used to extend some results on the representation of homogeneous random fields as moving averages.

We study the integrability of the supremum of ergodic ratios defined by means of a measure-preserving conservative ergodic transformation of a $\sigma$-finite measure space. Our result implies Ornstein's recent result for the supremum of ergodic averages.

Let $\{X_t\}, 0 \leqq t < \infty$, be a Markov process with state space $(E, \mathscr{E})$. Let $m$ be a $\sigma$-finite measure on $(E, \mathscr{E})$ and let the $L_\infty(E, \mathscr{E}, m)$ operator induced by the transition probability $P_t(x, A), x\in E, A\in \mathscr{E}$, be conservative and ergodic for all $t > 0$. Let $(m)$ abbreviate $m$ modulo 0. For fixed $\alpha > 0$, set $h^\alpha(x) = \lim_{t \rightarrow \infty} \|P_t(x, \bullet) - P_{t + \alpha}(x, \bullet)\|$, where $\|\bullet\|$ is the total variation. THEOREM. Either $h^\alpha(x) = 0(m)$ for $\operatorname{a.e} \alpha\in\mathbb{R}_+$ or $h^\alpha(x) = 2 (m)$ for $\operatorname{a.e} \alpha\in\mathbb{R}_+$. In particular, if $\{X_t\}, 0 \leqq t < \infty$, is a Markov process satisfying a Harris type recurrence condition, then $h^\alpha(x) = 0 (m)$ for $\operatorname{a.e} \alpha\in\mathbb{R}_+$.

The following generalizations of certain theorems due to G. Kallianpur and to Jamison and Orey are proved for an arbitrary Gaussian measure $P$ on a space of real functions: if the reproducing kernel Hilbert space $H$ is infinite dimensional then $P(H) = 0$; if a subgroup $G$ of the space of real functions (under addition) is measurable with respect to the $P$-completion of the Borel product sigma-algebra, then $P(G) = 0$ or $P(G) = 1$ and in the latter case $H \subset G$.

In this note we prove an identity due to Pollaczek (1952) and Wendel (1960). The identity describes the distributions of ordered partial sums of independent identically distributed random variables and thus generalizes Spitzer's identity. Our proof follows from a simple analytic argument applying a kind of Wiener-Hopf decomposition. We also give an extension of the Pollaczek-Wendel identity.

For each $k = 1, 2, \cdots$ let $n = n(k)$, let $m = m(k)$, and suppose $y_1^k, \cdots, y_n^k$ is an $m$-dependent sequence of random variables. We assume the random variables have $(2 + \delta)$th moments, that $m^{2 + 2/\delta}/n \rightarrow 0$, and other regularity conditions, and prove that $n^{-\frac{1}{2}}(y_1^k + \cdots + y_n^k)$ is asymptotically normal. An example showing sharpness is given.

An $L_1$-smoothing lemma is used to prove an $L_1$ version of the Chebyshev-Cramer asymptotic expansion for independent (identically distributed) random variables. The conditions imposed are exactly those demanded for the $L_\infty$ version.

A counterexample is given to a result of Perez which makes a statement about the convergence of a sequence of logarithms of Radon-Nikodym derivatives. The result, if true, would have been a generalization of the Shannon-McMillan theorem of information theory.