It is shown that all infinitely divisible distribution functions of class $L$ are unimodal.

Some questions concerning the orbits of a billiard ball in a polygon are studied. It is shown that almost all such orbits come arbitrarily close to a vertex of the polygon, implying that the entropy of the corresponding geodesic flow is zero. For polygons with rational angles, we show by using interval exchange transformations that almost all orbits are spatially dense. Two applications are given.

Let $L = \frac{1}{2} \sum^k_{i,j=1} a_{ij}(x)(\partial^2/\partial x_i \partial x_j) + \sum^k_{i=1} b_i(x)(\partial/\partial x_i)$ be an elliptic operator such that $a_{ij}(\bullet)$ are continuous and $b_i(\bullet)$ are measurable and bounded on compacts. Criteria for transience, null recurrence, and positive recurrence of diffusions on $R^k$ governed by $L$ are derived in terms of the coefficients of $L$.

This paper is concerned with the following question: if a characteristic function satisfies a certain property at the origin, what can be said about its behavior on the entire real line? If $k$ is an even integer and $f(u)$ is a characteristic function, then the existence of $f^{(k)}(0)$ implies the existence of $f^{(k)}(u)$ for all $u$. If $k$ is an odd integer, then it is possible to construct a characteristic function $f(u)$ such that $f^{(k)}(0)$ exists but $f^{(k)}(u)$ fails to exist for almost all $u$. However the existence of $f^{(k)}(0)$, when $k$ is odd, implies that $f(u)$ satisfies a $k$th order smoothness condition uniformly on the real line and thus $f(u)$ has many of the properties of a characteristic function with a continuous $k$th derivative. Several other results are obtained that show that if a characteristic function has a property $P$ at 0 then it either has the same property everywhere on the real line or comes close to having the property everywhere.

Let $(X_n: n \geqq 1)$ be a sequence of independent random variables, each having mean 0 and a finite variance. Under the Lindeberg condition and uniformity conditions on the characteristic functions, it is shown that the local limit theorem holds, i.e., if $S_n$ is the $n$th partial sum of the sequence, then $(2\pi \operatorname{Var} S_n)^{\frac{1}{2}}P(S_n \in (a, b)) \rightarrow b - a$. Under the assumption that the local limit theorem holds for each tail of $(X_n)$, and one other condition, it is then shown that the random walk generated by $(X_n)$ is recurrent if $\sum (\operatorname{Var} S_n)^{-\frac{1}{2}} = \infty$.

This paper considers the problems of linear prediction under the condition that the spectral structures of second-order stationary processes are vaguely specified. The approach adopted is closely in line with the theory of robust estimation due to Peter Huber. The paper shows that there exists a minimax one-step ahead predictor for the set of spectral distributions given as $\{H: H = (1 - \varepsilon)F + \varepsilon G, G \in \mathscr{D}_1\}$, where $F$ is a fixed probability distribution function and $\mathscr{D}_1$ is the set of all absolutely continuous probability distribution functions. That predictor turns out to be the optimal linear predictor for a spectral distribution which is derived by a suitable modification applied to $F$. Though there generally exists no minimax predictor for the set $\{H: H = (1 - \varepsilon)F + \varepsilon G, G \in \mathscr{D}_0\}$ ($\mathscr{D}_0$ is the set of all probability distribution functions), a linear predictor is explicitly constructed so that its maximal prediction error is arbitrarily close to the lower bound of the maximal prediction errors of possible linear predictors. The results obtained in this paper would have an important application in the errors-in-variable models.

Multiple Wiener integrals and stochastic integrals are defined for Gaussian processes, extending the related notions for the Wiener process. It is shown that every $L_2$-functional of a Gaussian process admits an adapted stochastic integral representation and an orthogonal series expansion in terms of multiple Wiener integrals. Also some results of Wiener's theory of nonlinear noise are generalized to noises other than white.

The paper concerns the Doob-Meyer increasing processes of simple point processes on the positive half line. It is shown that the weak convergence of such point processes to a simple Poisson process is implied by the pointwise weak convergence of their increasing processes, provided that the increasing processes satisfy a mild regularity condition. Conditions under which the regularity is satisfied are investigated. One condition is that the increasing process is that of the point process with its generated $\sigma$-fields. The Poisson convergence theorem is applied to superpositions of point processes.

We prove that an attractive nearest neighbor spin system on the integers has at most two extremal invariant measures, provided that its flip rates satisfy a mild positivity condition.

A characterization of type $p$ and cotype $p$ separable Banach spaces is given in terms of integrability properties of Levy measures. The following consequences are derived: (i) a separable Banach space is isomorphic to Hilbert space if and only if the set of Levy measures on it coincides with the set of Borel measures which integrate the function $\min (1, \|x\|^2)$; and (ii) the classical Levy-Khintchine representation of characteristic functions of infinitely divisible distributions holds in separable Banach spaces of cotype 2, in particular, in the separable $L_p$ spaces for $p \in \lbrack 1,2\rbrack$.

A definition of duality for pairs of stopping times of any random walk is motivated by the duality relation of ascending and descending ladder epochs $N, \bar{N}$ of random walk in $R^1$. Dual pairs share several of the properties of the pair $N, \bar{N}$.

Let $\bar{X}(t) = (X_1(t),\cdots, X_n(t))$ be standard $n$ dimensional Brownian motion. Results of the following nature are proved. If $\tau$ is a stopping time for $\bar{X}(t)$ then $|\bar{X}(\tau)|$ and $(n\tau)^{\frac{1}{2}}$ are relatively close in $L^p$ if $n$ is large. Also, if $n$ is large most of the moments $EX_i(\tau)^k, i = 1,2,\cdots, n$, are about what they would be if $\bar{X}(t)$ were independent of $\tau$.

In this paper we consider a stationary sequence $\{X_n, n \geqq 1\}$ satisfying weak dependence restrictions similar to those recently introduced by Leadbetter. Suppose $a_n$ and $b_n > 0$ are norming constants for which $\max\{X_{n1},\cdots, X_{nn}\}$ converges in distribution, where $X_{nk} = (X_k - b_n)/a_n$. Define a sequence of planar processes $I_n(B) = \sharp\{j: (j/n, X_{nj}) \in B, j = 1,2,\cdots, n\}$, where $B$ is a Borel subset of $(0, \infty) \times (-\infty, \infty)$. Then the $I_n$ converge weakly to a nonhomogeneous two-dimensional Poisson process possessing the same distribution as for independent $X_j$. Applying the continuous mapping theorem to this result generates a variety of further results, including, for example, weak convergence of the order statistics of the $X_n$ sequence. The dependence conditions are weak enough to include the Gaussian sequences considered by Berman.

Here there is derived a condition on sequences $\varepsilon_n \downarrow 0$ which implies that $P\lbrack W(n^\bullet)/(2n \log \log n)^\frac{1}{2} \not\in K^\varepsilon n \mathrm{i.o.}\rbrack = 0$, where $W$ is the Wiener process and $K$ is the compact set in Strassen's law of the iterated logarithm. A similar result for random walks is also given.

Let $X_1, X_2,\cdots$ be i.i.d. random variables with mean 0 and variance 1. Let $S_n = X_1 + \cdots + X_n$, and let $\{H_n\}$ be the standard partial sum processes on $\lbrack 0, \infty)$ defined in terms of the $S_n$'s and normalized as in Strassen. Each function of the "tail" behavior of the process $H_n$ is the dual of a function of the "initial" behavior of the process $H_n$, the duality being induced by the time inversion map $R$. The dual role of "initial" and "tail" functions is used to exploit an extension of Strassen's invariance theorem for the law of the iterated logarithm due to Wichura, and thereby obtain limit theorems for a variety of functions of the "tail" behavior of the sums $S_n$. For example, with probability one, $$\lim \sup_{n\rightarrow \infty} (n/2 \log \log n)^\frac{1}{2} \max_{n\leqq k < \infty} (k^{-1}S_k) = 1$$ and $$\lim \sup_{n\rightarrow \infty} n^{-1} \max \{k \geqq 1: k^{-1}S_k \geqq \theta(2 \log \log n/n)^\frac{1}{2}\} = \theta^{-2}.$$

Let $R_n$ be the number of distinct elements among $X_0, X_1,\cdots, X_n$, where $\{X_n\}$ is an irreducible recurrent Markov chain. It is shown that, under an appropriate condition, $n^{-1}R_n \rightarrow 0$ a.s. $(P_a)$ where $a$ is any state and $P_a$ is conditional probability measure given $X_0 = a$. We prove that any recurrent random walk satisfies our condition, so that the result contains the well-known random walk case. We also give an example of an irreducible recurrent chain for which the result fails to hold.

Among functions between manifolds those which transform probability distributions with densities to probability distributions with densities are characterised as almost submersions. As an application, conditions are given on sample size for sample means of distributions with density on Stiefel manifolds and Grassmannians to have densities.

Linear regression of a random variable against several functions of time is considered. Limit processes are obtained for the sequences of partial sums of residuals. The limit processes, which are functions of Brownian motion, have covariance kernels of the form: $$K(s, t) = \min (s,t) - \int^t_0 \int^s_0 g(x, y) dx dy.$$ The limit process and its covariance kernel are explicitly stated for each of polynomial and harmonic regression.

In this note we give a very simple and elementary proof of a slightly more general form of the conditional Borel-Cantelli lemma.