A brief overview is presented of the relation of the properties of real polymers to the problem of self-avoiding random walks. The self-consistent field method is discussed wherein the non-Markovian continuous self-avoiding polymer is replaced by a self-consistent Markovian approximation. An outline is presented of the method of solution of the resultant nonlinear integrodifferential equations. A description is also presented of the scaling theories which provide a means for deducing some exponents in the asymptotic dependence of walk properties on the length of the walk in the limit of infinite length walks.

A form of the central limit theorem for vector valued Markov chains is given, which is applicable to models arising in polymer chemistry.

If $(X_t)_{t \geq 0}$ is a Brownian motion on the real line, started at zero, if $M_t \equiv \max\{X_s; s \leq t\}$ and if $Y_t \equiv 2M_t - X_t$ for $t \geq 0$, then $(Y_t)_{t \geq 0}$ is a homogeneous strong Markov process equal in law to the radial part of Brownian motion in three dimensions. This result was discovered by Pitman, and recently Rogers and Pitman have found other one-dimensional diffusions $X$ for which $2M - X$ is again a diffusion. This paper gives a complete characterisation of all such diffusions $X$.

A simple condition sufficient to ensure that a function of a time-homogeneous Markov process is again a time-homogeneous Markov process is proved. This result is then used to study a number of diffusions; in particular, an extension of a result of Pitman is proved, from which it is possible easily to deduce the path decompositions of Williams.

Let $t(x, y)$ be the passage time from $x$ to $y$ in $Z^2$ in a percolation process with passage time distribution $F$. If $x \in R^2$ it is known that $\int (1 - F(t))^4 dt < \infty$ is a necessary and sufficient condition for $t(0, nx)/n$ to converge to a limit in $L^1$ or almost surely. In this paper we will show that the convergence always occurs in probability (to a limit $\varphi(x) < \infty$) without any assumptions on $F$. The last two results describe the growth of the process in any fixed direction. We can also describe the asymptotic shape of $A_t = \{y : t(0, y) \leq t\}$. Our results give necessary and sufficient conditions for $t^{-1} A_t \rightarrow \{x : \varphi(x) \leq 1\}$ in the sense of Richardson and show, without any assumptions on $F$, that the Lebesgue measure of $t^{-1} A_t\Delta\{x : \varphi(x) \leq 1\} \rightarrow 0$ almost surely. The last result can be applied to show that without any assumptions on $F$, the $x$-reach and point-to-line processes converge almost surely.

Circuit processes of order $r$ are defined using a finite class of weighted circuits in a finite set $S$. The probability of the next value of the process is made proportional to the total weight of those circuits in the class which pass through the value in question and the last $r$ values. The process is an order $r$ Markov chain in $S$, and the stationary distribution is easily calculated. Also, it is shown that all stationary order $r$ Markov chains in a finite set can be represented as circuit processes of that order.

Let $X_{n1}, \cdots, X_{nk_n}$ be independent random vectors in $\mathbb{R}^d$. Necessary and sufficient conditions are found for the existence of linear operators $A_n$ on $\mathbb{R}^d$ such that $\mathscr{L}(A_n(\sum^{k_n}_{j = 1} X_{nj})) \rightarrow N(\overset{\rightarrow}{0}, I)$, where $I$ is the $d \times d$ identity covariance matrix. These results extend the authors' previous work on sums of i.i.d. random vectors. The proof of the main theorem is constructive, yielding explicit centering vectors and norming linear operators.

Let $X_1, X_2, \cdots$ be i.i.d. random variables whose common distribution function $F$ is in the domain of attraction of a nonnormal stable distribution. A simple, probabilistic proof of the convergence of the normalized partial sums to the stable distribution is given. The proof makes use of an elementary property of order statistics and clarifies the manner in which the largest few summands determine the limiting distribution. The method is applied to determine the limiting distribution of self-norming sums and deduce a representation for the limiting distribution. The representation affords an explanation of the infinite discontinuities of the limiting densities which occur in some cases. Application of the technique to prove weak convergence in a separable Hilbert space is explored.

Let $S_n$ be the sum of $n$ independent and identically distributed random variables with zero means and unit variances. The central limit theorem implies that $P(S_n \leq 0) \rightarrow 1/2$, and the Spitzer-Rosen theorem (with refinements by Baum and Katz, Heyde, and Koopmans) provides a rate of convergence in this limit law. In the present paper we investigate the converse of this result. Given a certain rate of convergence of $P(S_n \leq 0)$ to 1/2, what does this imply about the common distribution of the summands?

In a sequential decision problem it is usually assumed that the available information is represented by an increasing family $\mathscr{F}$ of $\sigma$-algebras. Often a reduction, e.g., according to principles of sufficiency or invariance, is performed which yields a smaller family $\mathscr{G}$. The consequences of such a reduction for problems of optimal stopping are treated in this paper. It is shown that $\mathscr{G}$ is transitive for $\mathscr{F}$ (in the Bahadur sense) if and only if for any stochastic process adapted to $\mathscr{G}$ the value (i.e., maximal reward by optimal stopping) under $\mathscr{G}$ and the value under $\mathscr{F}$ are equal.

Consider $n$ points taken at random on the circumference of a unit circle. Let the successive arc-lengths between these points be $S_1, S_2, \cdots, S_n$. Convergence of the moment generating function of $\max_{1 \leq k \leq n} S_k - \ln n$ is proved. Let each point be associated with an arc, each of length $a_n$, and let the length of the circumference which is not covered by any arc, the vacancy, be $V_n$. Convergence of the vacancy after suitable scaling is obtained. The methods used are general and can, e.g., be used to obtain asymptotic results for other spacings and coverage problems.

Let $E$ be a locally convex Hausdorff linear topological space, $E'$ be the topological dual of $E$ and $\gamma$ be a nondegenerate, centered Gaussian-Radon measure on $E$. Then every nonnegative definite continuous functional on $E$ is the characteristic functional of a Borel probability measure on $E^\gamma$, the closure of $E'$ in $L_0(\gamma)$. In other words, identifying $E^\gamma$ with the reproducing kernel Hilbert space $\mathscr{H}_\gamma$ of $\gamma$, we may say that for every continuous nonnegative definite function $f$ on $E$ there exists a Borel probability $\mu$ on $\mathscr{H}_\gamma$ such that $f$ is the characteristic functional of $\mu$.

Bochner's theorem formulated by Xia Dao-Xing is established for an abstract Wiener space. Let $(\iota, H, E)$ be an abstract Wiener space. Then for every continuous cylinder set measure $\nu$ on $E'$, the image $\iota'(\nu)$ is a Radon measure on $H'$.

We obtain an expression for the distribution function of the random variable $\int ZdP$ where $P$ is a random distribution function chosen by Ferguson's (1973) Dirichlet process on $(R, B)$ ($R$ is the real line and $B$ is the $\sigma$-field of Borel sets) with parameter $\alpha$, and $Z$ is a real-valued measurable function defined on $(R, B)$ satisfying $\int |Z| d\alpha < \infty$. As a consequence, we show that when $\alpha$ is symmetric about 0 and $Z$ is an odd function, then the distribution of $\int ZdP$ is symmetric about 0. Our main result is also used to obtain a new result for convergence in distribution of Dirichlet-based random functionals.

Let $X_1, X_2, \cdots$ be a strictly stationary second order sequence which is "associated"; i.e., is such that any two coordinatewise nondecreasing functions of the $X_i$'s (of finite variance) are nonnegatively correlated. If $\sum_j \operatorname{Cov}(X_1, X_j) < \infty$, then the partial sum processes, $W_n(t)$, defined in the usual way so that $W_n(m/n) = (X_1 + \cdots + X_m - mE(X_1))/\sqrt n$ for $m = 1, 2, \cdots$, converge in distribution on $C\lbrack 0, T\rbrack$ to a Wiener process. This result is based on two general theorems concerning associated random variables which are of independent interest.

Let $\{X_n\}$ be a strictly stationary process satisfying some mixing conditions, including $\phi$-mixing condition. It is the aim of the present paper to give, using a slight modification of Stein's result, a rate $O(n^{-1/2} \log n)$ of the normal approximation for a sum $S_n = X_1 + \cdots + X_n$.

Let balls be thrown successively at random into $N$ boxes, such that each ball falls into any box with the same probability $1/N$. Let $Z_n$ be the number of occupied boxes (i.e., boxes containing at least one ball) after $n$ throws. It is well known that $Z_n$ is approximately normally distributed under general conditions. We give a remainder term estimate, which is of the correct order of magnitude. In fact we prove that $0.087/\max(3, DZ_n) \leqq \sup_x |P(Z_n < x) - \Phi((x - EZ_n)/DZ_n)| \leqq 10.4/DZ_n.$

We characterize the sequences $(\alpha_i)$ of real numbers such that $\sum^\infty_{i = 1} \alpha_i f_i$ exists a.e. or in $L_p$ for all sequences of independent identically distributed symmetric random variables with $p$th moment. Moreover, we also treat the case $\sup|\alpha_i f_i| < 0\infty$ a.e.

For random walk $S_k = \sum^k_{i = 1} \xi_i$ let $T$ be the hitting time of the lower half plane. By conditioning the process $\{S_k\}^n_{k = 1}$ relative to $\lbrack T > n\rbrack$ we create a "random walk conditioned to stay positive." Sample paths of such processes tend to grow rather quickly. In studying this growth we find that except for a set of probability $\epsilon$ all such sample paths have as lower bounds any sequence of the form $\{\delta k^\eta\}^n_{k = 1}$ where $\eta \in (0, 1/2)$ and $\delta < \delta(\epsilon, \eta)$. Applications of this result to sample path behavior of a random walk as it approaches or leaves the lowest of its first $n$ values are also given.

Let $\mathscr{P}$ be the set of all $k$-atom measurable partitions of a standard measurable space $(\Omega, \mathscr{F})$, and let $T$ be an isomorphism of $(\Omega, \mathscr{F})$ onto itself. Given $P \in \mathscr{P}$, each probability measure $\mu$ on $\mathscr{F}$ stationary and ergodic with respect to $T$ determines a joint distribution under $\mu$ of the $k$-state stochastic process $(P, T)$. We say that $P$ is universal for a property $S$ (depending on $\mu$) if the distribution of $(P, T)$ satisfies $S$ for all $\mu$. Theorems are given which assure the existence of a universal $P \in \mathscr{P}$.

Let $X_1, X_2 \cdots, X_n(n \geq 2)$ be independent standard normal. Then the random variable $U = X_1/V_n$ where $V_n = \exp_2\lbrack 2^{n - 2} - 1\rbrack X_2(X_3)^2 \cdots(X^2_n)^{2^{n - 3}} \quad n \geq 3 \\ = X_2 \text{for} n = 2$ has a symmetric stable distribution with index $2^{2 - n}$.