The new results are (C), (D) and (F) below. Let $X_0, X_1, \cdots$ be a Markov chain with countable state space $I$ and stationary transitions. Suppose $I$ is a positive recurrent class, with stationary probability vector $p$. Let $f$ be a real-valued function on $I$. Fix a reference state $s \varepsilon I$, and let $0 \leqq t_1 < t_2 < \cdots$ be the times $n$ at which $X_n = s$. Let $Y_j = \sum \{f(X_n):t_j \leqq n < t_{j + 1}\}$ and $U_j = \sum \{|f(X_n)|:t_j \leqq n < t_{j + 1}\}.$ Let $V_m = \sum^m_{j = 1} Y_j$ and $S_n = \sum^n_{j = 0}f(X_j)$. For (C) and (D) below, assume (A) $\sum_{i \varepsilon I} p_if(i) = 0$; and (B) $U^2_j$ has finite expectation. Then: (C) Theorem. $n^{-\frac{1}{2}} \max \{|S_j - V_{jp_s}|: 1 \leqq j \leqq n\} \rightarrow 0$ in probability; and (D) Theorem. $(n \log \log n)^{-\frac{1}{2}} \max \{|S_j - V_{jp_s}|: 1 \leqq j \leqq n\} \rightarrow 0$ almost everywhere. For (F), do not assume (A) and (B), but assume (E) $Y_j$ differs from 0 with positive probability. Let $v_m$ (respectively, $s_n$) be 1 or 0 according as $V_m$ (respectively, $S_n$) is positive or non-positive. Then (F) Theorem. $n^{-1} \sum \{s_j : 1 \leqq j \leqq n\} - p^{-1}_sn^{-1} \sum \{v_j : 1 \leqq j \leqq np_s\} \rightarrow 0$ almost everywhere. I do not believe the convergence in (C) is a.e., but have no counter-example.

$Z_1, Z_2, \cdots$ is a sequence of iid $k$-vectors with common distribution $P$. $G^\ast$ is a group of transformations $Z_n \rightarrow CZ_n + b, C \varepsilon G$, where $G$ is a Lie group of $k^2$ matrices, $\dim G \geqq 1, G$ closed in the group of all nonsingular $k^2$ matrices, and the totality of translation vectors $b$ is a subspace of $k$-space invariant under $G$. Let $\mathscr{N}$ be all $N(\mu, \Sigma)$ distributions, with $\Sigma k^2$ nonsingular. Let $U = (U_1, U_2, \cdots)$ be a maximal invariant under $G^\ast$ in the sample space, $\gamma = \gamma(\theta)$ a maximal invariant in $\mathscr{N}$, where $\theta = (\mu, \Sigma)$. For given $\theta_1, \theta_2 \varepsilon \mathscr{N}$ such that $\gamma(\theta_1) \neq \gamma(\theta_2)$ let $R_n$ be the probability ratio of $(U_1, \cdots, U_n)$. The limiting behavior of $R_n$ is studied under the assumption that the actual distribution $P$ belongs to a family $\mathscr{F} \supset \mathscr{N}$, defined as follows: the components of $Z_1$ have finite 4th moments, and there is no relation $Z'_1 AZ_1 + b'Z_1 =$ constant a.e. $P$, with $A$ symmetric, unless $A = 0, b = 0$. It is proved that $\mathscr{F}$ can be partitioned into 3 subfamilies, and for every $P$ in the first subfamily $\lim R_n = \infty$ a.e. $P$, in the second $\lim R_n = 0$ a.e. $P$, and in the third $\lim \sup R_n = \infty$ a.e. $P$ or $\lim \inf R_n = 0$ a.e. $P$. This implies that any SPRT based on $\{R_n\}$ terminates with probability one for every $P \varepsilon \mathscr{F}$.

Let $Z(t) = (X(t), Y(t)), t \geqq 0$, be the standard plane Brownian motion process. Let $(R(t), \theta_1(t))$ be the polar coordinates of $Z(t)$. Suppose that $R(0) = r > 0$ with probability 1, that is $Z(\cdot)$ starts away from the origin. We shall define the process $\theta(t), t \geqq 0$, as the total algebraic angle traveled on the continuous path $Z(s), 0 \leqq s \leqq t$; we take $\theta(0) = \theta_1(0)$; we have $\theta(\cdot) \equiv \theta_1(\cdot) \mod 2\pi$. Let $f = f(\theta), 0 \leqq \theta \leqq 2\pi$, be a bounded measurable function such that $f(0) = f(2\pi)$. For $r_1 > r$ let $\tau$ be the first passage time of $R(\cdot)$ to the point $r_1$. This note is devoted to the computation of the functional \begin{equation*}\tag{1.1}L(f) = E_{r,\theta}\{ \int^\tau_0 f(\theta_1(t)) dt\},\end{equation*} where $E_{r,\theta}\{\cdot \cdot\}$ is the expectation operator under the condition $R(0) = r, \theta_1(0) = \theta$. This is interpreted as the expected occupation time of a measurable subset of $\lbrack 0, 2\pi\rbrack$ if $f$ is the indicator function of the subset. We find an explicit formula for $L(f)$ as a linear functional on the Hilbert space $L_2\lbrack 0, 2\pi\rbrack$. A preliminary result of interest is presented in Section 2: the random variable $\lbrack \theta(\tau) - \theta(0)\rbrack/|\log (r_1/r)|$ has a Cauchy distribution for any positive numbers $r, r_1$ with $r \neq r_1$. This recalls the independent result of Spitzer that $\lbrack \theta (t) - \theta(0)\rbrack/\frac{1}{2} \log t$ has a limiting Cauchy distribution for $t \rightarrow \infty$ [4]. I thank the referee for his constructive remarks and for the alternate proof of Theorem 2.1 given in Section 6.

The force exerted on an object immersed in a flowing turbulent fluid is considered as a zero-memory, nonlinear transformation of the bivariate Gaussian process whose components are the fluid particle velocities and local accelerations that would be present at the location of the object if the object were not disturbing the fluid. A model often used in applications is assumed and the probability density and the moment generating function are derived and investigated. The covariance between the forces at two space-time points in the presence of a space-varying mean flow is developed and a series approximation outlined. Under suitable restrictions the partial sums of the series are used to obtain an easily computed approximation to the spectral density of the force at a fixed space location.

In [8] Hodges and Lehmann showed that robust estimates of location in the one and two sample problems could be obtained by inverting the known robust nonparametric tests and using as estimates the average of the resulting upper and lower 50 per cent confidence bounds. In particular, they proposed the use of the estimate derived from the Wilcoxon test, the median of averages of pairs of observations. Unfortunately, despite some short cuts viz. [8] and [9] computation of this estimate seems to require on the order of $n^2 \log_2 n$ steps, a prohibitive number. In [7] Hodges proposed a simple alternative estimate $D_n$ given by $D_n = \text{median}_i \frac{1}{2}(Z_{in} + Z_{i(n - i + 1)})$ where $Z_{1n} < \cdots < Z_{nn}$ are the order statistics of the sample under consideration. This procedure as was noted in [7] is related in the sense of Hodges and Lehmann to the one sample analogue of one of the oldest of non parametric tests, the Galton rank order test, viz. [6], [4a]. In this paper we derive the asymptotic theory of $D_n$ by employing an invariance principle due to Bickel [2] and thus relating the limiting distribution to that of certain functionals of Brownian motion. The necessary refinements of the stochastic process convergence results of [2], which may be of use in related problems of asymptotic theory, are gathered in Section 7. Unfortunately, we can only give explicit form to the limiting distribution of $D_n$ in the two cases of rectangular and Laplace parents. Although this limit is not normal we conclude that $D_n$'s scatter is quite close to that of the estimate proposed by Hodges and Lehmann. Using the same techniques we prove in Section 6 the consistency of the Galton test and characterize its power behaviour for alternatives close to the null hypothesis. Finally, Section 4 gives the small sample distribution of the Galton test statistic and of $D_n$ for a rectangular parent. The techniques of this paper carry over quite easily to the two sample situations. Although our evidence is incomplete it would seem that both $D_n$ and the Galton test are robust as well as easily computable non parametric procedures.

For comparative experiments with two or more treatments, rank methods posses, in their insensitivity to gross errors and extreme observations, a distinct advantage over the classical normal theory procedures (see [10]) (beside providing exact significance level when the form of the underlying distribution is unknown). First of the rank tests developed was the Wilcoxon-two-sample test, subsequently generalized to the $K$-sample problem by Kruskal and Wallis (see [12]). Both these tests have been shown to possess asymptotic (Pitman) efficiency equal to $3/\pi$ (against normal shift) relative to the classical $t$- and $\mathscr{F}$-tests respectively (see Andrews [1]). However, in many comparative experiments, it is desirable in the interest of increased precision to stratify the population or divide the experimental subjects into homogeneous (randomized) blocks. For such experimental designs, the first attempt at providing a rank test was made as far back as 1937 by Friedman [5] (for the one observation per cell case), who proposed a test based on independent rankings of observations within each block. This procedure, which we shall refer to in the sequel as the separate-ranking procedure, was extended subsequently to more general designs by Durbin [4] and Benard and van Elteren [2]. Van Elteren and Noether [22] computed the asymptotic efficiency of the separate-ranking procedure and showed (for the one observation per cell case) that relative to the normal theory $\mathscr{F}$-statistic its efficiency (against normal shift) is $3K/\pi(K + 1)$ (which takes the value $2/\pi$ for $K = 2$ and increases to $3/\pi$ as $K \rightarrow \infty)$. In 1962, however, Hodges and Lehmann [9] pointed out that the rather low efficiency of the separate-ranking procedure was due, presumably, to the absence of interblock comparisons and proposed a conditional test based on a combined ranking of all the observations after "alignment" (defined below) within each block (see also Mehra [17]). Subsequently, Lehmann in a series of papers [13], [14], [15] laid the foundations of an entirely new and remarkable approach to nonparametric inference parallel to the classical normal theory (parametric) analysis of variance. However the question of asymptotic efficiency of the test proposed in [9] was left essentially unanswered. It is the purpose of the present paper to study the asymptotic efficiency of the conditional test proposed in [9]. In Section 2, the asymptotic version of this test is discussed. In Section 3, limit distributions under translation alternatives are obtained. Section 4 contains a discussion of the asymptotic efficiency and Section 5 consists of certain concluding remarks.

A problem that seems to be of some practical importance is how to select the best one of $k$ experimental categories or populations when there is a fixed probability for each population that any measurement will be classified as a `success' and the best population is defined as the one with the greatest probability of a success. For example, we might be interested in determining which of $k$ new drugs offers the greatest probability of survival against a specified disease, or which of $k$ new production techniques has the greatest probability of producing a `good' item. A treatment of this problem using a fixed sample size approach was given by Sobel and Huyett [6]. To describe their formulation of the problem, denote the populations by $\prod_1, \prod_2, \cdots, \prod_k$, the corresponding probabilities by $p_1, p_2, \cdots, p_k$, and the ordered probabilities by $p_{\lbrack 1\rbrack} \geqq p_{\lbrack 2\rbrack} \geqq \cdots \geqq p_{\lbrack k\rbrack}$, and let $\prod_{\lbrack j\rbrack}$ be the population associated with $p_{\lbrack j\rbrack}$. Then [6] described a statistical procedure and gave tables for determining the common sample size required with each population so that population $\prod_{\lbrack 1\rbrack}$ will be selected with probability $\geqq P^\ast$ whenever $p_{\lbrack 1\rbrack} \geqq p_{\lbrack 2\rbrack} + d$, where $d$ and $P^\ast$ are constants selected in advance of the experiment. This formulation of the problem, which we will call the main formulation, seems satisfactory when nothing is known about the magnitude of $(p_1, p_2, \cdots, p_k)$ or if there is some a priori information available which indicates that $p_{\lbrack 1\rbrack}$ and $p_{\lbrack 2\rbrack}$ do not differ too much from .5, say .25 $\leqq p_{\lbrack 2\rbrack} \leqq p_{\lbrack 1\rbrack} \leqq .75$. An alternative formulation of the problem when the a priori information indicates that $p_{\lbrack 2\rbrack}$ and $p_{\lbrack 1\rbrack}$ differ substantially from .5 was given in [6] as follows: the sample size is determined so that population $\prod_{\lbrack 1\rbrack}$ is selected with probability $\geqq P^\ast$ whenever $p_{\lbrack 2\rbrack} \leqq p^\ast_{\lbrack 2\rbrack}$ and $p_{\lbrack 1\rbrack} \geqq p^\ast_{\lbrack 2\rbrack} + d$, where $p^\ast_{\lbrack 2\rbrack}$ is an additional constant determined in advance of the experiment on the basis of the a priori information about the probable value of $p_{\lbrack 2\rbrack}$. The present paper is based on a somewhat novel use of the Poisson distribution to obtain a random number of measurements from each population at every stage of experiment combined with the application of one-sided sequential confidence limits developed in [4]. Using these techniques we derive sequential procedures for selecting the best population both for the main formulation and for a generalization of the alternative formulation of the problem. Some Monte Carlo calculations which are summarized in Section 5 indicate that a substantial saving is possible with the sequential procedures.

In this note we consider the joint distribution of Student variates $(t_1, t_2)$, where $t_1$ corresponds to the $x$-observations and $t_2$ to $y$-observations from a bivariate normal distribution. No applications are suggested as Hotelling's $T^2$ is more appropriate whenever estimation of covariance matrix is necessary. Possibly on occasions, when the correlation coefficient, $\rho$, between $x$ and $y$ may be assumed known, for example from past records, the bivariate $(t_1, t_2)$ may be useful. The main interest in this distribution is theoretical. First, because this type of bivariate $(t_1, t_2)$ has never been worked out before while the joint distribution of $(\bar{x}, \bar{y}, s_1, s_2, r)$ is commonly known. Second, for degrees of freedom $n = 1$ (sample size $N = 2$) the bivariate $t$ distribution is an example of a bivariate Cauchy distribution. Lastly, the asymptotic approximation obtained in Section 3 is an application of the method of steepest descent, which has some methodological interest and can be used in other situations. There is no loss of generality, as far as the distribution of $(t_1, t_2, r)$ is concerned, in assuming the means of $x$ and $y$ to be zero and variances to be unity. The only parameter which enters into the joint distribution of $(t_1, t_2, r)$ or into that of $(t_1, t_2)$ is $\rho$. Because of the simplicity of the limiting distribution and the asymptotic approximation we will present them first, while the exact distribution are evaluated only for $n = 1$, and 3 $(N = 2$ and 4). The exact distribution for arbitrary $n$ can be worked out, in double or triple sums, following the method given for $n = 3$

A refinement of the Robbins-Monro procedure for estimating the root of a regression equation is given. The essential feature of the procedure is that it estimates the slope of the regression function at the root and employs this information to improve on the rate of convergence and the asymptotic variance of the Robbins-Monro procedure.

It is shown that the Keifer-Wolfowitz procedure--for functions $f$ sufficiently smooth at $\theta$, the point of minimum--can be modified in such a way as to be almost as speedy as the Robins-Monro method. The modification consists in making more observations at every step and in utilizing these so as to eliminate the effect of all derivatives $\partial^if/\lbrack\partial x^{(i)}\rbrack^j, j = 3, 5 \cdots, s - 1$. Let $\delta_n$ be the distance from the approximating value to the approximated $\theta$ after $n$ observations have been made. Under similar conditions on $f$ as those used by Dupac (1957), the results is $E\delta_n^2 = O(n^{-s/(s+1)})$. Under weaker conditions it is proved that $\delta_n^2n^{s/(s+1)-\epsilon} \rightarrow 0$ with probability one for every $\epsilon > 0$. Both results are given for the multidimensional case in Theorems 5.1 and 5.3. The modified choice of $Y_n$ in the scheme $X_{n+1} = X_n - a_nY_n$ is described in Lemma 3.1. The proofs are similar to those used by Dupac (1957) and are based on Chung's (1954) lemmas and, in Theorem 5.3, on a modification of one of these lemmas. The result of Theorem 5.3 is new also for the usual Kiefer-Wolfowitz procedure. The main and very simple idea, however, is in Lemma 3.1; it will suggest, to a reader acquainted with Dupac's Theorem 3 and its proof, the consequences elaborated in Theorem 5.1.

If $\{f(X_n): n = 1, 2, \cdots \}$ is a finite-state function of a finite-state Markov chain $\{X_n\}$, it is known that the distribution of $\{f(X_n)\}$ is determined by the distribution of $(f(X_1), f(X_2), \cdots, f(X_K))$ for suitable $K$, and a finite construction utilizing only the latter distribution exists in special cases yielding the probability structure of a chain $\{X'_n\}$ and a function $f'$ such that $\{f(X_n\}$ and $\{f'(X'_n)\}$ have the same distribution [7]. We obtain a finite construction when $(\{X_n\}, f)$ is such that if $i$ is a state of $\{X_n\}$ and $j$ is a state of $\{f(X_n)\}$, then there is at most one transition from $i$ to the set $f^{-1}(j)$ and the distribution of $X_1$ assigns positive probability to at most one state in each set $f^{-1}(j)$. Such "state-calculability" is a rather severe, but natural, structural restriction subsuming certain cases not previously treated. The corresponding finite construction is very simple and directly related to the representation of any finite-state process $\{Y_n\}$ by a function of a (possibly countable-state) Markov chain.

This paper is concerned with the properties of convex cones and their dual cones generated by points randomly distributed on the surface of a $d$-sphere. For radially symmetric distributions on the points, the expected number of $k$-faces and natural measure of the set of $k$-faces will be found. The expected number of vertices, or extreme points, of convex hulls of random points in $E^2$ and $E^3$ has been investigated by Renyi and Sulanke [4] and Efron [2]. In general these results depend critically on the distribution of the points. However, for points on a sphere, the situation is much simpler. Except for a requirement of radial symmetry of the distribution on the points, the properties developed in this paper will be distribution-free. (This lack of dependence on the underlying distribution suggests certain simple nonparametric tests for radial symmetry--we shall not pursue this matter here, however.) Our approach is combinatorial and geometric, involving the systematic description of the partitioning of $E^d$ by $N$ hyperplanes through the origin. After a series of theorems counting the number of faces of cones and their duals, we are led to Theorem 5 and its probabilistic counterpart Theorem 2', the primary result of this paper, in which the expected solid angle is found of the convex cone spanned by $N$ random vectors in $E^d$.

This paper is concerned with the distribution of the value of a game with random payoffs. Two types of games are considered: matrix games with iid matrix elements, and games of perfect information with iid terminal payoffs. Let $\| x_{ij}\|, i:1, 2, \cdots, m; j:1, 2, \cdots, n$, be the matrix of player I's payoffs in a zero-sum two-person game, and let $v(\|x_{ij}\|)$ be its (possibly mixed) value. Consider the random value $V_{m,n}(f) \equiv v(\| X_{ij}\|)$, where the $X_{ij}$ are $mn$ iid random variables, each distributed according to the density $f$. It is pointed out in Section 2 that the conditional distribution of $V_{m,n}$, given that it is pure, is that of the $n$th largest of $m + n - 1$ iid random variables, each distributed according to $f$. For $f$ uniform on (0, 1) (i.e., $f = u$), a method is given for determining the conditional distribution of $V_{2,n}(u)$, given that it is mixed. This leads to an elementary expression for the distribution of $V_{2,2}(u)$ and the asymptotic distribution of $V_{2,n}(u)$. Consider as well two players alternately choosing one of two alternative moves, with $n$ choices to be made in all by each. Corresponding to each of the $4^n$ possible sequences of moves, there are $4^n$ payoffs $x(i_1, i_2, \cdots, i_{2n})$ for player I, $i_k = 1$ or 2, where the odd and even locations indicate, respectively, the successive alternatives chosen by players I and II. The (pure) value $v(\{x(i_1, \cdots, i_{2n})\})$ of such a game is $\max_{i_1} \min_{i_2} \max_{i_3} \min_{i_4} \cdots \max_{i_{2n - 1}} \min_{i_{2n}} x(i_1, \cdots, i_{2n}).$ Now replace the $4^n$ numbers $x(i_1, \cdots, i_n)$ by independent uniformly distributed random variables $X(i_1, \cdots, i_{2n})$. The asymptotic behavior of the random value $V_n \equiv v(\{X(i_1, \cdots, i_{2n})\})$ is investigated in Section 3; it is shown that the asymptotic distribution $L$ of $V_n$ is everywhere continuous and monotone-increasing, and satisfies a certain functional equation; it is also shown that the moments of the normed $V_n$ converge to those of $L$. It is planned, in a subsequent paper, to explore games of perfect information in greater depth. After this paper was submitted, Thomas M. Cover drew our attention to [3] and [9]. The derivation in [9] of the expected value of a $2 \times n$ game, conditionally on there being a $2 \times 2$ kernel, is based on essentially the geometric considerations leading to our distribution (5); however, since the argument in [9] is not aimed at obtaining distributions, and is thus rather different in detail, a sketch of our derivation of (5) has not been deleted. In [3], the probability is computed, in the case of payoff distributions symmetric about zero, that an $m \times n$ game has positive value. Also, the work of Efron [4] and that of Sobel [8] pertain to Section 2, and that of Buehler [1] to Section 3. Finally, closely related to this paper, and indeed the source of our original interest in this area, is the work of Chernoff and Teicher [2].

An earlier paper [4] has been concerned with the distribution of the value of perfect information games with random payoffs of a certain very special type: two alternatives were assumed available to each player at every move, and the terminal payoffs were assumed to be iid and uniform. This paper considers a more general class of games, with $p$ and $q$ alternatives available, respectively, for players I and II at every move, and with the terminal payoffs arbitrarily distributed, though still iid. Specifically, consider a two-person zero-sum perfect information game, with player I and player II alternately choosing one of several alternative moves, with $n$ choices to be made in all by each. It is assumed that there are always $p$ and $q$ alternatives available respectively to players I and II. Corresponding to each of the $(pq)^n$ possible sequences of moves, there are $(pq)^n$ payoffs (to player $I$) $x(i_1, i_2, \cdots, i_{2n})$, where the indices $i_1, i_3, \cdots, i_{2n - 1}$, each with range $(1, 2, \cdots, p)$ indicate the successive alternatives chosen by player I, and the indices $(i_2, i_4, \cdots, i_{2n})$, each with range $(1, 2, \cdots, q)$, indicate the successive alternatives chosen by player II. The value $v(\{x(i_1, \cdots, i_{2n})\})$ of such a game is $\max_{i_1} \min_{i_2} \max_{i_3} \min_{i_4} \cdots \max_{i_{2n}} x(i_1, \cdots, i_{2n}).$ Now replace the $(pq)^n$ numbers $x(i_1, \cdots, i_{2n})$ by independent random variables $X(i_1, \cdots, i_{2n})$, each with cdf. $F$. This paper is concerned with the limiting behavior of the random values $V_n(F) \equiv v(\{X(i_1, \cdots, i_{2n})\})$. The limiting behavior of $V_n(F)$ is investigated in Section 2 for uniform $F (F = U)$. Analogous to the results for $p = q = 2$ obtained in [4], the limiting distribution for the sequence $\{V_n(U)\}$ is everywhere continuous and monotone increasing, and satisfies a certain functional equation. Limiting distributions arising from arbitrary $F$ are considered in Section 3. Section 4 is devoted to some results concerning norming sequences and domains of attraction. The final corollary of Section 4 establishes that all of the common cdf's lead to the same limiting distribution. This study bears a strong resemblance to Gnedenko's [1] study of extremes. Since, in Gnedenko's case, the limiting distributions for $Z_{(n)} = \max (Z_1, \cdots, Z_n)$, where $Z_n$ are independent identically distributed random variables, must in effect be limiting distributions for $Z_{(k^n)}$ for every positive integer $k$, Gnedenko's argument involves an infinite sequence of functional equations [1], p. 431, namely, one functional equation for every $k$. In the present treatment the limiting distributions for $V_n(F)$ must satisfy only the single functional equation (6). It may thus be worth noting that even a stronger resemblance would exist between this paper and a study of the limiting distributions for $Z_{(k^n)}, k$ fixed. I would like to acknowledge Professor H. T. David for his many helpful discussions concerning this research.

In a multinomial situation with observed proportions $Q_i (i = 1, \cdots, r + 1)$ and corresponding expected proportions $p_i$, it has been shown by Chapman [1] that for large samples $Y_i = \ln Q_i - \ln Q_{i + 1},\quad i = 1, \cdots, r,$ and $Y_j$ are independent for $j \neq i - 1, i, i + 1$. In this note the efficiency obtained when estimating the parameters of a distribution from these mutually independent odd (or even) $Y_i$'s is examined in the case of the geometric and Poisson distributions and it is shown that the resulting estimators are inefficient.