In [5] Karlin and McGregor analyzed a class of linear growth birth and death processes admitting a representation formula of the transition probability function in terms of classical orthogonal polynomials. Birth and death processes belong to the category of reversible stochastic processes (see [12] and [13]). In this case by invoking the spectral resolution of the identity for Hermitian operators one achieves the representation formula of Karlin and McGregor described in [3] and [4]. This theory is mostly restricted to the case of one-dimensional birth and death processes and diffusion processes. Some special higher dimensional birth and death processes motivated by certain applications in studies of population growth are reversible. For these processes there are representation formulae for the transition probability function that can be explicitly determined. In [9] and [10] Karlin and McGregor develop a discretized version of the classical technique of solving Laplace's equation in terms of spherical harmonics. This paper applies their method to determine the representation formula for the transition probability function of two two-dimensional (Sections 2 and 3) and the corresponding higher dimensional (Sections 4 and 5) linear growth birth and death processes. In each case the representation formula is expressed explicitly in terms of semi-direct products of classical orthogonal polynomials somewhat reminiscent of such products of spherical harmonics. In the Appendix the properties of the orthogonal polynomials are summarized and the method of derivation of the explicit form of the representation formula is outlined.

An empirical stochastic process for two-sample problems is defined and its weak convergence studied. The results are based upon an identity which relates the two-sample empirical process to the more usual one-sample empirical process. Based on this identity a relatively simple proof of a Chernoff-Savage theorem is obtained. The $c$-sample analogues of these results are also included.

We shall be concerned with the parametric problem of testing hypotheses concerning the value of one parameter when the values of other parameters (nuisance parameters) are not specified. Neyman [6] derived under certain conditions a locally most powerful two-sided test for this problem, i.e., he gave the form of the test maximizing the second derivative of the power function with respect to the parameter of interest at the point specified by the hypothesis. Generalizations of Neyman's results were given by Scheffe [7] and Lehmann [2], using the same technique as Neyman. They were also able to prove that the tests were UMP unbiased. A new technique for dealing with these problems was introduced by Sverdrup [9] and Lehmann and Scheffe [4] where the completeness of the sufficient statistics in an exponential family of densities is used to derive UMP unbiased tests. It is stated by Lehmann and Scheffe [4] that the conditions imposed earlier imply an exponential family of densities. When no UMP unbiased test exists we have little general theory. The problem is both one of principle and of technique. Most stringest tests exist under general conditions but are difficult to derive in particular cases. Lehmann [3] proposed maximin tests. Spjotvoll [8] has given an example of the form of a maximin test when no UMP unbiased and invariant test exists. This paper is an attempt to establish some results for testing hypotheses when the probability density of the observations does not constitute an exponential family under both the hypothesis and the alternative. The assumptions made in Section 2 are satisfied if we have an exponential family under the hypothesis, but do not say anything about the form of the density under the alternative. The results concern most powerful similar or unbiased tests, and under certain conditions the form of these tests for the particular family of densities studied, is given in Section 3. In Section 4 the theory in Section 3 is applied to the problem of testing serial correlation (not circular) in a first order autoregressive sequence. It is found that the usual tests are nearly UMP invariant. In Section 5 the problem of testing the value of the ratio of variances in the one-way classification variance components model is considered. Some numerical results are given for the power functions of the maximin test, the locally most powerful test and the standard $F$-test. The results indicate that the standard $F$-test performs well compared with the other tests.

A potential theory for supermartingales is presented below. It is much like the classical Newtonian potential theory and is a generalization of the potential theory for transient Markov chains. While dealing with stochastic processes with more general dependence relations the new theory retains what we believe to be the important features of the transient Markov chain theory. Briefly, a pure potential is a non-negative supermartingale $\{Z_n, F_n\}$ which satisfies the condition $E\lbrack Z_{n+k}\mid F_n\rbrack \rightarrow 0 \text{a.e. as} k \rightarrow \infty$ for every $n.$ The potential principles of domination, Riesz decomposition, lower envelope, balayage, equilibrium and minimum are proved for these potentials. It is shown how the corresponding results of the transient Markov chain theory can be derived from the new theory. Also, some applications to standard martingale theory are given.

Let $\mathscr{S}_1, \mathscr{S}_2$ be independent $m \times m$ matrices on $n_1, n_2$ degrees of freedom respectively, $\mathscr{S}_2$ having a Wishart distribution and $\mathscr{S}_1$ having a possibly non-central Wishart distribution with the same covariance matrix. Hotelling's generalized $T_0^2$ statistic is then defined [7] by \begin{equation*}\tag{1.1}T = n^{-1}_2T_0^2 = \operatorname{tr} \mathscr{S}_1\mathscr{S}^{-1}_2.\end{equation*} The complete distribution of this statistic is known only in particular cases. If $m = 1,$ then $(n_2/n_1)T$ is simply non-central $F$. In the case $n_1 = 1, T$ reduces to Hotelling's generalization of "Student's" $t,$ which also has a non-central $F$ distribution. When $m = 2$, Hotelling [7] has shown that in the null case the density function of $T$ is \begin{equation*}\tag{1.2}f(T) = \lbrack\Gamma(n_1 + n_2 - 1)/\Gamma(n_1)\Gamma (n_2 - 1)\rbrack(\frac{1}{2}T)^{n_1-1}(1 + \frac{1}{2}T)^{-(n_1+n_2)} \cdot_2F_1 (1, \frac{1}{2}(n_1 + n_2); \frac{1}{2}(n_2 + 1); v),\end{equation*} where $\nu = T^2/(T + 2)^2$, and $_2F_1$ is the Gaussian hypergeometric function. When $n_2$ becomes large, the distribution of $T_0^2$ approaches that of $\chi^2$ based on $mn_1$ degrees of freedom. Ito [9] has derived asymptotic expansions both for the cumulative distribution function (cdf) of $T_0^2,$ and for the percentiles of $T_0^2$ in terms of the corresponding $\chi^2_{mn_1}$ percentiles. Other approximations to the distribution requiring large $n_2$ for validity have been obtained by Pillai and Samson [12]. These authors have used the method of fitting a Pearson curve by means of moment quotients to tabulate upper 5% and 1% points for $m = 2, 3, 4$. The exact distribution of $T$ over the range $0 \leqq T < 1$ has been obtained in the general non-central case by Constantine [3], using the methods of zonal polynomials and hypergeometric functions of matrix argument developed by James and Constantine ([2] and [10], for example). Constantine's solution has the form \begin{equation*}\tag{1.3}f(T) = \lbrack\Gamma_m(\frac{1}{2}(n_1 + n_2))/\Gamma(\frac{1}{2}mn_1)\Gamma_m(\frac{1}{2}n_2)\rbrack T ^{\frac{1}{2}mn_1-1}\mathscr{P}(T),\end{equation*} where $\mathscr{P}(T)$ is a power series in $T$ convergent in the unit circle, and \begin{equation*}\tag{1.4}\Gamma_m(z) = \pi^{\frac{1}{4}m(m-1)} \prod^{m-1}_{i=0} \Gamma(z - \frac{1}{2}i).\end{equation*} In Section 2 of the present paper, it is shown that in the null case the density function $f(T)$ (or rather, its analytic continuation into the complex $T$-plane) satisfies an ordinary linear differential equation of degree $m$ of Fuchsian type, having regular singularities at $T = 0, -1, \cdots, -m$ and infinity. More specifically, an equivalent first-order system is obtained, and the problem is most conveniently treated in this form. Constantine's series (1.3) in the null case is shown in Section 3 to be the relevant solution for $f(T)$ in the neighborhood of the regular singularity at $T = 0$. The differential equations lead to convenient recurrence relations for the coefficients in $\mathscr{P}(T)$. In Section 4 an alternative derivation of Ito's asymptotic formula is presented. Preliminary results are then given (Section 5) for the regular singularity at $T = \infty$, and a heuristic treatment of the limiting distribution as $n_1 \rightarrow \infty$ is presented in Section 6. Finally, it is shown in Section 7 that the moments of $T$ may be obtained from the differential equations for the Laplace transform of $f(T)$ given in Section 1. One objective in deriving the differential equations for $f(T)$ has been to obtain a convenient exact method for computing the distribution and its percentiles. This work is in progress, and it is hoped that results will be available shortly.

Let $\mathbf{X}$ be a $p \times f$ matrix variate $(p \leqq f)$ whose columns are independently normally distributed with $E(\mathbf{X}) = \mathbf{M}$ and covariance matrix $\mathbf{\Sigma}$. Let $w_1, \cdots, w_p$ be the characteristic roots of $|\mathbf{XX}' - w\mathbf{\Sigma}| = 0,$ then the distribution of $\mathbf{W} = \operatorname{diag} (w_i)$ is given by [4], [5] \begin{equation*}\tag{1.1} e^{-\frac{1}{2}\operatorname{tr}\mathbf{\Omega}}_0F_1(\frac{1}{2}f; \frac{1}{4}\mathbf{\Omega}, \mathbf{W})\kappa (p, f) \cdot e^{-\frac{1}{2}\operatorname{tr}\mathbf{W}}|\mathbf{W}|^{\frac{1}{2}(f-p-1)} \mathbf{\prod}_{i>j} (w_i - w_j), 0 < w_1 \leqq \cdots \leqq w_p < \infty,\end{equation*} where \begin{equation*}\tag{1.2}\kappa(p, f) = \pi^{\frac{1}{2}p^2}/\{2^{\frac{1}{2}pf}\Gamma_p(\frac{1}{2}f)\Gamma_p (\frac{1}{2}p)\},\end{equation*} $\mathbf{\Omega} = \operatorname{diag} (\omega_i)$ where $\omega_i, i = 1, \cdots, p,$ are the characteristic roots of $|\mathbf{MM}' - \mathbf{\omega\Sigma}| = 0$ and $_0F_1$ is the hypergeometric function of matrix argument (see Section 2) defined in [5]. The above distribution of non-central means with known covariance matrix was obtained by James [4]. But (1.1) has also been shown, [5], to be the limiting distribution as $n \rightarrow \infty$ of $n\mathbf{R}^2 = \mathbf{W}$ such that $0 < n\mathbf{P}^2 = \mathbf{\Omega} < \infty,$ where $\mathbf{R}^2 = \operatorname{diag} (r_i^2)$ and $\mathbf{P}^2 = operatorname{diag} (\rho^2_i)$ and where the canonical correlation coefficients $r_1^2, \cdots, r_p^2$ between a $p$-set and a $q$-set of variates $(p \leqq q)$ following a $(p + q)$ variate normal distribution, are calculated from a sample of $n + 1$ observations and $\rho_1^2, \cdots, \rho_p^2$ are population canonical correlation coefficients. Further $q = f.$ In this paper, the first two non-central moments of $W_2^{(p)},$ the second elementary symmetric function (esf) in $\frac{1}{2}w_1, \frac{1}{2}w_2,\cdots, \frac{1}{2}w_p$ have been obtained first by evaluating certain integrals involving zonal polynomials, and then alternately in terms of generalized Laguerre polynomials, [2], [5]. These moments were used to suggest an approximation to the non-central distribution of $W_2^{(p)}$. The approximation is observed to be good even for small values of $f.$

Let $A$ and $B$ be two symmetric positive definite matrices of order $p,$ having the independent Wishart densities \begin{equation*}\tag{1.1}g(A) = C_1 \exp \{-\frac{1}{2} \operatorname{tr} A\}|A|^{(n-q-p-1)/2},\end{equation*} where \begin{equation*}\tag{1.2}C_1^{-1} = 2^{(n-q)p/2} \pi^{p(p-1)/4} \mathbf{\prod}^p_{i=1} \Gamma (\frac{1}{2}\lbrack n - q + 1 - i\rbrack),\end{equation*} and \begin{equation*}\tag{1.3}g(B) = C_2 \exp \{-\frac{1}{2} \operatorname{tr} B\}|B|^{(q-p-1)/2} \sum^\infty_{\alpha=0} (\lambda^2b_{11})^\alpha/\lbrack 2^{2\alpha}\alpha!\Gamma(\frac{1}{2}n + \alpha)\rbrack,\end{equation*} where \begin{equation*}\tag{1.4}C^{-1}_2 = \exp \{\frac{1}{2}\lambda^2\}2^{qp/2}\pi^{p(p-1)/4} \mathbf{\prod}^p_{i=2}\Gamma (\frac{1}{2}\lbrack q + 1 - i\rbrack).\end{equation*} Anderson [1] calls the density (1.3) as noncentral linear Wishart density. If we define the matrix $V$ by the relations \begin{equation*}\tag{1.5}A = CVC',\quad A + B = CC',\end{equation*} where $C$ is a lower triangular matrix of order $p.$ Then Kshirsagar [4] finds the noncentral multivariate linear beta density of $V$ to be \begin{equation*}\tag{1.6} g(V) = C_3|V|^{(n-q-p-1)/2}|I - V|^{(q-p-1)/2}\Phi(v_{11}),\end{equation*} where \begin{equation*}\tag{1.7}\Phi(v_{11}) = _1F_1\lbrack\frac{1}{2}n, \frac{1}{2}q; \frac{1}{2}\lambda^2 (1 - v_{11})\rbrack,\end{equation*} and \begin{equation*}\begin{align*}\tag{1.8}C_3 = \exp \{-\frac{1}{2}\lambda^2\} \mathbf{\prod}^p_{i=2} \Gamma(\frac{1}{2}\lbrack n + 1 - i\rbrack)\pi^{-p(p-1)/4} \\ \cdot\lbrack\mathbf{\prod}^p_{i=1} \Gamma(\frac{1}{2}\lbrack n - q + 1 - i\rbrack)\mathbf{\prod}^p_{i=2}\Gamma(\frac{1}{2}\lbrack q + 1 - i\rbrack)\rbrack^{-'}\end{align*}.\end{equation*} Kshirsagar [4] has used this distribution of $V$ to derive the distribution of the test criterion for testing the adequacy of a single hypothetical discriminant function as defined by Williams [6]. Continuing his earlier work, Kshirsagar [5] now uses the distribution of $V$ to obtain the distributions of the direction, and the collinearity factor of this single discriminant function. In case $\xi = \alpha'x$ denotes the discriminant function, then Bartlett [3] gives a factorization of $\Lambda = |V|$ as \begin{equation*}\tag{1.9}\Lambda = \Lambda_1\Lambda_2\Lambda_3,\end{equation*} where \begin{equation*}\begin{align*}\tag{1.10}\Lambda_1 = 1 - (\alpha'B\alpha)/(\alpha'(A + B)\alpha), \\ \Lambda_2 = \lbrack 1 - \alpha'B(A + B)^{-1}B\alpha/\alpha'B\alpha\rbrack/\Lambda_1, \text{direction factor}, \\ \Lambda_3 = \Lambda/\Lambda_1\Lambda_2,\text{the partial collinearity factor}\end{align*}.\end{equation*} Assuming $\alpha = (1, 0, 0, \cdots, 0)$ and factorizing the density of $V$ in terms of rectangular coordinates $T,$ where $V = TT'$ and $T$ lower triangular, Kshirsagar [5] expresses $\Lambda_1, \Lambda_2,$ and $\Lambda_3$ as function of the elements of $T,$ and thus obtains the densities of $\Lambda_1, \Lambda_2,$ and $\Lambda_3.$ Bartlett [3] gives an alternative factorization of $\Lambda$ as $\Lambda = \Lambda_1\Lambda_4\Lambda_5,$ where $\Lambda_4$ is called the collinearity factor and $\Lambda_5$ the partial direction factor, where $\Lambda_4$ and $\Lambda_5$ are certain functions of the elements of the matrices $A$ and $B.$ Kshirsagar expresses $\Lambda_4$ and $\Lambda_5$ as functions of the elements of $T$ and obtains their distributions. Kshirsagar's [5] derivations of these distributions, although elegant, are lengthy and involved, as he uses several lower triangular matrix transformations of the rectangular coordinates in his derivations. It might perhaps be of pedagogical interest to express $\Lambda_1, \Lambda_2, \Lambda_3, \Lambda_4$ and $\Lambda_5$ as functions of the elements of $V$ itself and thus derive their distributions. The present derivations are shorter and neater as they avoid most of the transformations used by Kshirsagar [5]. Since the distributions of $\Lambda$'s are well known beta distributions, they have been derived without the normalizing constants. We assume that all the integrals occurring in this paper are evaluated over appropriate ranges of the variable of integration.

In the multi-urn Ehrenfest model $N$ balls are distributed among $d + 1 (d \geqq 2)$ urns. If we label the urns $0, 1, \cdots, d,$ then the system is said to be in state $\mathbf{i} = (i_1, i_2, \cdots, i_d)$ when there are $i_j$ balls in urn $j (j = 1, 2, \cdots, d)$ and $N - \mathbf{1}\cdot\mathbf{i}$ balls in urn 0. (The vector $\mathbf{1}$ has all its components equal to 1 and $\mathbf{x} \cdot \mathbf{y}$ is the usual scalar product.) At discrete epochs a ball is chosen at random from one of the $d + 1$ urns; each of the $N$ balls has probability $1/N$ of being selected. The ball chosen is removed from its urn and placed in urn $i (i = 0, 1, \cdots, d)$ with probability $p^i,$ where the $p^i$'s are elements of a given vector, $(p^0, \mathbf{p})$, satisfying $p^i > 0$ and $\sum^d_{i=0} p^i = 1.$ We shall let $\mathbf{X}_N(k)$ denote the state of the system after the $k$th such rearrangement of balls. Our interest in this paper is to obtain limit theorems for the sequence of processes $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ as $N$ tends to infinity. For the classical Ehrenfest model $(d = 1, p^0 = p^1 = \frac{1}{2})$ Kac [7] showed that the distribution of $(X_N(\lbrack Nt\rbrack) - N/2)/(N/2)^{\frac{1}{2}}$ converges as $N \rightarrow \infty$ to the distribution of the Ornstein-Uhlenbeck process at time $t$ having started at $y_0$ at $t = 0,$ provided $X_N(0) = \lbrack(N/2)^{\frac{1}{2}}y_0 + N/2\rbrack.$ (The symbol $\lbrack x\rbrack$ denotes the integer part of $x$.) Recently, Karlin and McGregor [8] obtained a similar result for the continuous time version of the model with $d = 2$; in this version the random selection of balls is done at the occurrence of events of an independent Poisson process. In addition, they obtained a local limit theorem for the transition function. The proof in [7] depended on the continuity theorem of characteristic functions. On the other hand, the proof in [8] used the properties of the spectral representation of the $n$-step transition probabilities which is available for these processes. These results suggested the direction we shall follow in this paper. A preliminary calculation indicates that the process $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ is attracted to the pseudo-equilibrium state $N\mathbf{p}$ and that states far from $B\mathbf{p}$ will only occur rarely. Thus it is natural to consider the fluctuations of $\mathbf{X}_N(k)$ about $N\mathbf{p}$ measured in an appropriate scale. For our purposes the appropriate processes to consider are $\{\mathbf{Y}_N(k): k = 0, \cdots, N\}$, where $\mathbf{Y}_N(k) = (\mathbf{X}_N(k) - N\mathbf{P})/N^{\frac{1}{2}}.$ Next we define a sequence of stochastic processes $\{\mathbf{y}_N(t): 0 \leqq t \leqq 1\}$ which are continuous, linear on the intervals $((k - 1)N^{-1}, kN^{-1})$, and satisfy $\mathbf{y}_N(kN^{-1}) = \mathbf{Y}_N(k)$ for $k = 0, 1, \cdots, N.$ In other words we let $\mathbf{y}_N(t) = \mathbf{Y}_N(k) + (Nt - k)(\mathbf{Y}_N(k + 1) - \mathbf{Y}_N(k))$ if $kN^{-1} \leqq t \leqq (k + 1)N^{-1}$. Throughout this paper we shall let $X_N^i(0) = \lbrack N^{\frac{1}{2}}y_0^i + Np^i\rbrack,$ where $y_0 = (y_0^1, \cdots, y_0^d)$ is an arbitrary, but fixed, element of $R^d.$ (It will always be understood that $N$ is sufficiently large so that $0 \leqq X_N^i(0) \leqq N$ for all $i = 1, 2, \cdots, d$, where $X_N^i(\cdot)$ is the $i$th component of the vector $\mathbf{X}_N(\cdot). R^d$ is $d$-dimensional Euclidean space.) Observe that this initial condition implies that $|Y_N^i (0) - y_0^i| \leqq N^{-\frac{1}{2}}.$ With this initial condition and the Markov structure of the model, the processes $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ for $N = 1, 2, \cdots$ can be defined on a probability triple $(\Omega_N, \mathscr{F}_N, P_N).$ We shall let $C_d\lbrack 0, 1\rbrack$ denote the product space of $d$ copies of $C\lbrack 0, 1\rbrack$, the space of continuous functions on $\lbrack 0, 1\rbrack$ with the topology of uniform convergence, and endow $C_d\lbrack 0, 1\rbrack$ with the product topology. The topological Borel field of $C_d\lbrack 0, 1\rbrack$ will be denoted by $\mathscr{C}_d$. Clearly, the transformation taking the sequence $\{\mathbf{X}_N(k): k = 0, \cdots, N\}$ into $\{\mathbf{y}_N(t): 0 \leqq t \leqq 1\}$ is measurable and induces a probability measure on $\mathscr{C}_d$. We shall denote this induced measure by $\mu_N(\cdot; \mathbf{y}_0)$. The general notion of weak convergence of a sequence of probability measures is defined as follows. Let $S$ be a metric space and $\mathscr{S}$ be the Borel field generated by the open sets of $S$. If $\nu_N$ and $\nu$ are probability measures on $\mathscr{S}$ and if the $\lim_{N\rightarrow\infty} \int _sf d\nu_N = \int _sf d\nu$ for every bounded, continuous function $f$ on $S$, then we say that $\nu_N$ converges weakly to $\nu$ and write $\nu_N\Longrightarrow \nu$. The principal result of this paper is that $\mu_N(\cdot; \mathbf{y}_0) \Longrightarrow \mu (\cdot; \mathbf{y}_0)$ as $N \rightarrow \infty$, where $\mu(\cdot; y_0)$ is the probability measure on $\mathscr{C}_d$ of a $d$-dimensional diffusion process, $\mathbf{y}(\cdot)$, starting at the point $\mathbf{y}_0$. The limit process $\mathbf{y}(\cdot)$ is a $d$-dimensional analog of the Ornstein-Uhlenbeck process whose distribution at time $t$ is a multi-variate normal with mean vector $e^{-t}\mathbf{y}_0$ and covariance matrix $\Sigma$, where the elements of $\Sigma$ are \begin{equation*}\begin{split}\sigma_{ij} = (1 - e^{-2t})p^i(1 - p^i),\quad i = j, \\ = -(1 - e^{-2t})p^ip^j,\quad i \neq j.\end{split}\end{equation*} For applications it is useful to note that $\mu_N(\cdot; \mathbf{y}_0)\Longrightarrow\mu(\cdot;\mathbf{y}_0)$ is equivalent to the statement that $\lim_{N\rightarrow\infty} \mu_N(\{f(\mathbf{y}(\cdot)) \leqq \alpha\}; \mathbf{y}_0) = \mu(\{f(\mathbf{y}(\cdot)) \leqq \alpha\}; \mathbf{y}_0)$ for all functionals $f$ on $C_d\lbrack 0, 1\rbrack$ which are continuous almost everywhere with respect to $\mu(\cdot; \mathbf{y}_0)$. To establish weak convergence two steps are usually required. First, the convergence of the finite-dimensional distributions (fdd) of the approximating processes, $\{\mathbf{y}_N(t): t \geqq 0, N = 1, 2, \cdots\}$ in our case, to the corresponding fdd of the limiting process must be obtained. Second, the probability that the approximating processes can have large fluctuations between points at which these processes are determined by their fdd must be shown to be small. The notion of weak convergence is intimately related to the so-called invariance principles. An invariance principle was first introduced for the case of sums of independent, identically distributed random variables by Erdos and Kac [4] and generalized by Donsker [3]. The method of proof used by Erdos-Kac and Donsker was later modified by Billingsley [1] for dependent random variables. In carrying out the second step outlined above we shall follow Billingsley's argument in Theorems 2.3 and 3.1 of [1]. A program similar in nature to ours was carried out by Lamperti [10] for a particular class of Markov processes. This paper is organized into the following sections: Section 2 is devoted to our analog of the central limit theorem (clt), namely, that the distribution of $\mathbf{y}_N(t)$ converges to the distribution of $\mathbf{y}(t)$ at a single fixed value of $t$. In Section 3 the limit process $\mathbf{y}(t)$ is identified and the properties of the process needed here are discussed. Section 4 completes the proof of the convergence of the fdd of $\{\mathbf{y}_N(t)\}$ to those of $\{\mathbf{y}(t)\}$. The proofs in both Sections 2 and 4 are carried out using the Levy continuity theorem for characteristic functions. Section 5 provides the proof required to show weak convergence. The main tool here, in addition to Billingsley's theorems mentioned above, is the result of Stone (1961) on the weak convergence of random walks. Finally, in Section 6 applications are mentioned along with a suggestion as to how the multi-urn Ehrenfest model might be used to study certain problems in statistical mechanics, networks of queues, and epidemic theory.

The moment generating function (mgf) of Pillai's $V^{(s)}$ criterion in the central case has been considered by Pillai [8], [9] and James [3]. In the present paper, the mgf's of $V^{(s)}$ are considered in the following non-central cases: (i) MANOVA and (ii) canonical correlation. The lower order moments of $V^{(s)}$ for (i) and (ii) in the non-central case were obtained earlier by Pillai [10] for $s = 2$ in the linear case i.e. when there is only one non-zero population root. These results for (i) were extended by Khatri and Pillai [5], [6], [7] to general $s$ in the linear case and in the planar case i.e. when there are two non-zero population roots. Only the first two moments were considered in the planar case while the first four moments were obtained in the linear case. The results of this paper further facilitates the derivation of the general moments.

Let $Z_n$ be the maximum of $n$ independent identically distributed random variables each having the distribution function $F(x)$. If there exists a non-degenerate distribution function (df) $\Lambda(x)$, and a pair of sequence $a_n, b_n$, with $a_n > 0$, such that \begin{equation*}\tag{1.1}\lim_{n\rightarrow\infty}P\{a_n^{-1}(Z_n - b_n) \leqq x\} = \lim_{n\rightarrow\infty} F^n (a_nx + b_n) = \Lambda(x)\end{equation*} on all points in the continuity set of $\Lambda(x)$, we say that $\Lambda(x)$ is an extremal distribution, and that $F(x)$ lies in its domain of attraction. The possible forms of $\Lambda(x)$ have been completely specified, and their domains of attraction characterized by Gnedenko [5]. These results and their applications are contained in the book by Gumbel [6]. A natural question is whether the various moments of $a_n^{-1} (Z_n - b_n)$ converge to the corresponding moments of the limiting extremal distribution. Sen [9] and McCord [8] have shown that they do for certain distribution functions $F(x)$, satisfying (1.1). Von Mises ([10] pages 271-294) has shown that they do for a wide class of distribution functions having two derivatives for all sufficiently large $x$. In Section 2, the question is answered affirmatively for all distribution functions $F(x)$ in the domain of attraction of any extremal distribution provided the moments are finite for sufficiently large $n$. If there exists a sequence $a_n$ such that \begin{equation*}\tag{1.2}Z_n - a_n \rightarrow 0, \text{i.p.}\end{equation*} we say that $Z_n$ is stable in probability. If \begin{equation*}\tag{1.3}Z_n/a_n \rightarrow 1, \text{i.p.}\end{equation*} we say that $Z_n$ is relatively stable in probability. Necessary and sufficient conditions are well known for stability and relative stability both in probability (see Gnedenko [5]) and with probability one (see Geffroy [4], and Barndorff-Nielsen [1]). In Section 3 necessary and sufficient conditions are found for $m$th absolute mean stability and relative stability. The results of this work are valid for smallest values as well as for largest values.

In [5], B. Rosen showed that if $\{X_k: k = 1,2, \cdots\}$ is an independent sequence of identically distributed random variables with $EX_k = 0$ and $\operatorname{Var} X_k = \sigma^2, 0 < \sigma^2 < \infty$ and if $S_n = X_1 + \cdots + X_n$, then the series $\sum^\infty_{n=1} n^{-1} (P(S_n < 0) - \frac{1}{2})$ is absolutely convergent. This theorem was motivated by a result of Spitzer [6] who, under the same conditions, established the convergence of this series as a corollary to a result in the theory of random walks. Rosen's theorem was generalized by Baum and Katz [1] who showed that if $EX_k = 0$ and $E|X_k|^{2+\alpha} < \infty$ for $0 \leqq \alpha < 1$ then $\sum^\infty_{n=1}n^{-(1-\alpha/2)} |P(S_n < 0) - \frac{1}{2}| < \infty.$ These results led to the study of series convergence rate criteria for the central limit theorem and a partial solution of this problem was obtained for the case of identically distributed random variables in [2]. A more complete solution has been recently obtained by Heyde [4]. The first study of series convergence rates for $P(S_n < 0)$ in the case of independent but non-identically distributed random variables was made by Heyde [3]. Based on an extension of Rosen's theorem utilizing certain uniform bounds on the characteristic function of the $X_k$'s he concluded the absolute convergence of the series $\sum^\infty_{n=1}n^{-(1-\alpha/2)} (P(S_n < n^px) - \frac{1}{2})$ for $- \infty < x < \infty$ and $0 \leqq p < \frac{1}{2}(1 - \alpha), 0 \leqq \alpha < 1,$ thus obtaining what he termed small deviation convergence rates. In the present paper two more extensions of Rosen's theorem to independent but non-identically distributed random variables are given under different hypotheses than Heyde's. The first (Theorem 1) reduces to Rosen's theorem in the case of identically distributed random variables. The second (Theorem 2) results in a theorem similar to that of Baum and Katz [1] as required in Heyde's small deviation result. This will make it possible to obtain his conclusion by simply carrying out the last step in his proof. These results are obtained in Section 3. In Section 2 some preliminary results are stated and examples are given in Section 4 to show that the first two hypotheses of Theorem 1 cannot, in general, be relaxed.

Let $U_1, U_2, \cdots, U_N$ be a random sample from a population with a continuous distribution function and $R_i, i = 1, \cdots, N,$ be the rank of $U_i$ among the $N$ observations. Asymptotic normality is studied for the statistics of the type \begin{equation*}\tag{0.1} \sum^N_{i=1} \sum^N_{j=1} c_{ij}a_N(R_i/N, R_j/N),\end{equation*} where constants $c_{ij}$ satisfy certain negligibility conditions and the score function $a_N(\cdot, \cdot)$ is derived from a function $a(\cdot, \cdot)$ satisfying certain monotonicity and integrability conditions. It is shown that the statistic (0.1) is asymptotically equivalent to \begin{equation*}\tag{0.2} \sum^N_{i=1} \sum^N_{j=1} c_{ij}a(U_i, U_j),\end{equation*} so that the problem is reduced to a simpler one, viz. studying the asymptotic distribution of (0.2). Similar results are obtained for the two sample analog of (0.1) viz. \begin{equation*}\tag{0.3} \sum^N_{i=1} \sum^M_{j=1} c_{ij}a_{NM}(R_i/N, S_j/M)\end{equation*} where $S_j, j = 1, \cdots, M$, are the ranks corresponding to another independent random sample of size $M$ from some other population. Few more variants of the above and applications of these statistics are given. The present study is a generalization of a paper by Hajek (1961).

The statistic $V_{NM}$ is a two-sample statistic which may be used to test the null hypothesis $H_0$, that two samples, sizes $N$ and $M$, come from identical populations. It is an adaptation of the Kolmogorov two-sample statistic, and is defined by $V_{NM} = \sup_{-\infty<x<\infty} (F_N(x) - G_M(x)) - \inf_{-\infty<x<\infty} (F_N(x) - G_M(x))$ where $F_N(x), G_M(x)$ are the sample cumulative distribution functions. A single-sample analogue $V_N$ is defined by replacing $G_M(x)$ in the formula above by a hypothesised distribution function $F(x)$. For large values of $V_{NM}$ or $V_N, H_0$ will be rejected. These statistics are particularly useful for observations recorded as points on a circle; the value obtained for $V_N$ or $V_{NM}$, in contrast to that of the corresponding Kolmogorov statistic, does not depend on the choice of origin. Kuiper (1960) proved this result and suggested the use of the $V$ statistics for the circle. He also gave series approximations to the distributions of $N^{\frac{1}{2}}V_N$ and $N^{\frac{1}{2}}V_{NN}$, for large $N$, on the null hypothesis $H_0$. Reference to a distribution will henceforth be assumed to refer to the distribution on the null hypothesis. The $V_{NN}$ statistic had earlier been investigated by Gnedenko and co-workers (see Gnedenko (1954)), who used it for observations on a line; the $V$ statistics may be expected to be more powerful than the Kolmogorov statistics for certain alternatives. Gnedenko (1954) gives the exact distribution of $V_{NN}$. The exact distribution of $V_N$, in the upper and lower tails, has recently been given by Stephens (1965). In this paper we make the two-sample goodness-of-fit test available for a wide range of sample sizes by giving tables of the distribution of $V_{NM}$. In the next section a formula is given with which the $V_{NM}$ statistic may be calculated from the ranks of the two samples, and then the goodness-of-fit tests are set out. These are called exact or approximate tests, depending on whether the probabilities used are calculated from exact or approximate formulae. In Section 3 the construction of the tables is described. To find percentage points for large $N$, we develop in Section 3.3 a series expansion for the distribution of $N^{\frac{1}{2}}V_{NN}$ which differs from that given by Kuiper; the probabilities given by the two expansions are compared in Table 4, and the new series clearly gives the better results.

Let $\{Y_k\}$ be a stochastic process where either $k = 1,2, \cdots$, or $k = 0, \pm 1, \cdots$. Let $S$ and $T$ be measurable sets of sequences of states. Let $\epsilon$ be a state. Assume $P(Y_n = \epsilon) > 0$. Let $p_n(S\epsilon T) = P((\cdots, Y_{n-2}, Y_{n-1}) \varepsilon S, Y_n = \epsilon, (Y_{n+1}, Y_{n+2}, \cdots) \varepsilon T)$. We define the rank of $\epsilon$ at time $n$ to be maximal rank of matrices $(p_n(S_i\epsilon T_j); i, j = 1, \cdots, m)$ as $m$, the $S_i$ and the $T_j$ vary. In the stationary case, since the rank does not depend on $n$, we will refer to the rank of $\epsilon$. In this case, with finite state space, Gilbert [6] denoted the rank of $\epsilon$ by $n(\epsilon)$. A state which has rank 1 at time $n$ is a Markovian state at time $n$. A stochastic process all of whose states are Markovian at all times is a Markov process. Let $\{X_k\}$ be a second stochastic process indexed as $\{Y_k\}$. Gilbert proved (but stated in far less generality) that if $\nu_n(\epsilon)$ and $\mu_n(\delta)$ are the ranks at time $n$ of the states $\epsilon$ in $\{Y_k\}$ and $\delta$ in $\{X_k\}$, respectively, and if $Y_n = f(X_n)$, then \begin{equation*}\tag{1.1}\nu_n(\epsilon) \leqq \sum_{f(\delta) = \epsilon} \mu_n(\delta).\end{equation*} In the stationary finite state space case, Dharmadhikari [2] gave conditions under which if $\{Y_k\}$ has all states of finite rank it is possible to express $Y_k = f(X_k)$ where $\{X_k\}$ is stationary and Markovian. An example was given to show equality need not hold in (1.1). A similar proof can be used to show in the more general case that if $\epsilon$ is a state of finite rank in $\{Y_k\}$ and his conditions hold for $\epsilon$, then it is possible to write $Y_k = f(X_k)$ where $\epsilon = f(\epsilon_i)$ for a finite number of states $\epsilon_i$ of $\{X_k\}$ which are Markovian and $\delta = f(\delta)$ for $\delta \neq \epsilon_i$. See [4]. In Section 2 of the present paper we present an example showing that finite rank alone does not guarantee Dharmadhikari's result. In this case we have stationarity for $\{X_k\}$ and $\{Y_k\}$ and $\epsilon = f(\delta)$ for a countable set of states $\delta$ of $\{X_k\}$ which are Markovian. This is an example referred to by Dharmadhikari [2] and disproves a conjecture due to Gilbert. In Section 3 it is proved that if $\{Y_k\}$ has a state $\epsilon$ of finite rank at time $n$, then there exists a stochastic process $\{X_k\}$ such that $Y_k = f(X_k)$ where $\delta = f(\delta)$ if $\delta \neq \epsilon_i$ and $\epsilon = f(\epsilon_i)$ for a countable family $\{\epsilon_i\}$ of Markovian states of $\{ X_k\}$ at time $n$. However, we are unable to prove that the ranks of states at times other than $n$ are undisturbed. Trivially, no rank may be decreased. Alternative constructions are given to show Markovianess and rank 2 of states may be preserved in $\{X_k\}$. The latter is deferred to Section 4. Our construction is valid both for $n = 0, \pm 1, \cdots$ and for $n = 1,2, \cdots$ without any assumption of finiteness of the state space of $\{Y_k\}$. Trivially, if this state space is countable and $n = 1,2, \cdots$ it is possible to represent $Y_k = f(X_k)$ where $\{X_k\}$ is a countable state Markov process. In this case Carlyle [1] gave a particular construction of $\{X_k\}$ which, in the case that $\{X_k\}$ has a finite state space, yields the minimal state space. In Section 4 we consider a state $\epsilon$ of rank 2 at time $n$. It is proved that in this case the construction of the process $\{X_k\}$ can be carried out so that there are only two states, $\epsilon_1$ and $\epsilon_2$, for which $\epsilon = f(\epsilon_i)$. Furthermore, it is proved that ranks of all states $\delta \neq \epsilon$ are undisturbed. If, in addition, $\{Y_k\}$ is stationary, it is proved in Section 5 that a stationary $\{ X_k\}$ can be constructed with all these properties. The definition of rank can be readily extended to the case of densities. All results of this paper as well as Gilbert's and Dharmadhikari's results can be obtained in this case. These considerations are deferred to a later paper. The results of Section 4 have found applications to a stochastic process describing the temporal behavior of cloud cover [5]. The observable states of cloud cover are clear (under 5% of the sky covered by clouds), partly cloudy (between 5% and 50% covered), cloudy (between 50% and 95% covered) and overcast (over 95% covered). A nonstationary Markov chain does not fit the data. Certain matrices of observed frequencies are, approximately, of rank 2 for data from Boston. Assume the observable process is a function of a nonstationary Markov chain with eight states, each state of the observable process being the image of two Markovian states. This model fits the Boston data very well.

Within the class of models producing upper and lower probability systems, as discussed in Dempster (1967a), a simple and important subclass may be characterized by random intervals on the line. Detailed expressions are given here for the upper and lower probabilities of a general fixed closed interval determined by a general random closed interval. Such random closed intervals occur in the applications of the general class of models to statistical inference described in Dempster (1966, 1967b, 1968). The illustration given here concerns inference about binomial $p$ and stresses the flexibility allowed in the introduction of prior information.

The object of the present paper is to generalize the results of Hollander [4] (concerning rank tests for randomized blocks for ordered alternatives) to Chernoff-Savage [1] class of tests which includes his test as a special case. Allied efficiency results are also studied.

The study of common treatments between the blocks of statistical designs is not new. The block structure of BIB designs was most exhaustively studied by Connor [4], Shrikhande, Trehan, Chakrabarti [5] and others. Connor [3] extended this search to symmetrical Group Divisible (GD) designs restricting himself mainly to regular and semi-regular cases. Roy and Laha [6] were interested in partially balanced designs which are of the LB type. This paper deals with the study of treatments common to blocks of certain PBIB designs which are Kronecker product of certain known designs. The credit for showing that the Kronecker product of two BIB designs has the association scheme of the rectangular type goes to Vartak [8]. In Section 3 of this paper we have obtained the matrices of characteristic vectors of the treatment characteristic matrix $NN'$ of designs having rectangular association scheme. These are then used to set up bounds for the number of treatments common to any two blocks belonging to a group of blocks of the designs which are PBIB with three and two associate classes obtained by taking the Kronecker product of two BIB designs. Also the limits to the number of treatments common to some two blocks of a singular GD are determined.

A queueing system can be regarded as transforming one point process into another (as pointed out for example in Kendall (1964), Section 6), namely, the input or arrival process with inter-arrival intervals $\{T_n\}$ is acted on by a system comprised of a queue discipline and a service (or, delay) mechanism, producing the output or departure process with inter-departure intervals $\{D_n\}$. The object of this paper is to study the correlation structure of the sequence $\{D_n\}$ (and this sequence we shall for convenience call the output process of the system) when the input process is a renewal process and when the service times $\{S_n\}$ (assumed to be independently and identically distributed, and independent of the input process) are such that the system can and does exist in its stationary state. In particular, we shall be concerned with conditions under which the process $\{D_n\}$ is uncorrelated, by which we mean that $\operatorname{cov} (D_0, D_n) = E(D_0D_n) - (E(D_0))^2 = 0 (n = 1,2, \cdots)$. Schematically then, we study the mapping $\{T_n\} \overset{\{ S_n\}/1}{\longrightarrow} \{D_n\}$, and as consequences of the formal theorems of the paper the following statements can be justified $(T, S$, and $D$ denote typical members of $\{T_n\}, \{S_n\}$ and $\{D_n\})$. (i) $\operatorname{var} (D) \geqq \operatorname{var} (S)$, with equality only in the trivial case where both $\{T_n\}$ and $\{S_n\}$ are deterministic. (ii) Locally, the mapping can be made any of variance increasing, variance preserving, or variance decreasing (that is, all cases of $\operatorname{var} (D) >, =, < \operatorname{var} (T)$ are possible) by appropriate choice of $\{T_n\}$ and $\{S_n\}$. Globally however, the mapping is variance preserving, that is, $\operatorname{var} (D_1 + \cdots + D_n)/\operatorname{var} (T_1 + \cdots + T_n) \rightarrow 1\quad (n \rightarrow \infty)$. (iii) When $\{T_n\}$ is a Poisson process, the process $\{D_n\}$ is uncorrelated if and only if it is a Poisson process (and this occurs if and only if the $\{S_n\}$ are negative exponential). (iv) When the $\{S_n\}$ are negative exponential, $\{D_n\}$ is a renewal process if and only if it is a Poisson process (and this occurs if and only if $\{T_n\}$ is a Poisson process). However (cf. (iii)) $\{D_n\}$ can be uncorrelated without being a renewal process. If the $\{D_n\}$ are correlated then the terms $\operatorname{cov} (D_0, D_n)$ are of the same sign for all $n = 1,2, \cdots$ and converge to zero monotonically. (v) There exist $\{T_n\}$ and $\{S_n\}$ such that the serial covariances $\operatorname{cov} (D_0, D_n)$ are not of the same sign for all $n = 1,2, \cdots$ (see remark after Theorem 7).

Suppose for each $i = 1,2, \cdots, k$ the random variable $X_i$ has density function $f(x; \theta_i)$ where each of the parameters $\theta_1, \theta_2, \cdots, \theta_k$ is known to belong to some connected set $\Theta$ of real numbers. Independent random samples are taken from the distributions of $X_1, X_2, \cdots, X_k$ and we wish to find estimates $\hat{\theta}_1, \hat{\theta}_2, \cdots, \hat{\theta}_k$ of $\theta_1, \theta_2, \cdots, \theta_k$ which satisfy: \begin{equation*}\tag{1.1}\hat{\theta}_1 \geqq \hat{\theta}_2 \geqq \cdots \geqq \hat{\theta}_k.\end{equation*} Brunk [3] considered such a problem when $f(x; \theta)$ belongs to a certain exponential family of distributions which includes the binomial, the normal with fixed mean and variable standard deviation, the normal with fixed standard deviation and variable mean, and the Poisson distributions. A discussion of the history of this type of problem is given by Brunk [4]. In this paper we assume that the density function $f(x; \theta)$ has certain properties and develop a procedure for finding the restricted estimates. The above mentioned densities have those properties as do certain others. One in particular, not covered by Brunk's formulation, the bilateral exponential distribution (i.e. $f(x; \theta) = \frac{1}{2}e^{-|x-\theta|}$), is considered in Section Four. In Section Two we list those assumptions, describe our procedure for finding restricted estimates and prove that they are maximum likelihood estimates. In Section Three we give a representation theorem for our estimates and a theorem which implies that they are consistent in the special cases which we consider. In Section Five we describe an alternate method for obtaining restricted estimates and in Section Six we discuss another special case not considered by Brunk [3], which does not satisfy our regularity assumptions but for which the procedure described in Section Two clearly works.

Srivastava (1966) has proposed two classes of asymptotically efficient sequential procedures for selecting the population with the largest mean when nothing is assumed about the form of the distribution functions except for the finiteness of unknown variance. In order to choose between these two classes, it seems desirable to study the performance of these two classes of sequential procedures for moderate sample sizes. In this paper, assuming that these populations are normal, the average sample sizes and the error probabilities actually obtained are computed to compare the two procedures. Results of calculation show that Procedure B (see Section 2) is better than Procedure A in that the average sample size is smaller while the error probabilities are almost the same (the superiority of procedure B was suggested by Srivastava (1966) for a different intuitive reason). At the same time the calculation helps to show the closeness of the error probability (both cases) to $\alpha$. The calculation is on the lines of Ray (1957) and Robbins (1959) by generalizing a problem of the latter. Starr (1966) has recently carried out calculations to study the performance of a sequential procedure for finding fixed-width confidence bounds for the normal mean (see also Chow and Robbins, 1965).

In a previous paper [8] the asymptotic behavior of a class of invariant sequential probability ratio tests was studied to the extent of establishing termination with probability one. In this paper, under somewhat stronger conditions, certain bounds on the distribution of sample size will be obtained. The statistical framework is as follows: $Z_1, Z_2, \cdots$ is a sequence of independent and identically distributed (iid) random vectors with values in Euclidean $k$-space $E^k$. The common distribution is at first assumed to belong to the family $\mathscr{N}$ of all nondegenerate $k$-variate normal distributions. On $E^k$ there acts a group $G^\ast$ of affine transformations (precise assumptions on $G^\ast$ are given in Section 2). Let $V_1, V_2, \cdots$ be a maximal invariant sequence obtained from $Z_1, Z_2, \cdots$ under the application of $G^\ast$. Since every transformation in $G^\ast$ sends a member of $\mathscr{N}$ into a member of $\mathscr{N}, G^\ast$ also acts on $\mathscr{N}$. Let $\gamma$ be a maximal invariant function on $\mathscr{N}$, then the joint distribution of $V_1, V_2, \cdots$ depends only on $\gamma$. Denote the distribution of $(V_1, \cdots, V_n)$ by $P_{n\gamma}$. Let $\gamma_1, \gamma_2$ be two distinct values of $\gamma$, let $r_n = dP_{n\gamma_2}/dP_{n\gamma_1}$ (set equal to $\infty$ wherever $P_{n\gamma_2}$ is not absolutely continuous with respect to $P_{n\gamma_1}$) and put $R_n = r_n(V_1, \cdots, V_n)$. Then an invariant sequential probability ratio test, based on the sequence $\{R_n\}$, is defined by choosing stopping bounds $B < A$, letting the stopping variable $N$ be the smallest $n$ such that $R_n \leqq B$ or $\geqq A$, and accepting $\gamma_1$ or $\gamma_2$ according as $R_N \leqq B$ or $\geqq A$. Once the sequence $\{R_n\}$ has been defined we are at liberty to study its behavior when the actual common distribution of the $Z_n$ is not necessarily on the orbit of $\gamma_1$ or of $\gamma_2$ or, for that matter, is not even a member of $\mathscr{M}$. Still assuming $Z_1, Z_2, \cdots$ to be iid, the common distribution $P$ will be assumed to be a member of $\mathscr{P}$, to be defined later, where $\mathscr{P} \supset \mathscr{M}$. The joint distribution of $Z_1, Z_2, \cdots$ will also be denoted $P$. The object is to establish a bound on $P(N > n)$ as a function of $n$, for each $P \varepsilon \mathscr{P}$. Results on sample size distribution of invariant sequential probability ratio tests are very scarce. Ifram [5] considered a certain class of problems and obtained an exponential bound on $P(N > n)$, where $P$ is a member of the original model and should not belong to a certain exceptional set of distributions for which no results could be obtained. Sacks [6] also obtained an exponential bound in the case of the sequential $t$-test (as a by-product of other results), again excluding certain exceptional $P$'s. Savage and Sethuraman [7] obtained an exponential bound in a nonparametric problem, a sequential rank test. They allowed $P$ to be outside the original model, but again for a certain set of exceptional $P$'s no results could be obtained. In the present paper we shall establish, under Assumptions A and B (Section 2), an exponential bound of the form $P(N > n) < c\rho^n$ for some $\rho < 1$, except, again, for $P$'s in a certain set. For these exceptional $P$'s, however, we also obtain results, even though weaker, of the form $P(N > n) < c\rho^{n^{1/3}}$ (under an additional assumption on the function $\Phi$ of Section 2). This result is still strong enough to assert the existence of all moments of $N$, although not the existence of a moment generating function as is the case if $P(N > n)$ has an exponential bound. Thus, our results are more general and stronger in some respects than those of Ifram [5], but Ifram does give a precise value of the smallest possible $\rho$ in the exponential bound, whereas we have nothing comparable to offer.

Consider the following model, the practical applications of which will be discussed elsewhere. An urn contains an unknown number, $N$, of white balls, and no others. An estimate of $N$ is desired, based on the following sampling procedure. Balls are drawn at random, one at a time, from the urn. A white ball is colored black before it is returned, a black ball is returned unchanged. The ball is always returned before the next ball is drawn. We are interested in two problems: (i) what stopping rule $t$ to use to terminate sampling, and (ii) how to estimate $N$ after we stop. The present problem (also in a more general setup) has been considered by several authors, notably L. A. Goodman [5], Chapman [1], Darroch [3] and Darling and Robbins [2]. We shall refer to their results in the sequel. Let $w_i, b_i$ denote the (random) number of white balls, black balls, respectively, observed in the first $i$ draws $(w_i + b_i = i)$. We shall consider mainly the following stopping rules. RULE A. Let $A > 0$ be a fixed integer. $t_A = A$. RULE B. Let $B > 0$ be a fixed integer. $t_B = \inf \{i\mid b_i = B\}$. RULE C. Let $C > 0$ be fixed. $t_c = \inf \{i|b_i \geqq Cw_i\} = \inf \{i|i \geqq (C + 1)w_i\}$. RULE D. Let $- \infty < D < \infty$ be fixed. \begin{align*}t_D = \inf \{i\mid b_i \geqq \max (1, w_i \log w_i + w_iD)\} \\ = \inf \{i\mid i \geqq \max (w_i + 1, w_i \log w_i + w_i(D + 1))\}.\end{align*} RULE E. Let $\{D_j\}$ be such that $\lim D_j = \infty$. $t_E = \inf \{i\mid b_i \geqq \max (1, w_i \log w_i + w_iD_{w_i})\}.$ Since $w_i \leqq N$ each of these rules is bounded, and thus clearly stops with probability one. Of these rules, Rule B has been investigated most. See [5], [1] and [3]. Rule D has been considered in a recent paper [2] by Darling and Robbins, who show that for any $0 < \alpha < 1$ and a suitable choice of $D$ one can have $P_N(W_D = N) \geqq 1 - \alpha$ uniformly in $N$, where $W_D$ is the total of white balls observed before stopping. (See Section 6). The motivation for consideration of rules A to E stems from a theorem on the limiting distribution, as $N \rightarrow \infty$, of $w_i$, in a sample of fixed size $i$, for various relationships between $i$ and $N$. We restate the theorem here, since we shall need part of it in the sequel. Different parts of the theorem have been proved by various authors. See for example Renyi [8], where also proper references are given. Let $u_i = N - w_i =$ number of unobserved white balls in the sample of size $i (=$ number of white balls in the urn, after $i$ draws). Since there are linear relationships between $w_i, b_i, u_i$ we shall express the limiting distribution for one of these variables only. Let $\Phi$ denote the distribution function of a standard normal variable. We have, THEOREM 1. Let $N \rightarrow \infty$. Case $A$. If $i = (2N\lambda_N)^{\frac{1}{2}}$ where $\lambda_N \rightarrow 0$ then $P_N(b_i = 0) \rightarrow 1$. Case $B$. If $i = (2N\lambda_N)^{\frac{1}{2}}$ where $\lambda_N \rightarrow \lambda$ and $0 < \lambda < \infty$ then $P_N(b_i = k) \rightarrow e^{-\lambda}\lambda^k/k!\quad k = 0, 1, \cdots.$ Case $C$. If $i = a_NN$ where $\xi_NN^{-\frac{1}{2}} < a_N < \log N - \mu_N$ and $\xi_N \rightarrow \infty, \mu_N \rightarrow \infty$ then $P_N((w_i - Ew_i)/(\operatorname{Var} w_i)^{\frac{1}{2}} \leqq x) \rightarrow \Phi(x), - \infty < x < \infty.$ Case $D$. If $i = N \log N + Na_N$ where $a_N \rightarrow a$ and $- \infty < a < \infty$ then $P_N(u_i = k) \rightarrow e^{-\lambda}\lambda^k/k!, \quad k = 0, 1, \cdots \text{where} \lambda = e^{-a}.$ Case $E$. If $i = N \log N + Na_N$ where $a_N \rightarrow \infty$, then $P_N(w_i = N) \rightarrow 1$. We consider mainly the Maximum Likelihood Estimate (MLE) of $N$, denoted by $\hat{N}$. It turns out that if when we stop we have seen $w$ white and $b$ black balls, then $\hat{N} = \hat{N} (w, b)$ and does not depend on the stopping rule used, though the distribution of $\hat{N}$ clearly will depend on the stopping rule. The value of $\hat{N}(w, b)$ is discussed in Section 2. In Section 3 we briefly consider Rule A. It satisfies $P_N(\hat{N} = \infty) > 0$ for all $N \geqq A$. Rule B is discussed in Section 4 and it is shown that $2B\hat{N}/N$ has an asymptotic chi square distribution with $2B$ degrees of freedom, (to be denoted $\chi^2_{2B})$, as $N \rightarrow \infty$. In Section 5 Rule C is considered, and bounds on the distribution of $(\hat{N} - N)N^{-\frac{1}{2}}$ in terms of the normal distribution are given. Rules D and E are considered in Section 6. Let $\lbrack x\rbrack^\ast$ be the largest integer not exceeding $x$. For Rule D it is shown that $\hat{N} - N + \lbrack\lambda\rbrack^\ast$ has an asymptotic Poisson distribution with parameter $\lambda = \exp (-D - 1)$ and for Rule E $P_N(\hat{N} = N) \rightarrow 1$. The exact and asymptotic distributions of the corresponding $t$'s is also considered, and is closely related to the distribution of $\hat{N}$.

Let $\{Y_n, n \geqq 1\}$ be a stationary process with a finite state-space $J$. Let $\delta$ denote a state of $J$ and let $s, t$ denote finite sequences of states of $J$. If $s = (\delta_1, \cdots, \delta_n)$, let $p(s) = P\lbrack(Y_1, \cdots, Y_n) = s\rbrack$. The rank $n(\delta)$ of a state $\delta$ is defined to be the largest integer $n$ such that we can find $2n$ sequences $s_1, \cdots, s_n, t_1, \cdots, t_n$ such that the $n \times n$ matrix $|p(s_i\delta t_j)|$ is nonsingular. The number $n(\delta)$ was first defined by Gilbert [5] and the term rank was first used by Fox and Rubin [4]. A state $\delta$ is called Markovian if $n(\delta) = 1$. It is easy to check that $\delta$ is Markovian if, and only if, $p(s\delta t) = p(s\delta)p(\delta t)/p(\delta)$ for all $s$ and $t$. Suppose that $\mu$ is a fixed state of $J$. Let $J' = J - \{\mu\}$. Assume that $n(\mu) < \infty$. Fox and Rubin have shown that there exists a stationary process $\{X_n\}$ with a countable state-space $I = J' \cup J''$ and a function $f$ on $I$ onto $J$ such that (a) $f(i) = \mu$ if $i \varepsilon J''$ and $f(\delta) = \delta$ if $\delta \varepsilon J'$; (b) states of $J''$ are Markovian states of $\{X_n\}$; and (c) $\{Y_n\}$ and $\{f(X_n)\}$ have the same distribution. Gilbert [5] has shown that $J''$ must have at least $n(\mu)$ elements whereas Fox and Rubin [4] have given an example to show that $J''$ cannot always be chosen to be finite. For $\delta \varepsilon J'$ let $\nu(\delta)$ denote the rank of $\delta$ in $\{X_n\}$. In general $\nu(\delta) \geqq n(\delta)$. But Fox and Rubin have shown that $\{ X_n\}$ can be constructed in such a way that $\nu(\delta) = 1$ whenever $n(\delta) = 1$. Finally they have shown that, if $n(\mu) = 2$, then $\{X_n\}$ can be chosen in such a way that $J''$ has 2 elements and $\nu(\delta) = n(\delta)$ for all $\delta \varepsilon J'$. In this paper we give some conditions under which $J''$ can be chosen to be finite. These conditions are similar to those imposed in [2]. It is shown that $\{X_n\}$ can be constructed in such a way that, for $\delta \varepsilon J', \nu(\delta) = 1$ whenever $n(\delta) = 1$. Finally it is proved that if $N(\mu) = n(\mu)$, then $\nu(\delta) = n(\delta)$ for all $\delta \varepsilon J'$. This generalizes the result proved by Fox and Rubin for the case $n(\mu) = 2$. However, they have given results for the non-stationary case also. The results of this paper were partially reported in [3].

Let $x_1 < x_2 < \cdots < x_n$ be an ordered random sample of size $n$ from the absolutely continuous cdf $F(x)$ with positive density $f(x)$ having a continuous first derivative in a neighborhood of the $p$th population quantile $\nu_p(= F^{-1} (p))$. In order to convert the median or any other "quick estimator" [1] into a test we must estimate its variance, or for large samples its asymptotic variance which depends on $1/f(\nu_p)$. Siddiqui [4] proposed the estimator $S_{mn} = n(2m)^{-1}(x_{\lbrack np\rbrack+m} - x_{\lbrack np\rbrack-m+1})$ for $1/f(\nu_p)$, showed it is asymptotically normally distributed and suggested that $m$ be chosen to be of order $n^{\frac{1}{2}}$. In this note we show that the value of $m$ minimizing the asymptotic mean square error (AMSE) is of order $n^{\frac{1}{5}}$ (yielding an AMSE of order $n^{-\frac{4}{5}}$). Our analysis is similar to Rosenblatt's [2] study of a simple estimate of the density function.