The random distribution function $F$ and its law is said to be neutral to the right if $F(t_1), \lbrack F(t_2) - F(t_1) \rbrack/\lbrack 1 - F(t_1)\rbrack, \cdots, \lbrack F(t_k) - F(t_{k-1}) \rbrack/\lbrack 1 - F(t_{k-1}) \rbrack$ are independent whenever $t_1 < \cdots < t_k$. The posterior distribution of a random distribution function neutral to the right is shown to be neutral to the right. Characterizations of these random distribution functions and connections between neutrality to the right and general concepts of neutrality and tailfreeness (tailfreedom) are given.

Let $k$ be a positive integer, let $X, X_1, X_2, \cdots$ be i.i.d. random variables, and let $m_n^{(k)}$ be the $k$th largest of $X_1, \cdots, X_n$. Let $(M_n^{(k)}(t))_{0 < t < \infty}$ be the random process defined by $M_n^{(k)} (t) = m^{(k)}_{\lbrack nt\rbrack}. M_n^{(k)}$ takes values in the space $D$ of non-decreasing right-continuous functions on $(0, \infty)$. Let $D$ be endowed with the usual topology of weak convergence. We show that if $X$ is uniformly distributed over [-1,0], then wp 1 the sequence $(M_n^{(k)}/(\log_2 n/n))_{n\geqq 3}$ is relatively compact in $D$ and its limit points coincide with $\{x\in D: x(t) \leqq 0$ for all $t$, and $\int x(t) dt \geqq -1\}$. Also, we show that if $X$ is exponential with mean 1, then wp 1 the sequence $((M_n^{(k)} - \log n)/\log_2n)_{n\geqq 3}$ is relatively compact in $D$ and its limit points coincide with $\{x\in D: x(t) \geqq 0$ for all $t$, and $\lambda_k(x) \leqq 1\}$; here $\lambda_k(x) = \sup (\sum_{p < q} x(t_p) + kx(t_q))$, with the supremum being taken over all finite systems of points $\{t_p\}_{p \leqq q}$ over which $x$ is strictly increasing. Extensions of and corollaries to these results are given.

Let $\{X_n, n \geqq 1\}$ be a real-valued, stationary Gaussian sequence with mean zero and variance one. Let $M_n = \max_{1\leqq i\leqq n} X_i, r_n = E(X_{n+1}X_1); c_n = (2 \ln n)^{\frac{1}{2}}$ and $b_n = c_n - \frac{1}{2}\lbrack\ln (4\pi \ln n)\rbrack/c_n$. Define $U_n = 2c_n(M_n - c_n)/\ln\ln n$ and $V_n = c_n(M_n - b_n)$. If $r_n = O(1/\ln n)$ as $n \rightarrow \infty$ then (i) $p(\lim \inf_{n\rightarrow\infty} U_n = -1) = p(\lim \sup_{n\rightarrow\infty} U_n = 1) = 1$, and (ii) $E\{\exp(tV_n)\} \rightarrow E\{\exp (tX)\}$ as $n \rightarrow \infty$ for all $t$ sufficiently small where $X$ is a random variable with distribution function $e^{-e^{-x}}; -\infty < x < \infty$.

Small perturbations of certain species of optimization problems,--in particular of so-called casino problems with a fixed goal--cause only small perturbations of the optimal strategies.

Consider a sequence of dependent events, where each has uniformly small conditional probability given the past, and the sum of the conditional probabilities is approximately constant at $a$. Then the number of events which occur is approximately Poisson with parameter $a$. An explicit bound is given on the variation distance.

Let $\{X_n\}$ be a sequence of independent random variables each with a finite expectation and a finite variance. Write $Z_n$ for the standardized sum of $X_1, X_2, \cdots, X_n$ and suppose that for all large $n, Z_n$ has a probability density function which we denote by $h_n(x)$. It is well known that the usual assumptions of the Central Limit Theorem do not necessarily imply the convergence of $h_n(x)$ to the standard normal density $\phi(x)$. In this study, we find a set of sufficient conditions under which the relation $$\lim_{n\rightarrow\infty} |x|^k|h_n(x) - \phi(x)| = 0$$ holds uniformly with respect to $x\in (-\infty, +\infty), k$ being an integer greater than or equal to 2.

Let $S_n, n = 1,2,3, \cdots$, denote the partial sums of i.i.d. random variables with positive, finite mean. The first passage times $\min \{n; S_n > c\}$ and $\min \{n; S_n > c \cdot a(n)\}$, where $c \geqq 0$ and $a(y)$ is a positive, continuous function on $\lbrack 0, \infty)$, such that $a(y) = o(y)$ as $y \uparrow \infty$, are investigated. Necessary and sufficient conditions for finiteness of their moments and moment generating functions are given. Under some further assumptions on $a(y)$, asymptotic expressions for the moments and the excess over the boundary are obtained when $c \rightarrow \infty$. Convergence to the normal and stable distributions is established when $c \rightarrow \infty$. Finally, some of the results are generalized to a class of random processes.

Let $\{W(t): t$ a nonnegative real number$\}$ denote a stochastic process with right-continuous sample paths with probability one, independent increments which are statistically homogeneous, $E\{W(t)\} = 0$, and $E\{(W(t) - W(s))^2\} = \sigma|t - s|$ for some constant $\sigma$; let $T$ denote the set of stopping variables with respect to $W$; and let $c$ denote a non-increasing, right-continuous, square-integrable function on the nonnegative real line. Then $E\{\sup_{t\geqq 0} c(t)|W(t)|\}$ is shown to be finite which insures that $\sup_{\tau\in T} E\{c(\tau)W(\tau)\}$ is finite. Also, $\varepsilon$-optimal stopping variables are shown to exist with stopping points occurring only in discrete subsets of the nonnegative real line. These optimal stopping variables require observation of the process $W$ only at the possible stopping points.

For a constant $\beta > \frac{1}{2}$ and $W$ normalized Brownian motion with parameter space the nonnegative real line, the stopping variable $\lambda$ defined by $$\lambda = \sup \{t: W(s) < y_0(1 + s)^{\frac{1}{2}}, 0 \leqq s < t\}$$ where $y_0$ is the unique positive root of $$\int^\infty_0 x^{2(\beta-1)}e^{(yx-x^2/2)} dx = y \int^\infty_0 x^{(2\beta-1)}e^{(yx-x^2/2)} dx$$ is shown to be optimal in the sense that $E\{(1 + \lambda)^{-\beta}W(\lambda)\}$ is equal to the supremum of $E\{(1 + \tau)^{-\beta}W(\tau)\}$ over all stopping variables $\tau$ with respect to $W$. The values of $y_0$ for $\beta =$ 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 are given.

Let $S_n, n = 1,2, \cdots$ denote the partial sums of i.i.d. random variables with positive, finite mean and with a finite moment of order $r, 1 \leqq r < 2$. Let $Z_n, n = 1,2, \cdots$ denote the partial sums of i.i.d. random variables with a finite moment of order $r, 0 < r < 2$, and with mean 0 if $1 \leqq r < 2$. Let $N(c) = \min \{n; S_n > c\}, c \geqq 0$. Theorem 1 states that $N(c)$, (suitably normalized), tends to 0 in $r$-mean as $c \rightarrow \infty$. The first part of that proof follows by applying Theorem 2, which generalizes the known result $E|Z_n|^r = o(n)$, as $n\rightarrow \infty$ to randomly indexed partial sums.

Let $X_1, X_2, \cdots$ be a sequence of random variables, each having conditional mean zero given the past. Let $V_n$ be the conditional variance of $X_n$ given the past. Les $S_n = X_1 + \cdots + X_n$ and $T_n = V_1 + \cdots + V_n$. For simplicity, suppose $\sum V_i = \infty$ a.e. For which non-decreasing function $\phi$ does $S_n/\phi(T_n)$ necessarily lead to 0 a.e. as $n$ increases? It is necessary and sufficient that $\int^\infty 1/\phi(t)^2 dt < \infty$. That is, if the integral is finite, convergence to zero a.e. is guaranteed for all $X_i$ and $V_i$ satisfying the stated condition. If the integral is infinite, there is a sequence of independent, symmetric random variables $X_i$, each having variance 1, such that $S_n/\phi(n)$ oscillates between $\pm\infty$. The sufficiency is known, but a new proof is given.

Let $r(j)$ denote the $j$th autocorrelation based on a sample of $N$ consecutive observations on a stationary linear stochastic process. Under mild regularity conditions on the process, an iterated logarithm result is given for the convergence of $r(j)$ as $N \rightarrow \infty$ to the corresponding process autocorrelation $\rho (j)$.

If $T$ is an automorphism on a Lebesgue space and $P$ a finite generator for $T$, then $T$ is a Bernoulli shift if $$\sup \{|\mu(A \cap B) - \mu(A)\mu(B)|: A \in \vee^{-1}_{-\infty} T^j P, B \in \vee^\infty_k T^j P\}$$ is $o(|P|^{-a}k)$ where $a_k/k \rightarrow \infty$ as $k \rightarrow \infty$.

Let $\mathbf{Z}(t) = (\mathbf{Z}_1(t), \cdots, \mathbf{Z}_k(t)), t \geqq 0$, be a critical $k$-type, continuous time, Markov branching process. It is known that $\mathbf{Z}(t)/t$, conditioned on $\mathbf{Z}(t) \neq 0$, converges in distribution to $\mathbf{v}W$, where $\mathbf{v}$ is a vector determined by the mean matrix of the process, and $W$ is an exponentially distributed random variable. Thus if $\mathbf{\xi}$ is any fixed vector, then $(\xi \cdot \mathbf{Z}(t))/t$, conditioned on nonextinction, converges to $(\xi \cdot \mathbf{v})W$. If $\mathbf{\xi}$ is orthogonal to $\mathbf{v}$ then $t$ is not the right normalizing factor. We prove that in this case: (a) $\{(\mathbf{\xi} \cdot \mathbf{Z}(t))/(\mathbf{u} \cdot \mathbf{Z}(t))^{\frac{1}{2}} \mid \mathbf{Z}(t) \neq 0\}$ converges in distribution to a normal random variable, and (b) $\{(\mathbf{\xi} \cdot \mathbf{Z}(t))/t^{\frac{1}{2}}\mid\mathbf{Z}(t) \neq 0\}$ converges in distribution to a Laplacian random variable.

Using martingale theory, this note proves that Galton-Watson branching processes are a.s. convergent even when the assumption of homogeneity of reproduction is dropped.

Let $\{X_n, n \geqq 1\}$ be a stationary independent sequence of real random variables, $S_n = X_1 + \cdots + X_n$, and $\alpha_A$ the hitting time of the set $A$ by the process $\{S_n, n \geqq 1\}$, where $A$ is one of the half-lines $(0, \infty), \lbrack 0, \infty), (-\infty, 0 \rbrack$ or $(-\infty, 0)$. This note provides a simple proof of a known result in random walk theory on necessary and sufficient conditions for $E\{\alpha_A\}$ to be finite. The method requires neither generating functions nor moment conditions on $X_1$.

The convergence rate problem for probabilities of moderate deviations is completely solved by giving a necessary and sufficient condition for the existence of the absolute moment of order $c^2 + 2, c > 0$, in terms of probabilities of moderate deviations. Furthermore, a result on the rate of convergence of probabilities of moderate deviations is given under a weaker moment condition than in Rubin and Sethuraman (1965).