This paper proves almost-sure convergence of the empirical measure of the normalized singular values of increasing rectangular submatrices of an infinite random matrix of independent elements. The limit is the limit as both dimensions grow large in some ratio. The matrix elements are required to have uniformly bounded central $2 + \delta$th moments, and the same means and variances within a row. The first section (relaxing the restriction on variances) proves any limit-in-distribution to be a constant measure rather than a random measure, establishes the existence of subsequences convergent in probability, and gives a criterion for almost-sure convergence. The second section proves the almost-sure limit to exist whenever the distribution of the row variances converges. It identifies the limit as a nonrandom probability measure which may be evaluated as a function of the limiting distribution of row variances and the dimension ratio. These asymptotic formulae underlie recently developed methods of probability plotting for principal components and have applications to multiple discriminant ratios and other linear multivariate statistics.

We consider stochastic differential equations $dx = f(x) dt + g(x) dw$, where $x$ is a vector in $n$-dimensional space, and $w$ is an arbitrary process with continuous sample paths. We show that the stochastic equation can be solved by simply solving, for each sample path of the process $w$, the corresponding nonstochastic ordinary differential equation. The precise requirements on the vector fields $f$ and $g$ are: (i) that $g$ be continuously differentiable and (ii) that the entries of $f$ and the partial derivatives of the entries of $g$ be locally Lipschitzian. For the particular case of a Wiener process $w$, the solutions obtained this way turn out to be the solutions in the sense of Stratonovich.

A process is equivalent to a time change of Brownian motion if and only if it is a local semimartingale.

A Levy random measure is characterized by a conditional independence structure analogous to the Markov property. Here we introduce Levy random measures and present their basic properties. Preservation of the Levy property under transformations of random measures (e.g., change of variable, passage to a limit) and under transformations of the probability laws of random measures is investigated. One random measure is said to be a submeasure of a second random measure if its probability law is absolutely continuous with respect to that of the second. We show that if the second measure is a Levy random measure then the submeasure is Levy if and only if the Radon-Nikodym derivative satisfies a natural factorization condition. These results are applied to extend the theories of Gibbs states on bounded sets in $\mathbb{R}^\nu$ and $\mathbf{Z}^\nu$.

The concept of local nondeterminism introduced by Berman is generalized and applied to divided difference sequences generated by a Gaussian process. The resulting estimates are then used to find simple sufficient conditions for the finiteness of the moments of the number of crossings of level zero. In particular it is shown that under mild regularity conditions very little more is required to make all moments finite when the variance is finite. The results are extended to curves $\xi \in \mathscr{L}_2\lbrack 0, T\rbrack$. Finally examples are given in which the variance is finite but the third moment is infinite.

In a number of well-known applied probability models certain steady-state probabilities display an insensitivity property: they depend only on the means of certain lifetime distributions entering the definition of the model, not on their exact shapes. This phenomenon has been studied by Matthes and co-workers in a general framework. New and simple proofs are given for their essential results.

In this paper we derive the characteristic functions of multivariate stable distributions; specifically the canonical representation of symmetric stable laws is given. Based on that representation, we construct linear stable processes (which include autoregressive stable processes) and stable processes with spectral representation. A sufficient condition for linear stable processes to be regular is given; the complete regularity of autoregressive stable processes is proved. Furthermore, we derive the asymptotic distribution of the Fourier transform of a sample from stable processes with spectral representation.

Let $S_1, S_2,\cdots$ be partial sums of independent and identically distributed random variables and let $f(n)$ and $g(n)$ be increasing positive sequences. Nearly sharp bounds are presented for the probabilities $P\{S_i \geqq g(i), i = 1,\cdots, n\}$ and $P\{- f(i) \leqq S_i \leqq f(i), i = 1,\cdots, n\}$ under conditions on $f$ and $g$. The most difficult results are the lower bounds in the normal case. Results are obtained by an embedding method which approximates Brownian motion by sums of independent random variables taking on only two or three values.

If $X_i, i = 1,2,\cdots$ are independent and identically distributed vector valued random variables with distribution $F$, and $S$ is a class of subsets of $R^d$, then necessary and sufficient conditions are given for the almost sure convergence of $(1/n)D_n^s = \sup_{A\in S} |(1/n) \sum 1_A(X_i) - F(A)|$ to zero. The criteria are defined by combinatorial entropies which are given as the time constants of certain subadditive processes. These time constants are estimated, and convergence results for $(1/n)D_n^S$ obtained, for the classes of algebraic regions, convex sets, and lower layers. These results include the solution to a problem posed by W. Stute.

Osterwalder-Schrader (OS) positive symmetric stationary stochastic processes are discussed. A natural construction is given for the associated positive semigroup structure. Conversely, OS-positive symmetric stationary stochastic processes are constructed from positive semigroup structures. OS-positive processes are seen to be the natural generalization of Markov processes, positive semigroup structures being the natural generalization of positivity preserving semigroups. The inheritability of OS-positivity is discussed.

Choose a point at random, i.e., according to the uniform distribution, in the interval (0, 1). Next, choose a second point at random in the largest of the two subintervals into which (0, 1) is divided by the first point. Continue in this way, at the $n$th step choosing a point at random in the largest of the $n$ subintervals into which the first $(n - 1)$ points subdivide (0, 1). Let $F_n$ be the empirical distribution function of the first $n$ points chosen. Kakutani conjectured that with probability 1, $F_n$ converges uniformly to the uniform distribution function on (0, 1) as $n$ tends to infinity. It is shown in this note that this conjecture is correct.

The distortion rate function $D(R)$ is defined as an infimum of distortion with respect to a mutual information constraint. The usual coding theorems assert that, for ergodic souces, $D(R)$ is equal to $\delta(R)$, the least distortion attainable by block codes of rate $R$. If a source has ergodic components $\{\theta\}$ with weighting measure $dw(\theta)$, it has been shown by Gray and Davisson that $\delta(R)$ is the integral of the components $\delta_\theta(R)$ with respect to $dw(\theta)$. We show that $D(R)$ is the infimum of the integrals of $D_\theta(R_\theta)$ where the integral of $R_\theta$ is $R$. Our method of proof also gives a formula for the $\bar{d}$-distance in terms of ergodic components and shows that $D(R) = D'(R)$, which is defined as the infimum of distortion subject to an entropy constraint.

In this note the conditions on an invariance principle for triangular arrays of random variables contained in an earlier paper are weakened. Random norming by functions which are not stopping times is permitted, the $L^2$-boundedness conditions on the maximum of the summands relaxed, and joint convergence with an arbitrary sequence of random elements of some other metric space proved.

For a nondifferentiable random field $\{X_t: t \in \mathbb{R}^N\}$ with values in $\mathbb{R}^d$, it is often easy to check that with probability 1 $\lim \inf_{s\rightarrow t}\|X_s - X_t\|/\sigma(s, t) = 0$ and $\lim \sup_{s\rightarrow t}\|X_s - X_t\|/\sigma(s, t) = \infty$ for a.e. $t$, where $\sigma^2(s, t) = E\|X_s - X_t\|^2$. In this note we discuss the "proportion" of $s$'s near $t$ for which $\|X_s - X_t\|/\sigma(s, t)$ is small or large.

Necessary and sufficient conditions are given for a class of stationary Gaussian processes to be mixing in the sense of Kolmogorov.

Let $X_1, X_2,\cdots$ be a sequence of independent random variables, each with zero mean and finite variance. Define $S_n = \sum^n_{i=1} X_i, s_n^2 = E(S_n^2), t_n^2 = 2 \log \log s_n^2$ and $\Lambda = \lim \sup_{n\rightarrow\infty} S_n/(s_n t_n)$. Suppose that $|X_n| \leqq c_n s_n$ a.s. for all $n$ and some real sequence $\{c_n\}$ and that $s_n \rightarrow \infty$, and let $\nu = \lim \sup_{n\rightarrow\infty} t_nc_n$. By Kolmogorov's law of the iterated logarithm, $\Lambda = 1$ if $\nu = 0$. Egorov proved that $0 \leqq \Lambda < \infty$, in every case. In this paper it will be shown that, if $\nu < \infty$, then $0 < \Lambda \leqq 1 + \sum^\infty_{j=3} \nu^{j-2}/j!$. A similar result for certain classes of unbounded random variables will also be presented.

A sufficient condition for a sequence of random variables, $T_1, T_2,\cdots$, with cumulant generating functions, $\psi_1, \psi_2,\cdots$, to have a large deviation rate is that $n^{-1}\psi_n(\lambda)\rightarrow \psi(\lambda)$, where $\psi(\lambda)$ satisfies certain regularity conditions. Here it is shown that, when the large deviation rate exists and $T_1, T_2,\cdots$ are properly truncated, it is a necessary condition.

Let $T$ denote the life length of a series system of $n$ components having respective life lengths $T_1,\cdots, T_n$, not necessarily independent. We give necessary and sufficient conditions for the existence of a set of independent random variables $\{H_I\}, I$ a subset of $\{1,\cdots, n\}$, such that the life length of the original series system and the occurrence of its failure pattern (set of components whose simultaneous failure coincides with that of the system) have the same joint distribution as the life length of a derived series system of components having life lengths $\{H_I\}$ and the occurrence of the corresponding failure pattern of the derived system. We also exhibit explicitly the distributions of these independent random variables $\{H_I\}$. This extends the results of Miller while using more elementary methods.