Upper and lower functions are defined for the large values of $|X_d(t + u) - X_d(t - \nu)|$ as $(u + \nu) \downarrow 0$ where $X_d$ is a standard Brownian motion in $R^d$, and it is shown that the integral test for two-sided growth in $R^d$ is the same as that for one-sided growth in $R^{d+2}$. It is also shown that, for $d \geqq 4$, the lower asymptotic growth rate of $|X_d(t + u) - X_d(t - \nu)|$ for small $(u + \nu) = h$ is the same as the lower growth rate of $|X_{d-2}(t + h) - X_{d-2}(t)|$. Integral tests are also obtained for local asymptotic growth rates of the associated processes $P_d(a) = \inf_{t\geqq0} \{t: |X(t)| \geqq a\}$ and $M_d(t) = \sup_{0\leqq s\leqq t} |X_d(t)|$.

The various objects examined in this paper arise in the study of last exit times and balayage of additive functionals for standard Markov processes. The most important results concern the characterization of the Laplace transform of an entrance law, the relationship between the last exit distribution from a set and the capacity measure of the set, the characterization of projective sets and $d$-sets, and a last exit decomposition formula for finite sets $F$ which expresses the distribution of $X_t$ in terms of the last exit from $F$ prior to $t$.

Let $X(t)$ be a right continuous temporally homogeneous Markov process, $T_t$ the corresponding semigroup and $A$ the weak infinitesimal generator. Let $g(t)$ be absolutely continuous and $\tau$ a stopping time satisfying $$E_x(\int^\tau_0 |g(t)| dt) < \infty \text{and} E_x(\int^\tau_0|g'(t)| dt) < \infty$$. Then for $f \in \mathscr{D}(A)$ with $f(X(t))$ right continuous the identity $$E_xg(\tau)f(X(\tau)) - g(0)f(x) = E_x(\int^\tau_0 g'(s)f(X(s)) ds) + E_x(\int^\tau_0 g(s)Af(X(s)) ds)$$ is a simple generalization of Dynkin's identity $(g(t) \equiv 1)$. With further restrictions on $f$ and $\tau$ the following identity is obtained as a corollary: $$E_x(f(X(\tau))) = f(x) + \sum^{n-1}_{k=1} \frac{(-1)^{k-1}}{k!} E_x(\tau^k A^k f(X(\tau))) \\ + \frac{(-1)^{n-1}}{(n-1)!} E_x(\int^\tau_0 u^{n-1}A^nf(X(u)) du)$$ These identities are applied to processes with stationary independent increments to obtain a number of new and known results relating the moments of stopping times to the moments of the stopped processes.

We show that $L^1$-bounded semimartingales (quasi-martingales, $F$-processes) correspond to finite signed measures on the $\sigma$-field of previsible sets. This representation of semimartingales as signed measures is used to derive in a unified manner the main decomposition theorems for semi- and supermartingales.

Let $\mu$ be a probability measure on $(\mathscr{R}, \mathscr{B})$, where $\mathscr{R}$ is the real line and $\mathscr{B}$ the family of Borel sets on $\mathscr{R}$. A measurable set `$A$' is called $\mu$-invariant if $\mu(A + \theta) = \mu(A) \mathbf{\forall} \theta, -\infty < \theta < \infty$. Let $\mathscr{A}(\mu)$ denote the family of all $\mu$-invariant sets. Let $S(\mu)$ denote the set where the characteristic function of $\mu$ vanishes. In this paper we establish the following results concerning $\mu$-invariant sets. (i) Suppose $S(\mu) \cap \overline{\lbrack S(\mu) \oplus S(\mu) \rbrack}$ is compact. Then $A$ is $\mu$-invariant implies $\mu(A) = 0, \frac{1}{2}, 1$. (ii) Fourier series representations are developed to study $\mu$-invariant sets. (iii) Dependence of $\mathscr{A}(\mu \ast \nu)$ on $\mathscr{A}(\mu)$ and $\mathscr{A}(\nu)$ is examined and representations for $\mu \ast \nu$-invariant sets are derived. (iv) Dependence of $\mathscr{A}(\mu)$ on $S(\mu)$ is carefully examined. (v) $A$ conjecture that $\mathscr{A}(\mu) \subset \mathscr{A}(\nu)$ implies that $\mu$ is a factor of $\nu$ is shown to be false.

Let $g_1, g_2,\cdots$ be random elements of the Euclidean group of motions of $d$-dimensional Euclidean space $R^d (d \geqq 1)$, that are independent and identically distributed. The product $g_1\cdots g_n$ is represented in the form $t(n)r(n)$, where $t(n)$ is a translation and $r(n)$ is a rotation. In this paper it is shown that under natural conditions $r(n)$ and $n^{-\frac{1}{2}}t(n)$ jointly converge weakly as $n \rightarrow \infty$ to the product distribution of the Haar measure on a certain closed subgroup of the rotations group, and a normal distribution on $R^d$, with mean zero and covariance matrix $\sigma^2\mathbf{I}$ ($\mathbf{I}$ is the identity matrix), and the value of $\sigma^2$ is identified.

Let $X = \{X_t = (X_t^1,\cdots, X_t^d), t \geqq 0\}$ be an isotropic stochastic process with stationary independent increments having its values in $d$-dimensional Euclidean space, $d \geqq 2$. Let $R_t = |X_t|$ be the radial process. It is proved (except for a rather trivial exception) that the Markov process $\{R_t\}$ hits points if and only if the real process $\{X_t^1\}$ hits points; a simple analytic criterion for the latter possibility has been known now for some time. If $x > 0$, the sets $\{t: R_t = x\}$ and $\{t: X_t^1 = 0\}$ are then shown to have the same size in the sense that there is an exact Hausdorff measure function that works for both. Finally, if $X^1$ hits points, it is shown that then $X$ will hit any reasonable smooth surface.

The life distribution $H(t)$ of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities $P_k$ of not surviving the first $k$ shocks. Various properties of the discrete failure distribution $P_k$ are shown to be reflected in corresponding properties of the continuous life distribution $H(t)$. As an example, if $P_k$ has discrete increasing hazard rate, then $H(t)$ has continuous increasing hazard rate. Properties of $P_k$ are obtained from various physically motivated models, including that in which damage resulting from shocks accumulates until exceedance of a threshold results in failure. We extend our results to continuous wear processes. Applications of interest in renewal theory are obtained. Total positivity theory is used in deriving many of the results.

$\{X_n, n \geqq 1\}$ are $\operatorname{i.i.d.}$ random variables with continuous $\operatorname{df} F(x). X_j$ is a record value of this sequence if $X_j > \max \{X_1,\cdots, X_{j-1}\}$. We compare the behavior of the sequence of record values $\{X_{L_n}\}$ with that of the sample maxima $\{M_n\} = \{\max (X_1,\cdots, X_n)\}$. Conditions for the relative stability ($\operatorname{a.s.}$ and $\operatorname{i.p.}$) of $\{X_{L_n}\}$ are given and in each case these conditions imply the relative stability of $\{M_n\}$. In particular regular variation of $R(x) \equiv - \log (1 - F(x))$ is an easily verified condition which insures $\operatorname{a.s.}$ stability of $\{X_{L_n}\}, \{M_n\}$ and $\{\sum^n_{j=1} M_j\}$. Concerning limit laws, $X_{L_n}$ may converge in distribution without $\{M_n\}$ having a limit distribution and vice versa. Suitable differentiability conditions on $F(x)$ insure that both sequences have a limit distribution.

This paper describes the limit behavior of sub-critical, age-dependent branching processes for which the Malthusian parameter does not exist.

This paper shows that the epsilon entropy in the sup norm of a wide variety of processes with continuous paths on the unit interval is finite. In fact, the class coincides with the class of processes for which proofs of continuity have been given from a covariance condition. This suggests the conjecture that the epsilon entropy of any process continuous on the unit interval is finite in the sup norm of continuous functions. The epsilon entropy considered in this paper is defined as the minimum Shannon entropy of any partition by sets of diameter at most epsilon of the space of continuous functions on the unit interval, where the probability is the one inherited from the given process. The proof proceeds by constructing partitions and estimating their entropy using probability bounds.

Let $H$ be a real and separable Hilbert space, $\Gamma$ the Borel $\sigma$-field of $H$ sets, and $\mu_1$ and $\mu_2$ two probability measures on $(H, \Gamma)$. Several sufficient conditions for equivalence (mutual absolute continuity) of $\mu_1$ and $\mu_2$ are obtained in this paper. Some of these results do not require that $\mu_1$ and $\mu_2$ be Gaussian. The conditions obtained are applied to show equivalence for some specific measures when $H$ is $L_2\lbrack T \rbrack$.

This paper establishes a rate of convergence of a random walk to Brownian motion which is nearly best possible. The Skorokhod representation is employed in the proof.

A sequence of stopping or first passage times is utilized to derive the limiting distribution of the maximum of partial sums of independent, identically distributed random variables with mean zero and finite variance and concomitantly the limit distribution of the stopping times themselves. The result, due to Erdos and Kac, first appeared in the paper which launched the extremely fruitful invariance principle; reflection enters in the calculations relating to the choice of a specific distribution for the $\{X_n\}$. Moreover, it is noted when the $\{X_n\}$ are $\operatorname{i.i.d.}$ with mean $\mu > 0$ and variance $\sigma^2 < \infty$ that $\max_{1\leqq j\leqq n} S_j/j^\alpha$ has a limiting standard normal distribution for any $\alpha$ in [0, 1).

Consider a sample $X_1, \cdots, X_n$ from a Dirichlet process $P$ on an uncountable standard Borel space $(\mathscr{X}, \mathscr{A})$ where the parameter $\alpha$ of the process is assumed to be non-atomic and $\sigma$-additive. Let $D(n)$ be the number of distinct observations in the sample and denote these distinct observations by $Y_1, \cdots, Y_{D(n)}$. Our main results are (1) $D(n)/\log n \rightarrow_{\operatorname{a.s.}} \alpha(\mathscr{X}), n \rightarrow \infty$, and (2) given $D(n), Y_1, \cdots, Y_{D(n)}$ are independent and identically distributed according to $\alpha(\bullet)/\alpha(\mathscr{X})$. Result (1) shows that $\alpha(\mathscr{X})$ can be consistently estimated from the sample, and result (2) leads to a strong law for $\sum^{D(n)}_{i=1} Y_i/D(n)$.

The paper deals with the existence under the continuum hypothesis of the probability limit identification function, the definition of which has been presented by G. Simons in 1971.

It is proved that the class of stable probability measures and the class of self-decomposible probability measures on Banach spaces are closed under weak convergence.