A class of optimal stopping problems for conditioned random walk is discussed in terms of selling strategies for the stock market.

Limit theorems are proved for two stochastic models of molecular evolution in finite populations of fixed size. An additive fitness model is shown to be asymptotically neutral in the sense that the relative fitnesses converge in probability to one and the gene frequency distribution converges to the same limiting distribution as when all mutations are selectively neutral. A multiplicative fitness model is studied and weak convergence theorems are proved for the vector whose components are the fitnesses of individuals in the population.

Let $N$ be a strictly positive integer. Motivated by a certain discrete evasion game, we search for a $\{0, 1\}$-valued discrete time stochastic process whose conditional-on-the-past distributions of the sum of the next $N$ terms are as close to uniform as possible. A process is found for which none of the sums ever occurs with conditional probability more than $2e/(N + 1)$. The process is characterized by invariance under interchange of 0 and 1, and its waiting times between successive transitions, which are independently, identically, and uniformly distributed over $\{1,2, \cdots, N + 1\}$.

Let $\{\xi_k^n, k = 0, 1, \cdots\}$ denote a $R^r$ valued discrete parameter Markov process for each $n$. For each real $T > 0$, it is shown that suitable piecewise interpolations in $D^r\lbrack 0, T \rbrack$ converge weakly as $n \rightarrow \infty$, to the diffusion given by \begin{equation*}\tag{*} x(t) = x + \int^t_0 f(x(s), s) ds + \int^t_0 \sigma(x(s), s) dw(s),\end{equation*} under essentially the condition that the solution to $(^\ast)$ is unique in the sense of multivariate distributions, $f(\bullet, \bullet), \sigma(\bullet, \bullet)$ are bounded and continuous, and the scaled "infinitesimal" coefficients of the $\{\xi_k^n\}$ are close to $f(\bullet, \bullet)$ and $\sigma(\bullet, \bullet)$. It is not required that $f(\bullet, \bullet)$ or $\sigma(\bullet, \bullet)$ satisfy a uniform Lipschitz condition, nor that $\sigma(\bullet, \bullet)\sigma'(\bullet, \bullet)$ be positive definite. The result extends the result of Gikhman and Skorokhod (1969). Two examples arising in population genetics are given, where $\sigma(\bullet, \bullet)$ is not uniformly Lipschitz.

This paper is part of the constructive program, initiated by E. Bishop, of systematic examination of classical mathematics for their computational content. From this constructive standpoint, basic concepts in probability theory are studied. Positive proofs are then given to some important theorems: Ionescu-Tulcea's theorem, a submartingale convergence theorem, and the construction of a Markov process from a strongly continuous semi-group of transition operators.

A quantity $\int |E\mathscr{B} f| dP$ (or equivalently $\int|u - P(A: \mathscr{B})| dP, 0 < u < 1)$ associated with a $\sigma$-algebra $\mathscr{B}$ is shown to act as a criterion for a type of convergence of $\sigma$-algebras. This quantity also defines an ordering of $\sigma$-algebras, so that upper and lower limits can be defined in terms of this quantity. Another criterion for the convergence of $\sigma$-algebras is described based on the existence of these limits.

A martintote is a random sequence such that the asymptotic behavior of the process distribution, conditioned with respect to the past, remains the same along the sequence. In this respect the conditional distributions of a martintote behave similarly to the conditional expectations of a martingale. We give an optional sampling theorem for martintotes and a class of examples.

Wichura (1969) studied an invariance principle for partial sums of a multi-dimensional array of independent random variables. It is shown that a similar invariance principle holds for a broad class of generalized $U$-statistics for which the different terms in the partial sums are not independent. Weak convergence of generalized $U$-statistics for random sample sizes is also studied. The case of (generalized) von Mises' functional is treated briefly.

Gnedenko's work (1943) showing the existence of a limiting distribution for the maximum of the first $n$ terms in a sequence of suitably normalized independent identically distributed random variables is extended to sequences of random variables defined on a positive-recurrent denumerable Markov chain.

The central limit theorem and limit theorems for rarity require measures of normality and exponentiality for their implementation. Simple useful measures are exhibited for these in a metric space setting, obtained from inequalities for scale mixtures and power mixtures. It is shown that the Pearson coefficient of Kurtosis is such a measure for normality in a broad class $\mathscr{D}$ containing most of the classical distributions as well as the passage time densities $s_{mn}(\tau)$ for arbitrary birth-death processes.

This is a discussion how Levy's Continuity Theorem can be proved without idealistic compactness arguments, and how it can then be used to give constructive proofs to some limit theorems.

There are two kinds of vectorvalued measures which are involved in the theory of weakly stationary processes: orthogonal (Hilbert space valued) and multiplicative (projection-valued) measures. For both classes we show that weak convergence is equivalent with the convergence of integrals over bounded continuous functions. Moreover we prove continuity theorems for the Fourier transformation as well as for the Laplace transformation of such measures.

A size 2 generator of a measure space $(\mathbf{X}, \mathscr{F}, p)$ under a set of $\mathbf{S}$ of transformation of $X$ is a partition $\{A, A^c\}$ of $X$ such that $\mathscr{F}$ is the smallest $\sigma$-algebra containing $\{s^{-1}A: s\in S\}$ up to sets of $p$-measure zero. Let $S$ be a semigroup of invertible nonsingular measurable transformations on a separable measure space $(X, \mathscr{F}, p)$ with $p(X) = 1$. Suppose that $S$ does not preserve any finite invariant measure absolutely continuous with respect to $p$. Then $\mathscr{F}$ has a size 2 generator $\{A, A^c\}$ and the orbit of $A$ under $S$ is dense in $\mathscr{F}$.

For a stationary $\phi$-mixing sequence of stochastic $p(\geqq 1)$-vectors, weak convergence of the empirical process (in the $J_1$-topology on $D^p\lbrack 0, 1 \rbrack)$ to an appropriate Gaussian process is established under a simple condition on the mixing constants $\{\phi_n\}$. Weak convergence for random number of stochastic vectors is also studied. Tail probability inequalities for Kolmogorov Smirnov statistics are provided.

Let $X_1 \leqq X_2 \leqq\cdots \leqq X_m$ and $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ be independent samples of i.i.d. random variables from continuous distributions $F$ and $G$, respectively, and suppose $F(x) = \lbrack G(x)\rbrack^k$ or $F(x) = 1 - \lbrack 1 - G(x)\rbrack^k, k > 0.$ Let $R_i$ and $S_j$ denote the ranks of $X_i$ and $Y_j$, respectively, in the ordered combined sample. We express $P(R_i \leqq b_i$, all $i$) as the determinant of a simple $m \times m$ matrix. We also show that for increasing sequences $\{a_i\}$ and $\{b_i\}, P(a_i \leqq R_i \leqq b_i$, all $i\mid F, G) = P(\alpha_j \leqq S_j \leqq \beta_j$, all $j\mid F, G)$, where $\{\alpha_j\} = \{b_i\}^c$ and $\{\beta_j\} = \{a_i\}^c$ and complementation is with respect to the set $\{i\mid 1 \leqq i \leqq m + n\}$, for any pair of continuous distributions $F$ and $G$.

We give an elementary example, generalizing the one of Bernstein, of a set of pairwise independent random variables which are not jointly independent. It preserves, the "not enough emphasis" characteristic of Bernstein's example in which each random variable is a function of any two of them.

A new proof, based on the duality theorem of linear programming, is given of a theorem of V. Strassen, which states essentially that the minimum distance between random variables with given distributions equals the Prokhorov distance of their distributions.

Conditions are given which imply that the partial sums of a sequence of independent integer-valued random variables, suitably normalized, converge in distribution to a stable law of exponent $\alpha, 0 < \alpha < 2$, and imply as well that a strong version of the corresponding local limit theorem holds.

Let $F(x)$ be a probability distribution function and $\phi(t)$ its characteristic function. Define $\psi(t) = \Im \phi(t) = - \int^\infty_0 \sin tx dG(x)$, where $G(x) = 1 - F(x) - F(-x)$ is the tail difference of $F$. In general $G$ is known only to be of bounded variation on $\lbrack 0, \infty\rbrack$. The purpose of this paper is to relate differentiability of $\psi$ at 0 and Lipschitz behavior of $\psi$ to asymptotic behavior of $G$. Let $C_\alpha$ be the class of real valued functions $g$ such that $\int^\infty_x|dg(u)| = o(x^{-\alpha})(x\rightarrow + \infty)$, and let $\mu_\alpha^\ast$ be the symmetric moment of order $\alpha$. Then it is shown that $G \in C_1$ implies $\psi'(0) = \mu_1^\ast$ if either exists, while $G \in C_\alpha$ implies $\psi \in \operatorname{Lip} \alpha$ for $0 < \alpha < 1$. Extensions of these results to higher derivatives and values of $\alpha > 1$ are indicated. Finally it is shown that, for $1 < \alpha < 3, G(x) = o(x^{-\alpha}) (x \rightarrow + \infty)$ and $t^{-\alpha-1}(\psi(t) - t\mu_1^\ast) \in L(0,1)$ imply that $\mu_alpha^\ast$ exists.

It is well known that under certain conditions, binomial distributions converge to Poisson distributions. Simons and Johnson (1971) showed that the convergence is actually much stronger than in the usual sense. In this note the author shows that the result of Simons and Johnson is also true for Poisson binomial distributions which include binomial distributions as special cases.

This note complements a recent article by Thomas (1972). We prove a conjecture of Thomas which, in a way, makes his results complete. The proof uses the fact that for the sequences of random variables involved, the weak convergence property is conserved when taking certain transformations of the random variables.