Using a device which approximates stationary Gaussian processes by $M$-dependent processes, we find conditions on the covariance function to insure that the number of zero crossings, after centering and rescaling, has an asymptotically normal distribution. This device is then used to obtain central limit theorems for integrals of functions of stationary Gaussian processes.

For a countable set $S$ and strictly positive matrix $Q = (Q(x, y))_{x, y\in S}$ let $\mathscr{G}(Q)$ be the set of all probability measures $\mu$ on $\Omega \equiv S^\mathbb{Z}$, strictly positive on cylinder sets, and with the following "two-sided Markov property": $\mu \{\omega_n = x\mid\omega_l, l \neq n\} = \lbrack Q^2(x, z) \rbrack^{-1}Q(y, x)Q(x, z)$ a.e. on the set $\{\omega_{n - 1} = y, \omega_{n + 1} = z\}$. In other words, for every $\mu\in\mathscr{G}(Q)$, the conditional distribution of $\omega_n$ given all other $\omega_l$ depends on $\omega_{n - 1}$ and $\omega_{n + 1}$ only, and "behaves as if $\{\omega_n\}_{n\in \mathbb{Z}}$ is a Markov chain with transition probability matrix $Q$." $\mathscr{G}_0(Q)$ denotes the set of those $\mu\in \mathscr{G}(Q)$ which are in addition translation invariant. We establish a conjecture of Spitzer's [9] that either $\mathscr{G}_0(Q) = \varnothing$ or $\mathscr{G}_0(Q)$ consists of one element only, which is then necessarily a stationary Markov chain on $\Omega$. We also give a condition for $\mathscr{G}(Q) = \varnothing$.

This paper continues the development of a stochastic calculus for two-parameter martingales, and particularly for the two-parameter Wiener process. Whereas in an earlier paper we showed that two types of stochastic integrals were necessary for representing functionals and martingales of a Wiener process, introduction of two mixed area integrals is necessary to complete the stochastic calculus. These mixed integrals are weak martingales in the sense of Cairoli and Walsh, and are necessary in a general representation for weak martingales and transformations of weak martingales. Stopping times are introduced for two-parameter processes, and a characterization of strong martingales in terms of stopping times is given.

Let $X_1, X_2, \cdots$ be independent random variables with values in a Banach space $E$. It is then shown that Chung's version of the strong law of large numbers holds, if and only if $E$ is of type $p$. If the $X_n$'s are identically distributed, then it is shown that the central limit theorem is valid, if and only if $E$ is of type 2. Similar results are obtained for vectorvalued martingales.

Suppose that $(X_{nj})$ is a triangular array of random variables taking values in a Banach space $E$ and that $(B_n)$ is the corresponding sequence of random paths in $E$. Conditions are considered under which the distributions of $B_n$ converge to a Gaussian measure on $C(\lbrack 0, 1 \rbrack; E)$. Under stronger conditions on the array it is shown that if $E$ is of type 2 the paths enjoy certain regularity properties, which are reflected in the convergence. The technique here is to factorise the integration procedure by which one passes from the array to the sequence of paths, using fractional integrals.

In this paper, we prove a conjecture of Strassen on the set of $r$-quick limit points of the normalized linearly interpolated sample sum process in $C\lbrack 0, 1 \rbrack$. We give the best possible moment conditions for this conjecture to hold by finding the $r$-quick analogue of the classical law of the iterated logarithm and its converse. The proof is based on an $r$-quick version of Strassen's strong invariance principle and a theorem on the $r$-quick limit set of a semi-stable Gaussian process. Some applications of Strassen's conjecture are given. We also consider the notion of $r$-quick convergence related to the law of large numbers and outline some statistical applications to indicate the usefulness of this concept.

In this paper, we prove an analogue of the classical renewal theorem for the case where there is no drift. Our proof depends on a uniform version of Spitzer's well-known theorem on ladder epochs and ladder variables, and we obtain this uniform result by using uniform Tauberian theorems. Some further applications of these uniform Tauberian theorems to other problems in renewal theory and first passage times are also given.

Let $P$ be a distribution on $R$ with a positive, finite mean. A constructive, unified version of the renewal theorem is proved. A routine method is provided with which one can compute, in principle at least, the point $x_0$ where the renewal measure $Q$ settles down. As a corollary, it is shown that $x_0$ depends continuously on $P$.

A multidimensional renewal theorem is proven which generalizes the well-known result of Pyke on Markov renewal processes. The result is a "natural" setting of the multidimensional renewal theorem, and has applications to age-dependent branching processes.

Oscillations are defined for $n$ steps of the random walk formed by partial sums of variables with continuous cdf. When the summands are independent, identically and symmetrically distributed, several distribution free results are obtained relative to the number of oscillations and their lengths. Analogy with the behavior of records in a random sequence is used to obtain limit laws.

An attempt is made to enumerate the distinct antisymmetric comparative probability relations on sample spaces of $n$ atoms. The results include an upper bound to the total number of such relations and upper and lower bounds to the size of the subset of the comparative probability relations admitting an agreeing probability measure as representation. The theoretical results are supplemented by computer enumerations for $n \leqq 6$. The upper and lower bounds for the case of agreeing probability measures are both $$O(3^{\alpha n^2})\quad \text{for} \log_2(3^{\frac{1}{2}}) \leqq \alpha \leqq 1.$$

A theorem due to Tweedie (1974), showing the existence and uniqueness of an $R$-invariant measure for $R$-recurrent Markov chains, is derived by an alternative and direct method.

Let $P_n$ be the $n$th Bessel polynomial. Kelker (1971) showed that the Student $t$-distribution of $k = 2n + 1$ degrees of freedom is infinitely divisible if and only if $\varphi_n(x) = P_{n-1}(x^{\frac{1}{2}})/P_n(x^{\frac{1}{2}})$ is completely monotonic. Kelker and Ismail proved that $\varphi_n$ is indeed completely monotonic for some small values of $n$ and conjectured that this is always the case. This conjecture is proved here by a twofold application of Bernstein's theorem and the use of some special properties of the zeros of the Bessel polynomials. The same conclusion follows for $Y_k = (\chi k^2)^{-1}$, where $\chi k^2$ is a chi-square variable with $k$ degrees of freedom.

There exists a sequence of i.i.d. random variables taking values in the infinite dimensional Banach space $c_0$ satisfying the law of the iterated logarithm and failing to obey the central limit theorem.

Let $E$ be a separable Banach space with norm $\|\bullet\|$. Let $\{X_n\}$ be a sequence of $E$-valued independent, identically distributed random variables, and $S_n = X_1 + \cdots + X_n$. If $\{n^{-\frac{1}{2}}S_n\}$ converges in the sense of weak convergence of the corresponding measures in $E$, and $E$ is the real line, then it is well known that $\mathscr{E}\lbrack X_1 \rbrack = 0$ and $\mathscr{E}\lbrack\|X_1\|^2\rbrack < \infty$; consequently, the Hartman-Wintner law of the iterated logarithm also holds. We give an example here, with $E = C\lbrack 0, 1\rbrack$, such that the above convergence does not imply $\mathscr{E}\lbrack \|X_1\|^2 \rbrack < \infty$, nor does it imply the law of the iterated logarithm.

The author shows that the formula given by B. Weesakul [3] for the absorption probability in a random walk between a reflecting and an absorbing barrier does not hold true in general. The correct formula in the general case is given.