In this paper we treat the problem of comparison of translation experiments. The "convolution divisibility" criterion for "being more informative" by Boll (1955) [2] is generalized to a "$\epsilon$-convolution divisibility" criterion for $\epsilon$-deficiency. We also generalize the "convolution divisibility" criterion of V. Strassen (1965) [13] to a criterion for "$\varepsilon$-convolution divisibility." It is shown, provided least favorable "$\varepsilon$-factors" can be found, how the deficiencies actually may be calculated. As an application we determine the increase of information--as measured by the deficiency--contained in an additional number of observations for a few experiments (rectangular, exponential, multivariate normal, one way layout). Finally we consider the problem of convergence for the pseudo distance introduced by LeCam (1964) [8]. It is shown that convergence for this distance is topologically equivalent to strong convergence of the individual probability measures up to a shift.

The quantile process was shown by Bickel to converge in the uniform metric on intervals $\lbrack a, b\rbrack$ with $0 < a < b < 1$. By introducing appropriate new supremum metrics, this result is extended to all of (0, 1). Hence a natural process of ordered spacings from the uniform distribution converges in certain supremum metrics. This is used to establish the limiting normality of a large family of statistics based on ordered spacings, which can be used in testing for exponentiality. The non-null case is also considered.

Sequential tests of separated hypotheses concerning the parameter $\theta$ of a Koopman-Darmois family are studied from the point of view of minimizing expected sample sizes pointwise in $\theta$ subject to error probability bounds. Sequential versions of the (generalized) likelihood ratio test are shown to exceed the minimum expected sample sizes by at most $M \log \log \underline{\alpha}^{-1}$ uniformly in $\theta$, where $\underline{\alpha}$ is the smallest error probability bound. The proof considers the likelihood ratio tests as ensembles of sequential probability ratio tests and compares them with alternative procedures by constructing alternative ensembles, applying a simple inequality of Wald and a new inequality of similar type. A heuristic approximation is given for the error probabilities of likelihood ratio tests, which provides an upper bound in the case of a normal mean.

Let $\Omega$ be a finite set of states and $\Xi$ the class of prior distribution on $\Omega$. A nonnegative, continuous, concave function on $\Xi$ is called an uncertainty function and if $X = (\mathscr{X}, \mathscr{A}; \mathbf{P}_\theta, \theta \in \Omega)$ and $Y = (\mathscr{y}, \mathscr{B}; \mathbf{Q}_\theta, \theta \in \Omega)$ are two experiments $X$ is called at least as informative as $Y$ with respect to $U$ if $U(\xi |X) \leqq U(\xi | Y)\quad \text{for all} \xi \in \Xi$ where $U(\xi \mid X)$ is the expected posterior uncertainty for an observation on $X$ when the prior is $\xi \in \Xi$. Any such $U$ induces a partial ordering on the class of all experiments. The paper characterizes (i) the class of functions $U$ which lead to a total ordering of the class of experiments and (ii) the class of transformations of a function $U$ which preserve its induced ordering.

In this paper, confidence bounds on the reliability of a serial system composed of exponential subsystems are considered. Both the classical and the Bayesian analyses are discussed. The main result is that for the case in which there are no previous data, then there are no prior distributions on the subsystem reliabilities that are independent of current data and that yield the uniformly most accurate unbiased confidence bounds available through classical techniques.

An appealing approach to the problem of estimating the regression coefficients in a linear model is to find those values of the coefficients which make the residuals as small as possible. We give some measures of the dispersion of a set of numbers, and define our estimates as those values of the parameters which minimize the dispersion of the residuals. We consider dispersion measures which are certain linear combinations of the ordered residuals. We show that the estimates derived from them are asymptotically equivalent to estimates recently proposed by Jureckova. In the case of a single parameter, we show that our estimate is a "weighted median" of the pairwise slopes $(Y_j - Y_i)/(c^j - c^i)$.

Suppose that for each real number $t$ in [0, 1] we have a distribution with distribution function $F_t(\bullet)$, mean $\mu(t)$ and median $m(t) (\mu(t)$ and $m(t)$ are referred to as regression functions). Consider the problems of estimating $\mu(\bullet)$ and $m(\bullet)$. In this paper we propose and discuss an estimator, $\hat{m}(\bullet)$, of $m(\bullet)$ which is monotone. This estimator is analogous to the estimator $\hat{\mu}(\bullet)$ of $\mu(\bullet)$ which was explored by Brunk (1970) (Estimation of isotonic regression in Nonparametric Techniques in Statistical Inference, Cambridge University Press, 177-195). Rates for the convergence of $\hat{m}(\bullet)$ to $m(\bullet)$ are given and a simulation study, where $\hat{m}(\bullet), \hat{\mu}(\bullet)$ and the least squares linear estimator are compared, is discussed.

Say that a probability distribution $\{p_i; i \in I\}$ over a finite set $I$ is in "product form" if (1) $p_i = \pi_i\mu \prod^d_{s=1} \mu_s^{b_si}$ where $\pi_i$ and $\{b_{si}\}$ are given constants and where $\mu$ and $\{\mu_s\}$ are determined from the equations (2) $\sum_{i \in I} b_{si} p_i = k_s, s = 1, 2, \cdots, d$; (3) $\sum_{i \in I} p_i = 1$. Probability distributions in product form arise from minimizing the discriminatory information $\sum_{i \in I} p_i \log p_i/\pi_i$ subject to (2) and (3) or from maximizing entropy or maximizing likelihood. The theory of the iterative scaling method of determining (1) subject to (2) and (3) has, until now, been limited to the case when $b_{si} = 0, 1$. In this paper the method is generalized to allow the $b_{si}$ to be any real numbers. This expands considerably the list of probability distributions in product form which it is possible to estimate by maximum likelihood.

Estimation-preceded-by-testing is studied in the context of estimating the mean vector of a multivariate normal distribution with quadratic loss. It is shown that although there are parameter values for which the risk of a preliminary-test estimator is less than that of the usual estimator, there are also values for which its risk exceeds that of the usual estimator, and that it is dominated by the positive-part version of the Stein-James estimator. The results apply to preliminary-test estimators corresponding to any linear hypothesis concerning the mean vector, e.g., an hypothesis in a regression model. The case in which the covariance matrix of the multi-normal distribution is known up to a multiplicative constant and the case in which it is completely unknown are treated.

In this paper an asymptotic expansion is derived for the power function of the likelihood ratio criterion for testing independence between two sets of variates for the case when the population canonical correlation coefficients are small. The method used can theoretically give the expansion up to any order of $N$ where $N$ is the sample size. Here the expansion is given up to $N^{-3}$ and is an extension of an expansion obtained independently by Sugiura [10] using a different method. The theorem in Section 3 summarizes the final result. In Section 4 the expansion is compared numerically with a different approximation obtained by Sugiura and Fujikoshi [11] and with exact results obtained by Pillai and Jayachandran [9].

A parametrization of the rotation group $O^+(p)$ of $p$ by $p$ orthogonal matrices with determinant $+1$ in terms of their skew symmetric parts is used to derive, for $p = 3$, an explicit expansion for $_0F_0^{(p)}(Z, \Omega)$, a hypergeometric function of two matrix arguments appearing in the distribution of the eigenvalues of a $p$ by $p$ Wishart matrix. On the basis of a numerically derived simplification of the low order terms of this series, an asymptotic expansion of $_0F_0^{(3)}$ in terms of products of ordinary confluent hypergeometric series is conjectured. Limited numerical exploration indicates the new series to be several orders of magnitude more accurate than the series from which it was derived.

We consider $(X_1, Y_1), (X_2, Y_2), \cdots, (X_n, Y_n)$, random sample from an absolutely continuous bivariate, population with symmetric density $f(x, y)$ and test $H_0: f(x, y)$ symmetric about (0,0) against $H_1:$ all possible location alternatives. Hotelling's $T^2$ statistic is often used for this test. We denote a form of this statistic by $T_n^{(2)}$ and make an exact Bahadur efficiency comparison of $T_n^{(2)}$ with respect to three of its competitors: a new bivariate Wilcoxon signed rank test $T_n^{(1)}$, Hodges' bivariate sign test $T_n^{(3)}$, and Blumen's bivariate sign test $T_n^{(4)}$. When a bivariate normal alternative with parameter $\Delta = \mu'\sum^{-1}\mu$ obtains, it is shown that the exact Bahadur slopes of $T_n^{(1)}, T_n^{(2)}$, and $T_n^{(3)}$ are identical to the exact slopes of their univariate analogues with a univariate normal alternative with parameter $\Delta = \mu/\sigma$ obtains. In this case, the exact Bahadur efficiency of $T_n^{(1)}$ is uniformly better than either the exact Bahadur efficiency of $T_n^{(3)}$ or $T_n^{(4)}$ with respect to $T_n^{(2)}$.

The concepts of balance for ordered and for unordered pairs of treatments are introduced. Methods for constructing multistage experimental designs which are Youden designs at each stage, are given. In the construction of these designs an attempt was made to accommodate as much orthogonality and balance (both in our sense and in the classical sense) as is possible. These constructions are presented in several theorems. In one theorem, we give a uniform method of converting $t$ mutually orthogonal Latin squares of order $n$ into a $t$-stage balanced $(n - 1) \times n$ Youden designs. Since the known methods of constructing orthogonal Latin squares of order $n = 4t + 2$, for $t > 1$, are not uniform, a uniform method for constructing two-stage $(n - 1) \times n$ Youden designs for all even $n$ has been developed. In another theorem, a method of constructing $(2\lambda + 1)$-stage balanced $(2\lambda + 1) \times (4\lambda + 3)$ Youden designs, for $4\lambda + 3$ a prime power, is presented. A method of constructing $(p^\alpha - 1)$-stage balanced $(\nu - 1)/2 \times \nu$ Youden designs is given in another theorem for the case when $v = 4\lambda + 3$ and is the product of twin primes, i.e., $\nu = p^\alpha q^\beta, q^\beta = p^\alpha + 2$. Difference sets based on the elements of Galois fields were utilized for these constructions. Other miscellaneous results are given.

Kronecker product and reduced designs were defined by M. N. Vartak (1955). The main purpose of this paper is to give necessary and sufficient conditions for partially balanced incomplete block (PBIB) designs with $l$ associate classes constructed by the Kronecker product of some balanced incomplete block (BIB) designs to be reducible to PBIB designs with $l_1$ associate classes for a positive integer $l_1$ which is less than $l$.

The paper develops one-sided analogs to Scheffe's two-sided confidence bounds for a function $f(\mathbf{x}), \mathbf{x} \in R^n$. If the domain $X\ast$ of $f$ is a subset of $R_+^n = \{\mathbf{x}: x_i \geqq 0, \forall i\}$, then the upper Scheffe bounds are conservative upper confidence bounds, which can be sharpened, often to great practical advantage. This sharpening, accomplished by a non-trivial extension of Scheffe's method, is developed by the geometry-probability argument of Section 2. Section 3 derives coverage probabilities for general 2- and 3-parameter functions and illustrates savings by the sharp bounds in two examples.

For any collection of exchangeable events $A_1, A_2, \cdots, A_k$ the Bonferroni inequalities are usually stated in the form $\max \{N_0, N_2, \cdots, N_{k_e}\} \leqq \mathbf{P}\{\mathbf{\cup}^k_{i=1} A_i\} \leqq \min \{N_1, N_3, \cdots, N_{k_0}\}$ where $N_0 = 0, k_e(k_0)$ is the largest even (odd) integer $\leqq k$, $N_\nu = \sum^\nu_{\alpha=1} (-1)^{\alpha-1}\binom{k}{\alpha}\mathbf{P}_\alpha \quad (\nu = 1, 2, \cdots, k)$ and $\mathbf{P}_\alpha = \mathbf{P}\{A_{i_1} A_{i_2} \cdots A_{i_\alpha}\}$ for any collection of $\alpha$ events. We may regard $N_\nu$ as being of the $\nu$th degree because it involves $\mathbf{P}_1, \mathbf{P}_2, \cdots, \mathbf{P}_\nu$; hence the lower and upper bounds above are never of the same degree. In this paper we develop improved lower and upper bounds of the same degree. For degree $\nu = 2, 3$, and 4 these results are given explicitly. A related problem is to get lower and upper bounds for the probability of the intersection of events, $\mathbf{P}_k$, for large $k$ in terms of $\mathbf{P}_1, \mathbf{P}_2, \cdots, \mathbf{P}_\nu$. These are also derived and given explicitly for $\nu = 2, 3,$ and 4. Applications of these inequalities to incomplete Dirichlet Type $I$-integrals and to equi-correlated multivariate normal distributions are indicated.

Let $S_n = f + X_1 + \cdots + X_n$ be an expectation-decreasing semimartingale with values in the unit interval, and let $V_n$ be the conditional variance of $X_n$ given the past. Then $E(\sum V_n)$ is less than $f(2 - f)$, and this bound is sharp. Sharper bounds are available if the process $S_0, S_1, \cdots$ satisfies suitable additional constraints.

Ranga Rao [10] developed a version of the Edgeworth asymptotic expansion for $\mathrm{Pr}(X_n \in B)$, where $X_n = n^{-\frac{1}{2}} \sum^n_{i=1} Z_i, \lbrack Z_n\rbrack$ is a sequence of independent random vectors in $R_k$ having a common lattice distribution with mean vector zero and nonsingular covariance matrix $\not\sum$, and $B$ is a Borel set. Use of this expansion is very difficult, except for the distribution function of $X_n$. In this paper, Ranga Rao's expansion is used to obtain a different expansion, when $B$ is convex. This new expansion is much simpler to evaluate. In the special case when $B = \lbrack x \mid x^T \not\sum^{-1} x < c\rbrack$, the new expansion assumes its simplest form. The first partial sum is the usual multivariate normal approximation, and Esseen ([6] pages 110-111) determined the order of magnitude of its error, i.e., $\mathrm{Pr}(X_n \in B) = K_k(c) + O(n^{-k/(k+1)})$ where $K_k(c)$ is the chi-square distribution function with $k$ degrees of freedom. Note that the order of magnitude of the error is $n^{-\frac{1}{2}}$ for $k = 1$ and approaches $n^{-1}$ as $k$ increases. The second partial sum is $\mathrm{Pr}(X_n \in B) = K_k(c) + (N(nc) - V(nc)) \frac{\exp(-c/2)}{(2\pi n)^{k/2}|\not\sum|^{\frac{1}{2}}} + O(n^{-1})$ where $N(nc)$ is the number of integer vectors $m$ in the ellipsoid $(m + na)^T \not\sum^{-1}(m + na) < nc$ having center at $-na$, and $V(nc)$ is the volume of this ellipsoid. This provides a new expansion for the distribution function of the quadratic form $X_n^T \not\sum^{-1}X_n$. When $Z_i$ has a multinomial distribution with parameters $N = 1, p_1, \cdots, p_m, \sum^m_{i=1} p_i = 1, X_n^T \not\sum^{-1} X_n$ is the chi-square goodness-of-fit statistic, and the new expansion (with $k = m - 1$) provides very accurate approximations for its distribution function. The accuracy of the first several partial sums, and of the Edgeworth approximation under the (inappropriate) assumption that $Z_i$ has a continuous distribution, is examined numerically for a number of multinomial distributions. It is concluded that the Edgeworth approximation assuming a continuous distribution should never be used when $Z_i$ has a lattice distribution, and that the second partial sum of the new expansion is much more accurate than the normal approximation for all multinomial distributions examined.

Savage (1954) has shown that fine and tight qualitative probabilities are realizable by finitely additive probability measures. His proof for this result is, however, in need of a correction. Fine qualitative probabilities are either atomless or equivalent to the union of $n$ equivalent atoms. Tight qualitative probabilities are always atomless. Qualitative probability structures, which are equivalent to the union of $n$ equivalent atoms, are realizable by a unique probability measure. Fine qualitative probabilities are almost realizable. With these results, the proof for Savage's theorem can be worked out and a theorem of Villegas (1964) can be strengthened.

Let $F_n$ denote the distribution function of the $n$th row sum of a triangular array of infinitesimal, rowwise independent random variables, and let $F^\ast$ denote the limiting infinitely divisible distribution function. Bounds are obtained for $\sup_{-\infty < x < \infty} |F_n(x) - F^\ast(x)|$ in the case that the means are finite and also for the attraction to a stable law with exponent $\alpha \leqq 1$. Conditions for convergence of these bounds are given.

We give a mathematical definition of stationary, asymptotically memoryless, continuous-time channels, and prove a coding theorem and its converse for such channels.

We consider the problem of maximizing the expectation of the discounted total reward in Markovian decision processes with arbitrary state space and compact action space varying with the state. We get the existence theorem for a $(p, \epsilon)$-optimal stationary policy, and the relation between the optimality of a policy and the optimality equation. Assuming the action space is a compact subset of $n$-dimensional Euclidean space, the existence of an optimal stationary policy is established, and an algorithm is obtained for finding the optimal policy. The last two facts are based on the Borel implicit function lemma given in this paper.

Consider two GI/G/k queueing systems in which the second system has stochastically larger interarrival times and/or stochastically shorter service times than the first system. In this paper we find sufficient conditions under which this implies stochastically shorter waiting time, virtual waiting time, queue size, and imbedded queue size, for the second system as compared to the first.

A necessary and sufficient condition is given for a family of multivariate normal distributions with zero mean vectors to admit a complete sufficient statistic.

An axiomatization for one-way expected utility, $P \succ Q \Rightarrow E(u, P) > E(u, Q)$, is given which places no restriction on the cardinality of the consequence set. In comparison with a previous axiomatization [Ann. Math. Statist. $\mathbf{42}$ (1971) 572-577] for finite consequence sets, the new axioms strengthen the Archimedean property and weaken the order-independence conditions. In particular, the new theory avoids the criticism of indifference due to threshold phenomena that affects the previous axiomatization.

In this paper we have established the maximum number of constraints in a two-level orthogonal array of 192 runs of resolution VI.

The marginal distributions of the latent roots of the multivariate beta matrix are shown to constitute a complete system of solutions of an ordinary differential equation (d.e.), which is related to the author's d.e.'s for Hotelling's generalized $T_0^2$ and Pillai's $V^{(m)}$ statistics. Results may be derived for the latent roots of the multivariate $F$ and Wishart matrices $(\Sigma = I)$. Pillai's approximations to the distributions of the largest and smallest roots are interpreted as exact solutions, the contributions of higher order solutions being neglected.

In this paper a general method is proposed for deriving the asymptotic noncentral distribution of Hotelling's generalized $T_0^2$ statistic for the case of large error degrees of freedom $n_2$. The approximation is given here to order $n_2^{-2}$ and further terms in the expansion can be readily obtained.

In the independent sampling model, Rao-Blackwell distribution function estimators $\tilde{F}_n(x)$ obtained by conditioning on sufficient statistics $T_n(X_1, \cdots, X_n)$ are considered. If for each $n \geqq 1, T_n$ is symmetric in $X_1,\cdots, X_n$ and $T_{n+1}$ is $\mathscr{B}(T_n, X_{n+1})$ measurable, it is shown that $\tilde{F}_n(x)$ converges strongly to the corresponding $F(x)$ and uniformly in $x$. This is a direct generalization of the Glivenko-Cantelli theorem.

Let $\{X_j, j \geqq 1\}$ be a linear process defined by the relation $X_j = \sum^\infty_{r=0} g_v Y_{j-v}$, where $\{Y_j; j = 0, \pm 1, \cdots\}$ is a sequence of i.i.d. random variables which possess a $\operatorname{mgf} M_Y(t)$ over an open interval $I = (-c,c) (c > 0)$. Let $a$ be a fixed positive constant and denote the $\operatorname{mgf}$ of $Z_1 = Y_1 - a$ by $M_z(t)$. Assume that $E\{Y_1\} = 0, |g_v| \leqq C\rho_0^v (v \geqq 0)$ for some finite positive constants $\rho_0 (< 1), C$ and $\sum^\infty_{v = 0} g_v \neq 0$ (we can take $\sum^\infty_{v = 0} g_v = 1$ without loss of generality). Further assume that there exists a constant $\tau \in I_0, I_0 = (-cA^{-1}, cA^{-1}), A = \sum^\infty_{v=0}|g_v|$, such that $\rho = M_z(\tau) = \inf_{t \in I} M_z(t) < 1$ and $M_z'(\tau) = 0$. Then it is proved that for each $u = 0,1,2, \cdots$ we can find a bounded sequence $\{\beta_{u,n}\}$ of constants such that for any integer $r \geqq 3 P(\sum^n_{j=1}X_j/n \geqq a) = (\rho^n \lambda_n/\tau\sigma_n(2_\pi)^{\frac{1}{2}})\lbrack\sum^{r-3}_{u=0} \beta_{u,n} \sigma_n^{-u} + O(\sigma_n^{-(r - 2)})\rbrack$ as $n \rightarrow \infty$, where $\{\lambda_n\}$ and $\{\sigma_n\}$ are sequences of positive constants, and, as $n \rightarrow \infty, \lambda_n$ is bounded away from 0 and $\infty$ and $n^{-1}\sigma_n^2$ approaches a finite positive constant.

For $N = 1,2,\cdots$ let $S_N = \sum_k a_{N,k}X_k$ where $a_{N,k}$ is a real number for $N,k = 1,2, \cdots$ and $\{Xk\}$ is a sequence of not necessarily independent random variables. For the case $0 < t < 1$, with assumptions closely related to $E|X_k|^t < \infty$ it is shown that the rate of convergence of $P(|S_N| > \varepsilon)$ to zero is related to $\sum_k |a_{N,k}|^t$. The theorems presented here extend some of the results in the literature to not necessarily independent sequences $\{X_k\}$.

Let $\{X_n, n \geqq 1\}$ be a sequence of independent identically distributed random variables taking values in the $d$-dimensional integer lattice $E_d$, and let $S_0 = 0, S_n = X_1 + \cdots + X_n$. The range of the random walk $\{S_n, n \geqq 0\}$ up to time $n$, denoted by $R_n$, is the number of distinct lattice points visited by the random walk up to time $n$. Let $p$ be the probability that the random walk never returns to the origin. It is known that $n^{-1}R_n \rightarrow p$ a.s. and that for $p < 1$ if the genuine dimension is $d \geqq 4$ or if the random walk is strongly transient then there is a positive constant $\sigma^2$ such that $\operatorname{Var} R_n \sim \sigma^2n$. In the present note we shall prove that in these two cases $\lim \sup_{n\rightarrow\infty}\frac{R_n - np}{(2\sigma^2n \log \log n)^{\frac{1}{2}}} = 1 \mathrm{a.s}.$ and the $\lim \inf$ of the same sequence is almost surely $-1$.

Assume that $n$ balls are randomly distributed into $N$ equiprobable cells. The ball is presumed to have probability $p, 0 < p < 1$ of staying in the cell and $(1 - p)$ of falling through. Let $S_0$ denote the number of empty cells. In this note we establish the asymptotic normality of $S_0$ as $n$ and $N$ tend to infinity so that $np/N \rightarrow c > 0, np/N^{\frac{5}{6}} \rightarrow \infty$ and $n/N \rightarrow 0$, or $3np/N - \log N \rightarrow - \infty$ and $n/N \rightarrow \infty$. We accomplish this by estimating the factorial cumulants of $S_0$.

Let $\{p_n\}_0^N$ be a discrete distribution on $0 \leqq n \leqq N$ and let $g(u) = \sum^\infty_0 p_n u^n$ be its $\operatorname{pgf}$. Then for $0 \leqq t < \infty g_t(u) = g(u + t)/g(1 + t) = \sum^N_0 p_n(t)u^n$ is a family of $\operatorname{pgf}$'s indexed by $t$. It is shown that there is a unique value $t^\ast$ such that $\{p_n(t)\}_0^N$ is $\log$-concave $(PF_2)$ for all $t \geqq t^\ast$ and is not $\log$-concave for $0 < t < t^\ast$. As a consequence one finds the infinite set of moment inequalities $\{\mu_{\lbrack r\rbrack}/\mathbf{r}!\}^{1/r} \geqq \{\mu_{\lbrack r+1\rbrack}/(r + 1)!\}^{1/r+1} \mathbf{r} = 1,2,3,\cdots$ etc. where $\mu_{\lbrack r\rbrack}$ is the $\mathbf{r}$th factorial moment of $\{p_n\}_0^N$ when the lattice distribution is $\log$-concave. The known set of inequalities for the continuous analogue is shown to follow from the discrete inequalities.

Let $N$ be the stopping time of a sequential probability ratio test of composite hypothesis, based on the i.i.d. sequence $Z_1, Z_2, \cdots$ with common distribution $P$. If for every choice of stopping bounds there exist constants $c > 0, 0 < \rho < 1$ such that $P\{N > n\} < c\rho^n n = 1, 2, \cdots$, we say that $N$ is exponentially bounded under $P$; otherwise $P$ is called obstructive. A theorem is proved giving sufficient conditions for $P$ to be obstructive. By virtue of this theorem it is possible to exhibit families of obstructive distributions in several examples, including the sequential $t$-test.

We show that under mild conditions the hitting time distributions of a general stochastic process are the unique solutions of an abstract Dirichlet problem.

Let $\{X_n, n \geqq 1\}$ be a sequence of random variables such that for suitably chosen constants $a_n > 0$ and $b_n, n \geqq 1, \{(X_n - b_n)/a_n\}$ converges in distribution to a nondegenerate random variable $X$. Let $\{N_m, m \geqq 1\}$ be a sequence of positive, integer-valued random variables distributed independently of the sequence $\{X_n\}$ and converging to infinity in probability as $m\rightarrow \infty$. If $\{a_n\}$ and $\{b_n\}$ are the normalizing constants computed from a $\operatorname{cdf} F$ which is in the domain of attraction of one of the extreme value distributions and if the $\operatorname{cdf}$ of $X$ satisfies a condition determined by the domain of attraction to which $F$ belongs, then conditions on the limiting distribution of $\{N_m/m\}$ are obtained which are necessary and sufficient for the convergence in distribution of the sequence $\{(X_{N_m} - b_m)/a_m\}$ to a nondegenerate random variable $Y$. The $\operatorname{cdf}$ of $Y$ is either a location or a scale mixture of the $\operatorname{cdf}$ of $X$; and the $\operatorname{cdf} F$ is often unrelated to the distribution of $\{X_n\}$. These results extend a theorem stated by Berman; however, the method of proof is conceptually simpler.

Let $(S, \mathscr{I})$ be a measurable space, $M$ the set of all finite measures on $\mathscr{I}, \mathscr{F}_M$ the $\sigma$-algebra generated by the family of all measurable cylindrical sets $\cap^k_{i=1} \{\mu \in M: \mu(A_i) \leqq a_i\}$. With each probability measure $P$ on $\mathscr{F}_M$ the family $\{P_{A_1}, \cdots, A_k\}$ of all finite-dimensional probability measures of the cylidrical sets is associated. The following problem is considered: Given a sequence $P^{(n)}$ of probability measures on $\mathscr{F}_M$ such that each sequence $P^{(n)}_{A_1}, \cdots, A_k$ converges weakly to a $k$-dimensional probability measure $P_{A_1}, \cdots, A_k$, does the family $\{P_{A_1}, \cdots, A_k\}$ generate a probability measure $P$ on $\mathscr{F}_M?$ It is proved that the answer is affirmative if $(S, \mathscr{I})$ is the Euclidean $n$-space with the $\sigma$-algebra of Borel sets.

In [7] ergodic properties of nonhomogeneous denumerable state space Markov chains were studied. It was noticed in [6] that the results obtained in [7] were easily extended to arbitrary state spaces. However, it was admitted in [6] that the transition mechanism of the chain was defined by means of transition density functions, thus restricting the generality of the approach and, moreover, introducing elements irrelevant to the problem. The aim of this note is to draw attention to the fact that ergodic properties of the most general nonhomogeneous Markov chains are easily obtained by using a theory developed by Dobrusin [2] in the middle fifties.

Grenander and Maruyama independently proved that a stationary Gaussian process on the line or the integers with continuous covariance function is ergodic if and only if its spectral measure has no atoms. In this paper this theorem is extended to processes parameterized by locally compact Abelian groups.