Let $H = -d^2/dx^2 + P(x)$ on $L^2(\mathbb{R}, dx)$ and let $E = \inf \operatorname{spec} (H)$. Let $\Omega$ be a normalized vector with $H\Omega = E\Omega$. Let $q(t)$ be the Markov process with generator $G = \Omega^{-1} (H - E)\Omega$, which is a Brownian motion with drift. We investigate behavior of $q(t)$ as $t \rightarrow \infty$ and in particular prove that if $P(x) = a_{2m} x^{2m} + \cdots + a_0; a_{2m} > 0$, then $$\lim \sup_{t\rightarrow\infty} \int^{t+1}_t q(s) ds/(\ln t)^{1/2m} = (a_{2m})^{-1/2m}$$ with probability one. These represent fluctuations in the sense that the $\lim \inf$ is $-(a_{2m})^{-1/2m}$. We obtain some weaker results for the $P(\phi)_2$ Euclidean field theory.

Let $Z$ be a finite or countable set, $\Xi$ the set of subsets of $Z, \{\xi_t\}$ a Markov process with state space $\Xi$. A process $\{\xi_t^\ast\}$ with the same state space is called associate to $\{\xi_t\}$ if $\mathbf{P}_\xi\{\xi_t \cap \eta \neq \varnothing\} = \mathbf{P}_\eta^\ast\{\xi_t^\ast \cap \xi \neq \varnothing\}$ whenever $\xi$ and $\eta$ are subsets of $\mathbf{Z}$, at least one of which is finite. Criteria are found for the existence of a process associate to a given one. Examples and applications are given.

Martingale problems associated with the generators of infinite spin flip systems are considered. The stochastic calculus of spin flip systems is developed and applied to the existence and uniqueness questions. Existence of solutions is proved under the assumption that the flip rates are continuous functions of the configurations. Uniqueness theorems are proved under two different conditions and a counterexample to uniqueness in complete generality is given. The techniques also yield ergodic theorems, including rates of convergence, and results concerning mutual absolute continuity of different processes.

For each positive integer $p$, let $R_n^p$ be the number of points visited exactly $p$ times by a random walk during the course of its first $n$ steps. We call the random variables $R_n^p$ the multiple range of order $p$ for the given walk. We prove that for two-dimensional simple walk, $R_n^p$ obeys the strong law of large numbers $\lim_{n\rightarrow\infty} R_n^p/(\pi^2 n/\log^2 n) = 1\mathrm{a.s.}$ The method of proof generalizes to yield a similar result for all genuine two-dimensional walks with 0 mean and finite $2 + \varepsilon$ moments $(\varepsilon > 0)$.

This paper deals with an Edgeworth-type expansion for the distribution of a sample quantile. As the sample size $n$ increases, these expansions establish a higher order approximation which holds uniformly for all Borel sets. If the underlying distribution function has $s + 2$ left and right derivatives at the true quantile, the error of the approximation is of order $O(n^{-(s+1)})$. From this result asymptotic expansions for the distribution functions of sample quantiles and for percentage points are derived.

Lester E. Dubins and Leonard J. Savage have shown how to define a large family of finitely additive probability measures on the lattice of open sets of spaces of the form $X \times X \times \cdots$, where $X$, otherwise arbitrary, is assigned the discrete topology. This lattice does not include many of the sets which occur in the usual treatment of such probabilistic limit laws as the martingale convergence theorem, and in some unpublished notes Dubins and Savage conjectured that there might be a natural way to extend their measures to such sets. We confirm their conjecture here by showing that every set in the Borel sigma-field can be squeezed between an open and a closed set in the usual manner. It is then possible to generalize to this finitely additive setting many of the classical countably additive limit theorems. If assumptions of countable additivity are imposed, the extension studied here, when restricted to the usual product sigma-field, agrees with the conventional extension.

A local Holder condition is obtained for the local time of a stationary Gaussian process with spectral density function proportional to $(a^2 + \lambda^2)^{-(\alpha +\frac{1}{2})}$. A lower bound for the Hausdorff measure of the zero set of the process is also obtained.

By making use of the martingale $\int^\infty_0 \exp (yW(t) - (t/2)y^2) dF(y)$, Robbins and Siegmund have evaluated the probability that the Wiener process $W(t)$ would ever cross certain moving boundaries. In this paper, we study this class of boundaries and make use of certain moment generating function martingales to obtain boundary crossing probabilities for sums of i.i.d. random variables. Invariance theorems for these boundary crossing probabilities are proved, and some applications to confidence sequences and power-one tests are also given.

A $\{0, 1\}$-valued discrete time stochastic process $\beta = \{\beta_n\}^\infty_{n=1}$ will be referred to simply as a walk. The notion of central (modal) term of a binomial distribution is generalized to the conditional-on-the-past distributions of $N$th partial sums of walks. The emphasis here is placed on the smallest possible central term $V_A(N)$ within a given class $A$ of walks. If $A$ consists of (i) all walks, (ii) all stationary independent walks, (iii) all stationary Markov walks which are invariant under interchange of 0 and 1, then, respectively, (i) $\{N \cdot V_A(N)\}^\infty_{N=1}$, (ii) $\{N^{\frac{1}{2}} \cdot V_A(N)\}^\infty_{N=1}$, (iii) $\{N \cdot V_A(N)/(\log N)^{\frac{1}{2}}\}^\infty_{N=2}$ are bounded sequences which are bounded away from zero.

It is shown that certain estimators of the offspring and immigration means in a subcritical simple branching process with immigration are strongly consistent and obey the central limit theorem and law of the iterated logarithm under natural conditions.

Consider $m \geqq 2$ nonnegative, integer-valued random variables $X_1^{(n)},\cdots, X_m^{(n)}$ satisfying $\sum^m_{j=1} X_j^{(n)} \leqq n$. If $(X_1^{(n)},\cdots, X_m^{(n)})$ is one member of a family of random vectors, indexed by different values of the bound $n$, Darroch has proposed a definition of "independence except for the constraint," termed $F$-independence, which relates members of the family through their conditional distributions. In this paper we study the limit theory for sums of nonnegative, integer-valued variables, when the sums are bounded and the variables $F$-independent. The $F$-independence analogue of infinite divisibility, termed infinite $F$-divisibility, is defined and characterized, and it is shown that limit distributions of sums of $F$-independent, asymptotically negligible variables are infinitely $F$-divisible. Conditions under which the limit is binomial are given. Our results apply to families of random variables, induced by the $F$-independence definition, and their role in the theory is discussed.

Let $X_1, X_2,\cdots$ be a stationary sequence of $B$-valued random variables, where $B$ is a finite set. For each positive integer $n$, and number $\lambda$ such that $0 < \lambda < 1$, let $N(n, \lambda)$ be the cardinality of the smallest set $E \subset B^n$ such that $P\lbrack(X_1, X_2,\cdots, X_n) \in E\rbrack > 1 - \lambda$. An example is given to show that $\lim_{n\rightarrow\infty}n^{-1} \log N(n, \lambda)$ may not exist for some $\lambda$, thereby settling in the negative a conjecture of Parthasarathy.