A point process system is a random operator assigning a nonnegative integer-valued measure to a random nonnegative integer-valued measure. We define certain parameters for such a system and discuss the problem of estimating these parameters. We also consider the related problem of measuring the degree of association of two point processes.

Content process $X$ of a continuous store satisfies $X_t = X_0 + A_t - \int^t_0 r(X_s) ds, t \geqq 0$. Here, $A$ has nonnegative stationary independent increments, and $r$ is a nondecreasing continuous function. The solution $X$ is a Hunt process. Paper considers the local time $L$ of $X$ at $0$. $L$ may be the occupation time of $\{0\}$ if the latter is not zero identically. The more interesting case is where the occupation time of $\{0\}$ is zero but 0 is regular for $\{0\}$; then $L$ is constructed as the limit of a sequence of weighted occupation times of $\{0\}$ for a sequence of Hunt processes $X^n$ approximating $X$. The $\lambda$-potential of $L$ is computed in terms of the Levy measure of $A$ and the function $r$.

$\{X_n, n\geqq 1\}$ are i.i.d. rv's with df $F$. Set $M_n = \max\{X_1, \cdots, X_n\}$. As a basic assumption, suppose normalizing constants $a_n > 0, b_n, n \geqq 1$ exist such that $\lim_{n\rightarrow\infty} P\lbrack M_n \leqq a_n x + b_n \rbrack = G(x)$, nondegenerate. Define the random function $Y_n(t) = (M_{\lbrack nt \rbrack} - b_n)/a_n$. By considering weak convergence of underlying two dimensional point processes, an alternate proof of the original Lamperti result that $Y_n \Rightarrow Y$ is given where $Y$ is an extremal-$G$ process. From the convergence of the point processes, other weak convergence results are shown. Let $x(t)$ be nondecreasing and $Nx(I)$ be the number of times $x$ jumps in time interval $I$. Then $Y_n^{-1} \Rightarrow Y^{-1}, NY_n \Rightarrow NY, NY_n^{-1} \Rightarrow NY^{-1}$. From these convergences emerge a variety of limit results for record values, record value times and inter-record times.

Weak convergence of the multivariate point process of upcrossings of several high levels by a stationary Gaussian process is established. The limit is a certain multivariate Poisson process. This result is then used to determine the joint asymptotic distribution of heights and locations of the highest local maxima over an increasing interval. The results are generalized to upcrossings and local maxima of two dependent Gaussian processes. To prevent nuisance jitter from hiding the overall structure of crossings and maxima the above results are phrased in terms of $\varepsilon$-crossings and $\varepsilon$-maxima, but it is shown that under suitable regularity conditions the results also hold for ordinary upcrossings and maxima.

Let $\{X_i: i \geqq 1\}$ be a strong-mixing sequence of uniform $\lbrack 0, 1 \rbrack$ rv's and $\{C_i\}$ a sequence of constants, and define the generalized empirical process by $U_N(t) = (\sum^N_{i=1} C_i^2)^{-\frac{1}{2}} \sum^N_{i=1} C_i(I_{\lbrack X_i\leqq t \rbrack} - t), 0 \leqq t \leqq 1$. In this paper, the weak convergence, relative to the Skorohod metric, of $(U_N/q)$ to a certain Gaussian process $(U_0/q)$ is proved under certain conditions on the constants $\{C_i\}$, the strong-mixing coefficient and the function $q$ defined on $\lbrack 0, 1 \rbrack$. The class of functions $q$ considered in this paper include those of the type $q(t) = \lbrack t(1 - t) \rbrack^\eta, \eta > 0$. The earlier results of Fears and Mehra [7] concerning empirical processes for $\phi$-mixing sequences are also improved by weakening the conditions on the $\phi$-mixing coefficient and the function $q$.

An extension of the usual problem of bounding the total variation of the difference of two probability measures is considered for certain convolutions of probability measures on a measurable Abelian group. The result is a fairly general approximation theorem which also yields an $L_p$ approximation theorem and a large deviation result in some special cases. A limit theorem in equally general setting is proved as a consequence of the main theorem. As the convolutions of probability measures under consideration reduce to the Poisson binomial distribution as a special case, an alternative proof of the approximation theorem in this special case is discussed.

The class $\mathscr{J}$ of subexponential distributions is characterized by $F(0) = 0, 1 - F^{(2)} (x) \sim 2\{1 - F(x)\}$ as $x \rightarrow \infty$. New properties of the class $\mathscr{J}$ are derived as well as for the more general case where $1 - F^{(2)} (x) \sim \beta\{1 - F(x)\}$. An application to transient renewal theory illustrates these results as does an adaptation of a result of Greenwood on randomly stopped sums of subexponentially distributed random variables.

The joint distribution of the $n$th record value and the $n$th inter-record time is considered to obtain a transformation which has nondegenerate limit distribution.

Suppose $Y, Y_n$ are stochastic processes in $C\lbrack 0, 1 \rbrack$ and the finite-dimensional distributions of $Y_n$ converge vaguely to those of $Y$. Then a necessary and sufficient condition for the vague convergence of the distributions of $Y_n$ to that of $Y$ is an approximate equicontinuity of the sequence $\langle Y_n \rangle$. Dudley (1966) generalized this standard result. We generalize Dudley's result to the case when the values of $X_n$ are in an arbitrary metric space and extend the result also to the case of the Skorohod metric. In our situation vague compactness does not imply tightness and thus a different proof than Dudley's (1966) must be used. The proof we use is simple and of interest even when other proofs are available.

Suppose $S_n = \sum^n_1 X_j$, where $\{X_n\}$ is a sequence of random variables. Under progressively weaker hypotheses, Pyke and Root (1968), Chatterji (1969) and Chow (1971) have proved that $E|S_n - b_n|^r = o(n)$, where $0 < r < 2$ and $\{b_n\}$ is properly chosen. This paper gives a fairly elementary proof of Chow's result under further weakened hypotheses.

Let $(X, \mathscr{F}, P)$ be a probability space and $\mathscr{A}$ a sub-$\sigma$-lattice of $\mathscr{F}: \mathscr{A}$ is closed under countable union and countable intersection and contains $X$ and $\varnothing$. Let $P^\mathscr{A}$ denote conditional expectation given $\mathscr{A}$ (Barlow and coworkers, 1972), and for fixed $p \geqq 1$ let $M^\mathscr{A}$ denote conditional $p$-mean given $\mathscr{A}$ (Brunk and Johansen, 1970). Rogge showed (1974) that for $\sigma$-fields $\mathscr{A}$ and $\mathscr{B}, \sup \{\|P^\mathscr{A} f - P^\mathscr{B} f\|_2: 0 \leqq f \leqq 1\} \leqq 2\{\delta(\mathscr{A}, \mathscr{B})\lbrack 1- \delta(\mathscr{A}, \mathscr{B}) \rbrack\}^{\frac{1}{2}}$, where $\delta(\mathscr{A}, \mathscr{B}) \equiv \max\{\sup_{A\epsilon\mathscr{A}} \inf_{B\epsilon\mathscr{B}} P(A \triangle B), \sup_{B\epsilon\mathscr{B}} \inf_{A\epsilon\mathscr{A}} P(A \triangle B)\}:$ and that for $p \geqq 1$ the convergence to 0 of $\|P^\mathscr{F} n f - P^\mathscr{F} \infty f\|_p$ is uniform for $|f| \leqq 1$ if $\delta(\mathscr{F}_n, \mathscr{F}_\infty) \rightarrow 0$. In the present paper an inequality for conditional $p$-means given $\sigma$-lattices similar to Rogge's is obtained for $p \geqq 2$ and is applied to obtain uniformity of convergence to 0 of $\|M^\mathscr{A}_n f - M^\mathscr{A} \infty f\|_p$ when $\delta(\mathscr{A}_n, \mathscr{A}_\infty) \rightarrow 0$.

A generalization of the Shannon-McMillan Theorem ($L^1$ version) is obtained for the action of an amenable group on a probability space, thereby settling a conjecture of Pickel and Stepin. Interesting properties of the limit function are derived. The entropy of an action of an amenable group is defined.

In this paper, we extend known characterizations of normal and other distributions. Let $X(t), t \geqq 0$, be a continuous (in probability) homogeneous process, with independent increments. Let $g(s, t)$ and $h(s)$ be continuous functions on $\lbrack a, b \rbrack^2$ and $\lbrack a, b \rbrack, 0 \leqq a < b < \infty$. Define stochastic integrals $Y_1 = \int^b_a h(s)X(ds)$ and $Y_2 = \int^b_a \int^b_a g(s, t)X(ds)X(dt)$. It is known that $Y_1$ exists in the sense of convergence in probability. It is shown here that $Y_2$ exists at least in the sense of convergence in $L_2$, under the additional assumption that $X$ is of second-order. The main results of this paper are to obtain, under additional appropriate assumptions on $g$ and $h$, characterizations of a class of stochastic processes which include the Brownian motion, Poisson, negative binomial and gamma processes, based on the linear regression of $Y_2$ on $Y_1$.

Let $\{X_n\}$ be a sequence of independent, identically distributed random variables with $EX_1 = 0, EX_1^2 = 1$. Define $S_n = X_1 + \cdots + X_n$, and $A_n = \max_{1\leqq k\leqq n} |S_k|$. We prove that $\lim \inf A_n(n/\log \log n)^{-\frac{1}{2}} = \pi/8^{\frac{1}{2}}$ with probability one. This result was proved by Chung under the assumption of a finite third moment and under progressively weaker moment assumptions by Pakshirajan, Breiman, and Wichura. Chung posed the problem of whether it is enough to assume only the finiteness of the second moment in his review of Pakshirajan's paper in 1961. We showed earlier that $(n/\log \log n)^{\frac{1}{2}}$ is the correct normalization but were unable to show that the constant is necessarily $\pi/8^{\frac{1}{2}}$.

Necessary and sufficient conditions are given for almost all paths of a Gaussian process to be absolutely continuous (with derivative in $L_p$). Also sufficient conditions for almost sure path absolute continuity of second order processes are derived, slightly generalizing those previously known.

Let $\{(\Omega, \mathscr{A}, \mu_n), n \geqq 1\}$ be a sequence of probability spaces. Blum and Pathak [Ann. Math. Statist. 43 (1972) 1008-1009] proved a zero-one law for permutation invariant sets $A \epsilon \mathscr{A}^\infty$; which includes the zero-one laws of Hewitt and Savage [Trans. Amer. Math. Soc. 80 (1955) 470-501] and Horn and Schach [Ann. Math. Statist. 41 (1970) 2130-2131] as special cases. The proper reason for this is shown to be the fact that the set of measures admitting the zero-one law of Blum and Pathak coincides with the set of all strong limit points of measures admitting the zero-one law of Horn and Schach.

Cells of certain variety live a random length of time and then split into two new cells. Let $t_1 < t_2 < t_3 < \cdots$, be an increasing sequence of positive numbers such that any given cell has probability $\lambda/n$ with $\lambda > 0$, that its life span be at least $t_n$ units of time. Starting with one cell, the $n$th generation will have $2^n$ cells and for each one we count the number of its ancestors and itself whose life span was at least $t_n$ units of time. These numbers determine an empirical distribution (the $n$th empirical distribution). It is shown that for almost all cell cultives (starting each time with one cell) the sequence of these empirical distributions converges to the Poisson distribution with parameter $\lambda$.