Let $G$ be a closed subgroup of the orthogonal group $O(n)$ acting on $R^n$. A real-valued function $f$ on $R^n$ is called $G$-monotone (decreasing) if $f(y) \geqq f(x)$ whenever $y \precsim x$, i.e., whenever $y \in C(x)$, where $C(x)$ is the convex hull of the $G$-orbit of $x$. When $G$ is the permutation group $\mathscr{P}_n$ the ordering $\sim$ is the majorization ordering of Schur, and the $\mathscr{P}_n$-monotone functions are the Schur-concave functions. This paper contains a geometrical study of the convex polytopes $C(x)$ and the ordering $\precsim$ when $G$ is any closed subgroup of $O(n)$ that is generated by reflections, which includes $\mathscr{P}_n$ as a special case. The classical results of Schur (1923), Ostrowski (1952), Rado (1952), and Hardy, Littlewood and Polya (1952) concerning majorization and Schur functions are generalized to reflection groups. It is shown that a smooth $G$-invariant function $f$ is $G$-monotone iff $(r'x)(r'\nabla f(x))\leqq 0$ for all $x \in R^n$ and all $r \in R^n$ such that the reflection across the hyperplane $\{z\mid r'z = 0\}$ is in $G$. Furthermore, it is shown that the convolution (relative to Lebesgue measure) of two nonnegative $G$-monotone functions is again $G$-monotone. The latter extends a theorem of Marshall and Olkin (1974) concerning $\mathscr{P}_n$, and has applications to probability inequalities arising in multivariate statistical analysis.

One-sided iterated logarithm laws of the form $\lim \sup (1/b_n) \sum^n_1 X_i = 1$, a.s. and $\lim \sup (1/b_n) \sum^n_1 X_i = -1$, a.s. are obtained for asymmetric independent and identically distributed random variables, the first when these have a vanishing but barely finite mean, the second when $E|X|$ is barely infinite. In both cases, $\lim \inf (1/b_n) \sum^n_1 X_i = -\infty$, a.s. The constants $b_n/n$ are slowly varying, decreasing to zero in the first case and increasing to infinity in the second. Although defined via the distribution of $|X|, b_n$ represents the order of magnitude of $E|\sum^n_1 X_i|$ when this is finite. Corresponding weak laws of large numbers are established and related to Feller's notion of "unfavorable fair games" and in the process a theorem playing the same role for the weak law as Feller's generalization of the strong law is proved.

Let $\{(S_{nj}, \mathscr{F}_{nj}), 1 \leqq j \leqq k_n\}$ be a square-integrable martingale for each $n = 1,2,3,\cdots$. Define $X_{nj} = S_{nj} - S_{n,j-1} (S_{n0} = 0), U^2_{nj} = \sum^j_{i=1} X^2_{ni}, U^2_n = U^2_{nk_n}$, and for each $z \in \lbrack 0, 1\rbrack$ let $\xi_n(z) = U^{-1}_n \sum^{k_n}_{j=1} X_{nj} I(U^{-2}_n U^2_{nj} \leqq z)$ and $\eta_n(z) = \sum^{k_n}_{j=1} X_{nj} I(U^{-2}_n U^2_{nj} \leqq z); \xi_n$ and $\eta_n$ are random elements of $D\lbrack 0, 1\rbrack$. Sufficient conditions are given for $\xi_n$ to converge in distribution to Brownian motion and for $\eta_n$ to converge to a mixture of Brownian motion distributions. We give several applications and examples.

This paper is concerned with the almost everywhere convergence of martingales indexed by countable filtering sets. It is shown that the convergence is a consequence of the maximal inequality as it is in the classical case. It also contains some results about the law of large numbers when the index belongs to a sector and an optimal condition assuring the almost everywhere convergence of martingales in these sectors.

In this paper we discuss characterizations, basic properties and applications of a partial ordering, in the set of probabilities on a partially ordered Polish space $E$, defined by $P_1 \prec P_2 \operatorname{iff} \int f dP_1\leqq \int f dP_2$ for all real bounded increasing $f$. A result of Strassen implies that $P_1 \prec P_2$ is equivalent to the existence of $E$-valued random variables $X_1 \leqq X_2$ with distributions $P_1$ and $P_2$. After treating similar characterizations we relate the convergence properties of $P_1 \prec P_2 \prec \cdots$ to those of the associated $X_1 \leqq X_2 \leqq \cdots$. The principal purpose of the paper is to apply the basic characterization to the problem of comparison of stochastic processes and to the question of the computation of the $\bar{d}-$distance (defined by Ornstein) of stationary processes. In particular we get a generalization of the comparison theorem of O'Brien to vector-valued processes. The method also allows us to treat processes with continuous time parameter and with paths in $D\lbrack 0, 1\rbrack$.

Let $X_n$ be the standardized mean of $s$ observations obtained by simple random sampling from the $n$ numbers $a_{n1},\cdots, a_{nn}$ and let $b_n$ be the maximum deviation of these numbers from their mean. If $b_n$ tends to zero then the distribution function of $X_n$ tends uniformly to the normal distribution function. However this approximation is not adequate at the tails of the distribution. Here we obtain limit theorems for $P(X_n > x)$ in the two cases when $x = o(b_n^{-1})$ and $x = O(b_n^{-1})$. These are related to similar results for sums of independent random variables.

The normalized probability function of the Wilcoxon rank sum statistic is shown to converge to the normal density under very mild conditions on the two sample distributions. This is done by studying the conditional distribution of the rank sum given the first sample and by a rather heavy use of characteristic functions.

A Berry-Esseen bound of order $n^{-\frac{1}{2}}$ is established for linear combinations of order statistics. The theorem requires a "smooth" weight function, and the underlying distribution function must not have "too much weight in the tails." The distribution function need not be continuous.

It is shown in this paper that the sample paths of a Robbins-Monro process with harmonic coefficients may be approximated by weighted sums of independent, identically distributed random variables. A law of iterated logarithm and a weak invariance principle follow from this result.

New results are presented on the problem of source coding subject to a fidelity criterion for stationary, measurable, continuous-time stochastic processes. The main result is established without an ergodicity requirement and for quite general reproduction alphabets. This is then specialized to ergodic sources to prove a coding theorem for ergodic continuous-time sources. The approach is to obtain coding theorems for continuous-time sources from the coding theorem for nonergodic discrete-time sources.

Let us observe an infinite sequence $z_1 = r_1 + \varepsilon_1, z_2 = r_2 + \varepsilon_2, \cdots$ where $r_1, r_2,\cdots$ are the partial sums of independent and identically distributed random variables and the sequence of random variables $\varepsilon_k$ (the errors) is bounded by a function $f(k)$. Knowing the sequence $z_n$ we want to determine the distribution function of the summands. We will show that this problem cannot be solved in general even if $f(k)$ is constant.

For standard $p$-functions, an upper bound for $M = p(1)$, for a given value $m$ of $m(p) = \min\{p(t), 0 < t \leqq 1\}$, was proved in a previous paper by the author. The bound implied that $\nu_0 \leqq .590, \nu_0$ being the constant defined by $$I_M = \inf\{m(p)|p(1) = M\},\quad \nu_0 = \inf\{M|I_M > 0\},$$ in which $p$ varies over the class of standard $p$-functions. In the present paper both of these upper bounds are sharpened by a refinement of the argument, the limit for $\nu_0$ being reduced to .560.

Let $X_1, X_2,\cdots$ be a sequence of independent, identically distributed random variables with $E(X_1) = 0, E(X_1^2) = 1$, and $E(X_1^4) < \infty$, and for $n = 1,2,\cdots$ let $P_n$ be the distribution of $n^-\frac{1}{2} \sum^n_{i=1} X_i$. If $f$ is a function with bounded uniformly continuous derivative of order 4, then $\int f dP_n$ has an asymptotic expansion in terms of $n^{-\frac{1}{2}}$ with a remainder term of $o(n^{-1})$. This remains true even if $P_1$ is purely discrete and nonlattice.

The supports of infinitely divisible measures on separable Hilbert spaces are characterized in terms of angular semigroups. Restricted to $\mathbb{R}^n$ this result extends results of Hudson and Mason. Restricted to $\mathbb{R}^1$ our result improves Tucker's result and Hudson and Tucker's results on such supports. Also investigated are the supports of stable measures on Hilbert space.

Let $X_n, n \in \mathbb{N}$, be independent and identically distributed random variables and $\tau_n$ be random summation indices such that $\tau_n/n \rightarrow \tau > 0$ in probability. It is shown that even if $\tau_n/n$ converges to $\tau$ as quickly as possible (i.e., $\tau_n/n = \tau$) no general approximation orders for suitably normalized random sums $\sum^{\tau_n(\omega)}_{\nu=1} X_\nu(\omega)$ are available. If, however, the limit function $\tau$ is independent of $X_n, n \in \mathbb{N}$, we give a positive approximation result.

Sample continuity is proved for extended stochastic integrals involving a two-parameter Wiener process.

Suppose $n$ balls are placed into $N$ cells with arbitrary probabilities. Limit distributions for the number of empty cells are considered when $N \rightarrow \infty$ and $n \rightarrow \infty$ in such a way that $n/N \rightarrow \infty$. Limit distributions for the number of balls to achieve exactly $b$ empty cells are obtained for $N \rightarrow \infty$ with $b$ fixed and for $b \rightarrow \infty$ with $b/N \rightarrow 0$.

The uniform empirical integral differences of Sethuraman's large deviation theorem are proved to converge arbitrarily slowly.

We define a concept of saturation for a sequence of integers $\{k_j\}$. In the main theorem we prove that if $\{k_j\}$ saturates and $T$ is any weakly mixing measure-preserving transformation on an arbitrary probability space, then there exists a dense set $\mathscr{D}_T \subset L^2$ such that for $f \in \mathscr{D}_T$ $$\lim_{N\rightarrow\infty} \frac{1}{N} \sum^N_{j=1} f(T^{k_j}x) = E(f) \mathrm{a.e.}$$ This has the following application to probability theory: Let $Y_1, Y_2,\cdots$ be independent and identically distributed positive (or negative) integer-valued random variables with $E(Y_1) < \infty$. Let $$k_j(\omega) = \sum^j_{l=1} Y_l(\omega) \quad j = 1,2,\cdots$$. Then there exists a set $C$ of probability one such that for $\omega \in C$ and for any weakly mixing measure preserving transformation $T$ on an arbitrary probability space $$\lim_{N\rightarrow\infty} \frac{1}{N} \sum^N_{j=1} f(T^{k_j(\omega)}x) = E(f) \mathrm{a.e.}$$ for all $f \in L^1$.