Let $\{Z(t); t \geqq 0\}$ be a one dimensional age dependent branching process with offspring probability generating function (pgf) $h(s) \equiv \sum^\infty_{j=0} p_js^j$ and lifetime distribution function $G(t)$ (see Section 2 for definitions). If $m(t) \equiv EZ(t)$ is the mean function let $Y(t) = Z(t)/m(t)$. Our objective in this paper is to study the limiting behavior of the process $\{Y(t); t \geqq 0\}$. The main result is THEOREM 0. Assume $Z(0) \equiv 1, m = h'(1) > 1, G(0+) = 0$. (Here $\rightarrow_p$ and $\rightarrow_d$ mean convergence in probability and distribution respectively). Then: \begin{equation*}\tag{1}\sum^\infty_{j=2} j \log jp_j = \infty\quad\text{implies} Z(t)/EZ(t) \rightarrow_p 0\end{equation*} and \begin{equation*}\tag{2}\sum^\infty_{j=2} j \log jp_j < \infty\quad\text{implies} Z(t)/EZ(t) \rightarrow_d W\end{equation*} where $W$ is an nonnegative random variable such that (a) $EW = 1$, (b) $\varphi(u) = E(e^{-uW})$ for $u \geqq 0$ satisfies \begin{equation*}\tag{3}\varphi(u) = \int^\infty_0 h(\varphi(ue^{-\alpha y})) dG(y)\end{equation*} where $\alpha$ is the unique root of the equation $m \int^\infty_0 e^{-\alpha y} dG(y) = 1$ (c) $P(W = 0) = q$ the extinction probability (d) $W$ has an absolutely continuous distribution on the positive real axis and the density function is continuous. That is, there exists a nonnegative continuous function $g(x)$ defined for $x > 0$ such that for $0 < x_1 < x_2 < \infty$ \begin{equation*}\tag{4}P(x_1 < W < x_2) = \int^{x_2}_{x_1} g(x) dx.\end{equation*} Kesten and Stigum [4] proved the above result for the case when $G(x)$ is the step function \begin{equation*}\tag{5}\begin{align*}G(x) = 0 \text{if} x\leqq 1 \\ = 1\quad x > 1.\end{align*}\end{equation*} This is the Galton-Watson process in discrete time. They considered the multi-dimensional case. Athreya and Karlin [1] considered the case (here $0 < \lambda < \infty)$ \begin{equation*}\tag{6}\begin{align*}G(x) = 1 - e^{-\lambda x} \text{for} x > 0 \\ = 0\quad x \leqq 0.\end{align*}\end{equation*} This is the continuous time Markov branching process. Their approach was via split times. Levinson [6] established the law convergence of $Z(t)/EZ(t)$ under conditions slightly stronger than ours. Harris [3] claimed mean square convergence of $Z(t)/EZ(t)$ when $h'' (1) < \infty$ and the absolute continuity of $W$ when in addition to $h'' (1) < \infty, 1 - G(t) = O(e^{-cr})$ for some $c > 0$. Our result is the sharpest known in this direction in as much as (i) we establish the convergence of $Z(t)/EZ(t)$ without any conditions, (ii) we give a necessary and sufficient condition for the nondegeneracy of the limit random variable $W$ and (iii) when $W$ is nondegenerate we establish the absolute continuity without any extra assumptions. The methods employed in this paper are all extremely simple. Among them are a simplified and sharpened form of Levinson's [6] arguments and a simplification of Stigum's [7] idea to prove absolute continuity of $W$. One of the important ideas used here is the exploitation of the underlying Galton-Watson process constituted by the size $\{\zeta_n\}$ of the different generations. The key to the understanding of the moment condition $\sum_j j \log jp_j < \infty$ is the simple Lemma 1. Here is an outline of the rest of the paper. In Section 2 we describe the setting and introduce the necessary terminology and notation. The functional equation (3) is studied in detail in Section 3 where it is shown that a necessary and sufficient condition for (3) to have a nontrivial solution is the finiteness of $\sum j \log jp_j$. The next section explores the connection between the process $\{Z(t); t \geqq 0\}$ and the underlying Galton-Watson process $\{\zeta_n; n = 0, 1, 2, \cdots\}$ and shows that if $\sum j \log jp_j = \infty$ then $Z(t)/EZ(t) \rightarrow_p 0$. Assuming $\sum j \log jp_j < \infty$ the convergence in distribution of $Z(t)/EZ(t)$ to a nondegenerate random variable $W$ is shown in Section 5 while Section 6 takes up the proof of absolute continuity. The last section lists some open problems.

In this paper some Kolmogoroff-type inequalities are derived. One of them is best possible for martingales. Another holds for sums of independent random variables and for martingales in which conditional variances satisfy a certain property. But it does not hold, in general, for martingales. This elucidates an important difference between martingales and independent sums. Let $E$ denote expectation. Consider a sequence of random variables $X_1, \cdots, X_n$ and the following three assumptions: \begin{equation*}\tag{a}E(X_1) = 0, E(X_i\mid X_{i-1}, \cdots, X_1) = 0,\quad i = 2, \cdots, n.\end{equation*}\begin{equation*}\tag{b}|X_i| \leqq T \text{almost surely},\quad i = 1, \cdots, n.\end{equation*}\begin{equation*}\tag{c} E(X_1^2) \neq 0, E(X_i^2\mid X_{i-1}, \cdots, X_1) \neq 0 \text{almost surely}, i = 2, \cdots, n.\end{equation*} The first says that $\{X_i\}$ is absolutely fair, the second that $\{X_i\}$ is almost surely bounded, and the third, that conditional variances are positive. We shall deal with the following classes of sums. $M(n)$ is the class of all martingales $\{S_i\}$ of $n$ partial sums $S_i = X_1 + \cdots + X_i$ where $\{X_i\}$ satisfies (a). $B(n) \subset M(n)$ is the subclass in which $\{X_i\}$ satisfies (a), (b), (c), $V(n) \subset B(n)$ the subclass where, additionally, $E(S_n^2) = E(S_n^2\mid X_{n-1}, \cdots, X_1)$, and finally $I(n) \subset B(n)$ the subclass where, in addition, $\{X_i\}$ are independent. $M(n)$ is a pneumonic for martingale, $B(n)$ for bounded martingale, $V(n)$ for martingales where variance equals conditional variance, and $I(n)$ for independent sums. Write $\sigma_i^2 = E(X_i^2), s_i^2 = \sigma_1^2 + \cdots + \sigma_i^2, i = 1, \cdots, n, C_1^2 = E(X_1^2), C_i^2 = E(X_i^2\mid X_{i-1}, \cdots, X_1), i = 2, \cdots, n, C^2 = C_1^2 + \cdots + C_n^2$. Let $M_n = \max (S_1, \cdots, S_n), p(t) = \mathrm{Pr}\{M_n \geqq ts_n^2\}$ and $r(t) = \lbrack e/(1 + tT)\rbrack^{ts_n^2/T}\lbrack 1/(1 + tT)\rbrack^{s_n^2/T^2}.$ Bennett [1] and Hoeffding [4] showed independently that $\sup_{I(n)} \lbrack\mathrm{Pr}\{S_n \geqq ts_n^2\}\rbrack \leqq r(t).$ Steiger [7] extended this to the context of Kolmogoroff inequalities by showing that $p(t) \leqq r(t)$ for all sums $\{S_i\} \varepsilon I(n)$. In Section 3 we show that the inequality actually holds throughout the larger class, $V(n)$, and that it is best-possible there. However the inequality can be false in $B(n)$. This shows that the maxima of martingale partial sums may have larger tail probabilities than those of independent sums or of sums in $V(n) - a$ characteristic property. In Section 2 we prove two preliminary lemmas of independent interest. These are used in Section 3 to obtain Theorem 1, which gives a best-possible upper bound in $B(n)$ for $\operatorname{Pr} \{M_n \geqq tC^2\}$ which, when restricted to $V(n)$ gives $\sup_{V(n)} \lbrack p(t)\rbrack\leqq r(t)$, also best possible. These results are compared to known results of Marshall [6], Dubins and Savage [2], and Steiger [7]. Finally, in Section 4, a partial converse to Theorem 1 is given.

The purpose of this paper is to investigate the asymptotic normality of linear combinations of order statistics; that is, to find conditions under which a statistic of the form $S_n = \mathbf{\sum}^n_{i=1} c_{in}X_{in}$ has a limiting normal distribution as $n$ becomes infinite, where the $c_{in}$'s are constants and $X_{1n}, X_{2n}, \cdots, X_{nn}$ are the observations of a sample of size $n$, ordered by increasing magnitude. Aside from the sample mean (the case where the weights $c_{in}$ are all equal to $1/n$), the first proof of asymptotic normality within this class was by Smirnov in 1935 [19], who considered the case that nonzero weight is attached to at most two percentiles. In 1946, Mosteller [13] extended this to the case of several percentiles, and coined the phrase "systematic statistic" to describe $S_n$. Since the publication in 1955 of a paper by Jung [11] concerned with finding optimal weights for $S_n$ in certain estimation problems, interest in proving its asymptotic normality under more general conditions has grown. For example, Weiss in [21] proved that $S_n$ has a limiting normal distribution when no weight is attached to the observations below the $p$th sample percentile and above the $q$th sample percentile, $p < q$, and the remaining observations are weighted according to a function $J$ by $c_{in} = J(i/(n + 1))$, where $J$ is assumed to have a bounded derivative between $p$ and $q$. Within the past few years, several notable attempts have been made to prove the asymptotic normality of $S_n$ under more general conditions on the weights and underlying distribution. These attempts have employed three essentially different methods. In [1] Bickel used an invariance principle for order statistics to prove asymptotic normality when $\mathbf{\sum}_{i<tn} c_{in}$ converges to a function $J(t)$ of bounded variation and the underlying distribution $F$ has a continuous density, positive on the support of $F$. His method was quite successful in dealing with statistics which put no weight on observations below the $p$th and above the $q$th sample percentile, $p < q$, but in other cases he did not allow the more extreme observations to be weighted more than in the sample mean. More recently in [17], Shorack used a more high powered version of the same approach to obtain a stronger result, allowing much more weight on the extremes. Chernoff, Gastwirth, and Johns, in [3], employed a device of Renyi [14] and expressed $S_n$ as a linear combination of independent, exponentially distributed random variables plus a remainder term; they then showed that the sum of independent variables has a limiting normal distribution and the remainder is asymptotically negligible in the sense that it tends to zero in probability as $n$ tends to infinity. They proved asymptotic normality if the $c_{in}$'s give substantial weight to only a fixed number of percentiles and obey certain boundedness conditions elsewhere, and $F$ has a density which is continuous outside a set of measure zero, bounded away from zero on the interior of the support of $F$, and smooth in the tails. A different approach is used by Govindarajulu in [7]. He adopted a version due to LeCam of the method of Chernoff and Savage [4], which was used by Govindarajulu, LeCam, and Raghavachari in [8] to prove asymptotic normality of linear rank statistics, and for the case $c_{in} = J(i/(n + 1))$ he expressed $S_n$ as a linear combination of independent random variables plus a remainder term. Then if $J$ is absolutely continuous and both it and its derivative satisfy certain boundedness conditions at zero and one, the remainder term tends to zero in probability and $S_n$ has a limiting normal distribution. While his conditions on the weights are very restrictive (for example he does not allow substantial weight to be put on sample percentiles), his conditions on the underlying distribution are the weakest yet obtained, requiring only that the inverse of $F$ does not grow very rapidly at zero or one. Recently in [12], Moore gave a short proof of asymptotic normality, also along the lines of Chernoff and Savage [4], which permits quite general $F$, but again at the expense of stringent conditions on $J$. In this investigation we use yet another method of attacking this problem. Using a procedure due to Hajek [9], who applied it to linear rank statistics, we will represent the statistic $S_n$ as a linear combination of independent random variables, to which the usual central limit theory can be applied, plus a remainder term, and then, under quite general conditions on the weights, prove that the remainder converges to zero in mean square, rather than in the weaker sense of convergence in probability as in [3], [7], and [12]. This is accomplished by first approximating the statistic $T_n = \mathbf{\sum}^{n-b_n}_{i=b_n} c_{in}X_{in}$ by a sum of independent random variables, where $b_n$ is a sequence of integers tending to infinity slower than $n$ but faster than $\log n$ as $n$ increases, and then finding conditions under which $T_n$ approximates $S_n$ in mean square. The result is essentially stronger than that of Bickel, in that much more weight is allowed on the extreme observations; however, a smoothness condition on the distribution similar to that of Chernoff, Gastwirth, and Johns is required. For statistics of the form $T_n$ the result is essentially stronger than that of Chernoff, Gastwirth, and Johns, although slightly more restrictive conditions are required to prove mean square equivalence of $T_n$ and $S_n$. We treat a much more general class of weights than does Govindarajulu or Moore, but our conditions on the underlying distribution are stronger than their conditions. In the course of the proof we derive asymptotic expressions for the covariances of the order statistics and the variance of $S_n$. For the case of the variance of a single order statistic $X_{in}$, the result is proved under weaker conditions on $F$ and for a wider range of $i$ than by Sen [16], Van Zwet [20], Bickel [1], or Blom [2], although the speed of convergence is slower than in [16], [20], or [2]. The asymptotic expression for covariances is similarly improved over Blom [2]. The first section of the paper describes the method to be used and states two previously known propositions which will be useful in the following sections. Section two contains the calculation of an approximation for an order statistic by a sum of independent random variables and an exact expression for the covariance of two such approximations. Asymptotic expressions for this covariance and the covariance of two order statistics are derived in section three, as well as an expression for the variance of $S_n$. Section four contains the proof of the asymptotic normality of $S_n$ when the extremes are not included and no weight is allowed for $i/(n + 1)$ near a point where the derivative of the inverse of $F$ misbehaves, and conditions are given under which these restrictions can be dropped. Finally, in section five we discuss the limitations of the method used and extend the results to the slightly more general class of statistics of the form $\mathbf{\sum}c_{in}h(X_{in})$, also considered in [3], [7], and [17].

$\{\zeta_n\}$ is a sequence of $\operatorname{iid}$ "environmental" variables in an abstract space $\Theta$. Each point $\zeta \varepsilon \Theta$ is associated with a $\operatorname{pgf} \phi_\zeta(s)$. The branching process $\{Z_n\}$ is defined as a Markov chain such that $Z_0 = k$, a finite integer, and given $Z_n$ and $\zeta_n, Z_{n+1}$ is distributed as the sum of $Z_n \operatorname{iid}$ random variables, each with $\operatorname{pgf} \phi_{\zeta_n}(s)$. Set $\xi(\zeta_n) = \phi'_{\zeta_n}(1)$ and assume that $E|\log\xi(\zeta_n)| < \infty$. Then: (i) $P\{Z_n = 0\} \rightarrow 1$ if $E \log \xi(\zeta_n) \leqq 0$; (ii) $qk = _{\operatorname{def}} \lim P\{Z_n = 0\} < 1$ if $E \log \xi(\zeta_n) > 0$ and $E|\log (1 - \phi_{\zeta_n}(0))| < \infty$. Furthermore $\{q_k\}, k = 1, 2, \cdots$, forms a moment sequence.

Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random variables drawn according to a probability measure $\mathscr{P}$. The two-hypothesis testing problem $H_0: \mathscr{P} = \mathscr{P}_0 \operatorname{vs.} H_1: \mathscr{P} = \mathscr{P}_1$ is investigated under the constraint that the data must be summarized after each observation by an $m$-valued statistic $T_n\varepsilon \{1, 2, \cdots, m\}$, where $T_n$ is updated according to the rule $T_{n+1} = f_n(T_n, X_{n+1})$. An algorithm with a four-valued statistic is described which achieves a limiting probability of error zero under either hypothesis. It is also demonstrated that a four-valued statistic is sufficient to resolve composite hypothesis testing problems which may be reduced to the form $H_0:p > p_0 \operatorname{vs.} H_1:p < p_0$ where $X_1, X_2, \cdots$ is a Bernoulli sequence with bias $p$.

In [4] the rank of a state of a stochastic process was defined, although the notion, without the name, is originally due to Gilbert [5]. Let $I = \{1, 2, \cdots\}$. The purpose of the present paper is to prove the THEOREM. Let $\{Y_k\}$ have state space $U_k$ at time $k(k = 1, 2, \cdots$ or $k = 0, \pm 1, \pm 2, \cdots)$. Let $N$ be a finite subset of the index set and, for each $n \varepsilon N$, let $V_n \subset U_n$ be a set of states of finite rank at time $n$. Without loss of generality, assume $(U_n \sim V_n) \mathbf{\cap} (V_n \mathbf{\times} I) = \varnothing$. Then, there exists a process $\{X_k\}$ such that (i) $\{X_k\}$ has state space $U_k$ at time $k \not\in N$ and state space $(U_n \sim V_n) \mathbf{\cup} (V_n \mathbf{\times} I)$ at time $n \varepsilon N$; (ii) The states $(\epsilon, i)$ for $(\mathbf{\epsilon}\varepsilon V_n, i \varepsilon I$ and $n \varepsilon N$ are Markovian; and (iii) $Y_k = F_k (X_k)$ where $F_k(\delta) = \delta$ if $\delta \varepsilon U_k \sim V_k$ (take $V_k = \varnothing$ for $k \not\in N)$ and $F_n (\epsilon, i) = \epsilon$ if $\epsilon\varepsilon V_k$. This theorem is a generalization of Theorem 1 of [4]. Its proof is in Section 2. Section 3 contains corollaries which are the analogues of the corollaries to Theorems 1 and 2 of [4]. These show that if, in addition to the $\epsilon\varepsilon V_n$ for $n \varepsilon N$, there are states of rank 1 (Markovian states) or 2, then $\{X_k\}$ can be constructed so as to preserve the ranks of these states. Section 4 contains a third corollary giving conditions under which $N$ may be infinite. In particular, under these conditions $N$ may be the whole index set and, for each $n \varepsilon N$, we may let $V_n$ be the set of all states of finite rank at time $n$. Corollary 4, also in Section 4, states that, under the conditions of Corollary 3, stationarity in $\{Y_k\}$ may be preserved in $\{X_k\}$. Dharmadhikari [1], [2], [3] has given conditions under which $\{X_k\}$ can be constructed to be a stationary, finite Markov chain. In [2] he requires the condition, among other, that each state of $\{Y_k\}$ be of finite rank. We have weakened our conditions by not insisting on stationarity or finiteness of the state space of $\{Y_k\}$ and by imposing finiteness of rank only on some states. We have completely dropped his condition that certain cones be polyhedral. This last is the reason that we have countably many Markovian states mapping into a single state of $\{Y_k\}$ instead of finitely many.

This paper studies the product epsilon entropy of mean-continuous Gaussian processes. That is, a given mean-continuous Gaussian process on the unit interval is expanded into its Karhunen expansion. Along the $k$th eigenfunction axis, a partition by intervals of length $\epsilon_k$ is made, and the entropy of the resulting discrete distribution is noted. The infimum of the sum over $k$ of these entropies subject to the constraint that $\mathbf{\sum} \mathbf{\epsilon}_k^2 \leqq \mathbf{\epsilon}^2$ is the product epsilon entropy of the process. It is shown that the best partition to take along each eigenfunction axis is the one in which 0 is the midpoint of an interval in the partition. Furthermore, the product epsilon entropy is finite if and only if $\mathbf{\sum} \lambda_k \log \lambda_k^{-1}$ is finite, where $\lambda_k$ is the $k$th eigenvalue of the process. When the above series is finite, the values of $\mathbf{\epsilon}_k$ which achieve the product entropy are found. Asymptotic expressions for the product epsilon entropy are derived in some special cases. The problem arises in the theory of data compression, which studies the efficient representation of random data with prescribed accuracy

Let $\Pi_1, \Pi_2, \cdots, \Pi_k$ be $k$ populations. The random variable $X_i$ associated with $\Pi_i$ has a continuous distribution $F_i, i = 1, 2, \cdots, k$. We are primarily interested in selecting a subset such that the probability is at least $P^\ast$ that the selected subset includes the population with the largest (smallest) quantile of a given order $\alpha (0 < \alpha < 1)$. We assume each $F_i$ has a unique $\alpha$-quantile, $\xi_{\alpha i}$. Let $F_{\lbrack i\rbrack}(x) = F_{\lbrack x\rbrack}$ denote the cumulative distribution function of the population with the $i$th smallest $\alpha$-quantile. In the following, we consider families of distributions ordered in a certain sense with respect to a specified continuous distribution $G$ and propose and study a selection procedure which is different from the non-parametric procedure of Rizvi and Sobel (1967). We assume (a) $F_{\lbrack i\rbrack} (x) \geqq F_{\lbrack k\rbrack}(x), i = 1, 2, \cdots, k$ and all $x$. (b) $\mathbf{\exists}$ a continuous distribution $G \ni F_{\lbrack i\rbrack} \underset{\sim}{\prec} G, \mathbf{\forall}i = 1, 2, \cdots, k$, where $\underset{\sim}{\prec}$ denotes a partial ordering relation on the space of distributions. A relation $\underset{\sim}{\prec}$ on the space of distributions is a partial ordering if \begin{equation*}\begin{split}F \underset{\sim}{\prec} F\quad \mathbf{\forall} \text{distributions} F \\ F \underset{\sim}{\prec} G,\quad G \underset{\sim}{\prec} H \text{implies} F \underset{\sim}{\prec} H.\\ \end{split}\end{equation*} Note that $F \underset{\sim}{\prec} G$ and $G \underset{\sim}{\prec} F$ do not necessarily imply $F \equiv G$. Various special cases in addition to stochastic ordering are: (i) $F \prec_\ast G \operatorname{iff} F(0) = G(0) = 0$ and $G^{-1}F(x)/x$ is nondecreasing in $x \geqq 0$ on the support of $F$. (ii) $F \prec_c G \operatorname{iff} G^{-1}F(x)$ is convex on the support of $F$. (iii) $F \prec_r G \operatorname{iff} F(0) = G(0) = \frac{1}{2}$ and $G^{-1}F(x)/x$ is increasing (decreasing) for $x$ positive (negative) on the support of $F$. (iv) $F \prec_s G \operatorname{iff} F(0) = G(0) = \frac{1}{2}$ and $G^{-1}F$ is concave-convex about the origin, on the support of $F$; i.e., $\{x\mid 0 < F(x) < 1\}$. If $G(x) = 1 - e^{-x}$ for $x \geqq 0$, then (i) defines the class of IFRA distributions studied by Birnbaum, Esary and Marshall (1966) while (ii) defines the class of IFR distributions studied by Barlow, Marshall and Proschan (1963). For any distribution $G, F \prec_\ast G \operatorname{iff} F(x)$ crosses $G(\theta x)$ at most once and from below if at all as a function of $x$ for all $\theta > 0$. If $G(x) = 1 - \exp (-x^\lambda)$ for $x \geqq 0$ and $\lambda > 0$, then $F \prec_\ast G$ implies that $F$ is "sharper" than the family of Weibull distributions with shape parameter $\lambda$. Implications of orderings defined by (iii) were studied by Lawrence (1966). Van Zwet (1964) studies orderings defined by both (ii) and (iv). Clearly $\prec_c$ ordering implies $\prec_\ast$ ordering and $\prec_s$ ordering implies $\prec_r$ ordering. If $\mathbf{X}_i = (X_{i1}, X_{i2}, \cdots, X_{in})$ is the observed sample from the $i$th population, then we restrict ourselves to the class of statistics $T_i = T(\mathbf{X}_i)$ that preserve both ordering relations (a) and (b), i.e., $(a') P_{F_\lbrack i\rbrack}\{T(\mathbf{X}) \leqq x\} \geqq P_{F_\lbrack k\rbrack}\{T(\mathbf{X}) \leqq x\}$ for all $x$ and $i = 1, 2, \cdots, k$. $(b') F_{T(X_i)} \precsim G_{T(\mathbf{Y})}, i = 1, 2, \cdots, k$, where $F_{T(\mathbf{X}_i)}$ represents the cdf of $T(\mathbf{X}_i)$ under $F_{\lbrack i\rbrack}$ and $G_{T(\mathbf{Y})}$ is the cdf of $T(\mathbf{Y})$ under $G, \mathbf{Y} = (Y_1, Y_2, \cdots, Y_n)$ being a random sample from $G$. In Section 2 of this paper, we propose and study procedures $R (R')$ for selecting the population with the largest (smallest) $\alpha$-quantile for distributions which are $\prec_\ast$ ordered with respect to a specified distribution $G$. The infimum of the probability of a correct selection is obtained in Theorem 2.1 and asymptotic evaluation is given in Theorem 2.2. Section 3 deals with quantile selection procedures for the class of IFRA distributions. In Section 4, we study the efficiency of procedure $R$ with respect to a procedure studied by Rizvi and Sobel (1967) under scale type slippage configurations. Asymptotic relative efficiency of $R$ with respect to a selection procedure for the gamma populations proposed by Gupta (1963) is also investigated. Section 5 deals with selection procedures for the median for distributions that are $\prec_r$ ordered with respect to a specified $G$. In Section 6 we propose a selection procedure with respect to the means for distributions that are $\prec_c$ ordered with respect to $G(x) = 1 - e^{-x}$. Application to the selection of gamma populations is also given in Section 6.

Let $X_i(i = 0, 1, \cdots, p)$ be $(p + 1)$ independent and identically distributed nonnegative random variables each representing the $j$th order statistic in a random sample of size $n$ from a continuous distribution $G(x)$ of a nonnegative random variable. Let $G_{j,n}(x)$ be the cumulative distribution function of $X_i(i = 0, 1, \cdots, p)$. Consider the ratios $Y_i = X_i/X_0 (i = 1, 2, \cdots, p)$. The random variables $Y_i (i = 1, 2, \cdots, p)$ are correlated and the distribution of the maximum and the minimum is of interest in problems of selection and ranking for restricted families of distribution. The distribution-free subset selection rules using the percentage points of these order statistics are investigated in a companion paper by Barlow and Gupta (1969). In the present paper, we discuss the distribution of these statistics, in general, for any $G(x)$ and then derive specific results for $G(x) = 1 - e^{-x/\theta}, x > 0, \theta > 0$. Section 2 deals with the distribution of the maximum while Section 3 discusses the distribution of the minimum. Some asymptotic results are given in Section 4, while Section 5 describes the tables of the percentage points of the two statistics.

Asymptotic non-null distribution of the likelihood ratio criterion for testing the linear hypothesis in multivariate analysis is obtained up to the order $N^{-2}$, where $N$ means the sample size, by using the characteristic function expressed in terms of hypergeometric function with matrix argument. This result holds without any assumption on the rank of non-centrality matrix. Asymptotic non-null distribution of the likelihood ratio criterion for independence between two sets of variates is also obtained up to the order $N^{-1}$.

Models of the second kind, described in Section 2, are applied to define inferences about single parameter families having monotone density ratio structure. A definition and some properties of this structure are given in Section 3 along with basic general formulas for inferences. Mixture families always have the monotone density ratio property, and some theory and illustrations are given in Section 4. Location and scale parameter families sometimes have the property, as shown and illustrated in Section 5 where the examples consist of uniform and normal location parameters and exponential and normal scale parameters.

Probability as measure on a Boolean algebra was presented by Kappos [5], but a treatment of conditional probability relative to a subalgebra is missing. The Stone space of a $\sigma$-complete Boolean algebra (see [10], p. 24) enables one to apply the concepts of conditional probability for a $\sigma$-algebra of subsets of some space (see [2], pp. 15-28), but the problem deserves closer attention. In this note we consider conditional probability with respect to a $\sigma$-subfield of the $\sigma$-field generated by the open-closed subsets of the Stone space of a Boolean $\sigma$-algebra. We show that there is always a regular conditional probability (see [4], p. 80) relative to a full $\sigma$-subalgebra of Baire sets. With a modified definition of probability on a Boolean algebra a treatment of conditional probability is possible without reference to the Stone space. For this a generalized integral is defined and the theory of integration is begun for it. A definition of conditional probability on a $\sigma$-complete Boolean algebra is given for which there is no regularity condition. We conclude the discussion with a study of the relationship of this theory with the conventional theory.

In section one, $L_1$ bounds are obtained for transforms of discrete parameter martingales having independent increments. Section two develops the theory of quadratic variation for continuous parameter martingales having independent increments. An application of the results of these two sections to the theory of stochastic integrals appears in section three.

A simple representation of a class of functions of Cauchy variates whose members are also Cauchy variates is given. This leads to a characterization of the Cauchy distribution.

In [2] it is proved that mixtures of exponential distributions are infinitely divisible (id). In [3] it is proved that the same holds for the discrete analogue, i.e. for mixtures of geometric distributions. In this note we show that these results imply that a density function $f(x)$ (or distribution $\{p_n\}$ on the integers) is id if the function $f(x)$ (or the sequence $\{p_n\}$ is completely monotone (cm). For the definition and properties of cm functions and sequences we refer to [1].