In the first nontrivial case, dimension $p = 3$ and sample size $N = 3$ or 4 (depending on whether or not the mean is known), it is proved that the classical level $\alpha$ normal test of independence of the first component from the others, based on the squared sample multiple correlation coefficient $R^2$, maximizes, among all level $\alpha$ tests, the minimum power on each of the usual contours where the $R^2$-test has constant power. A corollary is that the $R^2$-test is most stringent of level $\alpha$ in this case.

In [3] the author established necessary and sufficient conditions for asymptotic normality of estimates based on simple random sampling without replacement from a finite population, and thus solved a comparatively old problem initiated by W. G. Madow [8]. The solution was obtained by approximating simple random sampling by so called Poisson sampling, which may be decomposed into independent subexperiments, each associated with a single unit in the population. In the present paper the same method is used for deriving asymptotic normality conditions for a special kind of sampling with varying probabilities called here rejective sampling. Rejective sampling may be realized by $n$ independent draws of one unit with fixed probabilities, generally varying from unit to unit, given the condition that samples in which all units are not distinct are rejected. If the drawing probabilities are constant, rejective sampling coincides, with simple random sampling without replacement, and so the present paper is a generalization of [3]. Basic facts about rejective sampling are exposed in Section 2. To obtain more refined results, Poisson sampling is introduced and analyzed (Section 3) and then related to rejective sampling (Section 4). Next three sections deal with probabilities of inclusion, variance formulas and asymptotic normality of estimators for rejective sampling. In Section 8 asymptotic formulas are tested numerically and applications to sample surveys are indicated. The paper is concluded by short-cuts in practical performance of rejective sampling. The readers interested in applications only may concentrate upon Sections 1, 8 and 9. Those interested in the theory of mean values and variances only, may omit Lemma 4.3 and Section 7.

Many procedures have been proposed for handling outlying observations. Most authors use various statistics $T$ to reject one or more observations, if $T$ is too large (or too small), and compute percentage points of the distribution function of $T$. Some authors, as in [11], [13], [15], [16], find statistics with the optimum property of minimizing for certain alternative hypotheses the error of the second kind given the error of the first kind. The observations that are not rejected are used to estimate unknown parameters e.g. the mean. In this paper, we shall consider one of these procedures which gives rise to a one-parameter family of estimators for the mean and compare their risks with those of the Bayes solutions with respect to a one-parameter family of prior distributions. Two simplified models of outliers as will be specified in (2.1) and (2.2) and a quadratic loss function will be investigated. For sample sizes of 3, 6, and 10, the risks are tabulated and plotted. For Model (2.1) the risks of the first procedure are only a little larger than those of the Bayes solutions; in other words, to each admissible strategy there exists an estimator arising from the first procedure with approximately the same risk. Winsorizing the observations [18] should presumably result in an even better approximation; however, numerical results are not available. For Model (2.2), the risks differ considerably. Although this model is rather unrealistic it is included here because of its prominence in the literature. References for tables of percentage points and the distribution functions of various statistics are collected in the table guide of J. A. Greenwood and H. O. Hartley [7]. Bayes solutions are considered by S. Karlin and D. Truax [10], B. de Finetti [3], and T. E. Ferguson [4]. Some other important papers are [1], [12], [14], [18]. In this paper, "stragglers" are random variables that obey a probability law with a larger variance or with a larger or smaller expectation than the "typical" random variables. An "outlier" is an observation that because of its magnitude may be suspected of being a straggler. Thus the term "outlier" is not precisely defined.

For each $n$ let $t_n$ be an estimate (based on $n$ independent and identically distributed observations) of a real valued parameter $\theta$. Suppose that, for each $\theta, n^{\frac{1}{2}}(t_n - \theta)$ is asymptotically normally distributed with mean zero and variance $v(\theta)$. According to R. A. Fisher we then have \begin{equation*}\tag{(1)}v(\theta) \geqq I^{-1}(\theta),\end{equation*} where $I$ is the information contained in a single observation. It is known however that, in the absence of regularity conditions on the sequence $\{t_n\}$, (1) does not necessarily hold for each $\theta$. On the other hand, according to LeCam (1952, 1953, 1958) the set of points $\theta$ for which (1) fails is always of Lebesgue measure zero. This note gives a simple proof of the stated result of LeCam, along the following lines. First a sufficient condition for the validity of (1) at a given value of $\theta$, say $\theta^0$, is obtained. This is a little weaker than the condition that $t_n$ be asymptotically median-unbiased (i.e. $P(t_n < \theta \mid \theta) \rightarrow \frac{1}{2}$ as $n \rightarrow \infty$) uniformly for $\theta$ in some neighborhood of $\theta^0$. It is then shown that the sufficient condition is automatically satisfied at almost all $\theta^0$. The main propositions are stated in the following paragraphs of this section, and the proofs are given in Section 2. The proofs depend on the Neyman-Pearson lemma concerning the optimality of the likelihood ratio test of a simple hypothesis against a simple alternative. This lemma is made available in the present context by means of the well known considerations that an estimate of $\theta$ can provide a test of the value of $\theta$, and that the quality of the resulting test is heavily dependent on the quality of the estimate. A similar application of the Neyman-Pearson lemma to estimation theory is made in Bahadur (1960). It is shown there that if instead of asymptotic variances one considers quantities called asymptotic effective variances, Fisher's bound becomes valid for all $\theta$. Now let $X = \{x\}$ be a sample space of points $x, \mathscr{B}$ a $\sigma$-field of sets of $X$, and $\{P(\cdot \mid \theta): \theta \varepsilon \Theta\}$ a set of probability measures $P(\cdot \mid \theta)$ on $\mathscr{B}$, where $\theta$ is a real parameter and $\Theta$ is an open interval on the real line. It is assumed that the following conditions (i)-(iv) are satisfied. (i) There exists a $\sigma$-finite measure on $\mathscr{B}$, say $\mu$, such that, for each $\theta, P(\cdot \mid \theta)$ admits a probability density with respect to $\mu, f(x \mid \theta)$ say, i.e., \begin{equation*}\tag{(2)}P(B |mid \theta) = \int_B f(x | \theta) d\mu\quad\text{for all} B \varepsilon \mathscr{B}, \theta \varepsilon \Theta.\end{equation*} (ii) For each $x \varepsilon X$, \begin{equation*}\tag{(3)}L(\theta \mid x) = \log f(x \mid \theta)\end{equation*} is a twice-differentiable function of $\theta$ and the second derivative is continuous in $\theta$. (iii) With dashes on $L$ denoting partial differentiation with respect to $\theta$, \begin{equation*}\tag{(4)}0 < E(\{L'(\theta \mid x)\}^2 \mid \theta) \equiv I(\theta) < \infty,\quad E(L'(\theta \mid x \mid \theta) = 0,\end{equation*} and \begin{equation*}\tag{(5)}E(L"(\theta \mid x) \mid \theta) = -I(\theta)\end{equation*} for every $\theta$. (iv) For any given $\theta^0$ in $\Theta$, there exists a $\delta > 0$ and a $\mathscr{B}$-measurable function $M(x)$ such that $|L" (\theta \mid x)| \leqq M(x)$ for all $x \varepsilon X$ and all $\theta \varepsilon (\theta^0 - \delta, \theta^0 + \delta)$, and such that $E(M(x) \mid \theta^0) < \infty; \delta$ and $M$ are, of course, allowed to depend on the given $\theta^0$. The above conditions are a simplification, along the lines of LeCam (1953), of the conditions formulated by Cramer (1946) for his analysis of the likelihood equation. It may be added that the present regularity conditions are in a sense weaker than those of LeCam (1953, 1958), since the method of proof of the latter papers requires local conditions such as (ii)-(iv) and also the existence and consistency of maximum likelihood estimates based on independent observations on $x$. Let $x_1, x_2, \cdots$ denote a sequence of independent and identically distributed observations on $x$. For each $n = 1, 2, \cdots$ write $x^{(n)} = (x_1, \cdots, x_n)$. Let $X^{(n)}$ denote the sample space of $x^{(n)}$, and $\mathscr{B}^{(n)}$ the $\sigma$-field of sets of $X^{(n)}$ which is determined by the given $\mathscr{B}$ in the usual way. For any measure $Q$ on $\mathscr{B}$, the corresponding product measure on $\mathscr{B}^{(n)}$ will be denoted by $Q^{(n)}$. For simplicity, $Q^{(n)}$ is abbreviated to $Q$ in cases where the domain of $Q^{(n)}$ is plain from the context. Now let there be given a sequence $\{t_n\}$ such that $t_n$ is a $\mathscr{B}^{(n)}$-measurable function on $X^{(n)}$ into $\Theta (n = 1, 2, \cdots)$. It is assumed that for each $\theta$ in $\Theta$ there exists a positive constant $v(\theta)$ such that, as $n \rightarrow \infty, n^{\frac{1}{2}}(t_n - \theta)$ is asymptotically normally distributed with mean 0 and variance $v(\theta)$ when $\theta$ obtains. (For a treatment of the case when the present assumption $0 < v < \infty$ is weakened to $0 \leqq v < \infty$, cf. the last paragraph of Section 3.) The given sequence $\{t_n\}$ will remain fixed throughout. We note that \begin{equation*}\tag{(6)}\lim_{n \rightarrow \infty} P(t_n < \theta \mid \theta) = \frac{1}{2}\end{equation*} for each $\theta$ in $\Theta$. PROPOSITION 1. If $\theta^0$ is a point in $\Theta$, and if \begin{equation*}\tag{(7)}\lim\inf_{n \rightarrow \infty} P(t_n < \theta^0 + n^{-\frac{1}{2}} \mid \theta^0 + n^{-\frac{1}{2}}) \leqq \frac{1}{2},\end{equation*} then (1) holds for $\theta = \theta^0$. It follows from Proposition 1 by symmetry that if \begin{equation*}\tag{(8)}\lim\inf_{n \rightarrow \infty} P(t_n > \theta^0 - n^{-\frac{1}{2}} \mid \theta^0 - n^{-\frac{1}{2}}) \leqq \frac{1}{2},\end{equation*} then also (1) holds for $\theta = \theta^0$. Another consequence of Proposition 1 is that if (6) holds uniformly for $\theta$ in some open interval of $\Theta$ then (1) holds for each $\theta$ in that interval. A somewhat weaker conclusion concerning the sufficiency of uniform convergence for (1) has been obtained independently by Rao (1963). The sequence $\{t_n\}$ is said to be superefficient if $v(\theta) \leqq I^{-1}(\theta)$ for all $\theta$ and the inequality is strict for at least one $\theta$. Examples of superefficient estimates were discovered by J. L. Hodges, Jr. (cf. LeCam (1953)). General studies bearing on superefficiency, using methods different from the present ones, were carried out by LeCam (1953, 1958). An informal discussion along lines similar to those of LeCam was given independently by Wolfowitz (1953). It is shown in LeCam (1953) that if $\{t_n\}$ is superefficient then $v(\theta) = I^{-1}(\theta)$ for almost all $\theta$ in $\Theta$; the following more general conclusion is given in LeCam (1958): PROPOSITION 2. The set of all $\theta$ in $\Theta$ for which (1) does not hold is of Lebesgue measure zero. It was observed by Chernoff (1956) that the asymptotic variance of an estimate is always a lower bound to the asymptotic expected squared error; in view of Proposition 2, this observation yields: PROPOSITION 3. $\lim\inf_{n \rightarrow \infty} \{ nE\lbrack (t_n - \theta)^2 \mid \theta\rbrack\} \geqq I^{-1}(\theta)$ for almost all $\theta$ in $\Theta$. The conclusions stated in this section can be extended to the case when $\theta$ is a $p$ dimensional parameter; a brief account of these extensions is given in Section 3. An extension to sampling frameworks more general than the present one of independent and identically distributed observations is described in Section 4.

A solution is given to the problem of how to determine at which points in the interval $\lbrack -1, 1\rbrack$ observations should be taken and what proportion of the observations should be taken at each such point so as to minimize the variance of the predicted value of a polynomial regression curve at a specified point beyong the interval observations. The solution obtained states that the points are to be chosen to be Chebychev points and the number of observations are to be selected proportional to the absolute value of the corresponding Lagrange polynomial at the specified point. The preceding Chebychev solution becomes the minimax solution for the interval $(-1, t),$ provided $t > t_1 > 1$ where $t_1$ is a value satisfying a certain equation. Under the customary normality assumptions, the Chebychev solution to the prediction problem is used to construct a confidence band for a polynomial curve that will possess minimum width at any specified point beyond the interval of observations.

Let $\mathscr{F}$ be an arbitrary class of distribution functions and let $Y_1, Y_2, \cdots, Y_N$ be a sample of size $N$ from some $F \varepsilon \mathscr{F}$. For $0 < P, \gamma < 1$, a statistic $U_N = U(Y_1, \cdots, Y_N)$ is said to be a $(P, \gamma)$ upper tolerance limit for $F$ relative to $\mathscr{F}$, if \begin{equation*}\tag{0} P_F\lbrack 1 - F(U_N) \leqq P\rbrack \geqq \gamma\end{equation*} for all $F \varepsilon \mathscr{F}$. (The interval $(-\infty, U_N\rbrack$ would be called a $1 - P$ content tolerance limit at confidence level $\gamma$ in the terminology of [3].) For certain parametric classes of distributions such as the normal family or the exponential family, tolerance limits are available for all sample sizes greater than or equal to $N = 2$ for all $0 < P, \gamma < 1$. (See, e.g. [5] for normal tolerance limits. In [4] exponential and double exponential tolerance limits are obtained based on the concept of exponential content.) The forms of the tolerance limits in these cases are heavily dependent upon the particular parametric class under consideration. Consequently, if tolerance limits are desired for larger classes of distributions it is necessary to abandon these results in favor of statistics $U_N$ for which the distribution of $1 - F(U_N)$ can be appropriately bounded for all $F$'s in the larger class. Until now, the statistics used for this purpose were certain of the sample order statistics, $X_K = K$th smallest of $Y_1, Y_2, \cdots, Y_N, K = 1, 2, \cdots, N$. These are the traditional non-parametric tolerance limits. Fraser [2] and Robbins [7] have shown that they have desirable uniqueness and optimality properties when $\mathscr{F}$ is the class of distributions absolutely continuous with respect to Lebesgue measure. The non-parametric tolerance limits have one unfortunate disadvantage; namely, for given $P$ and $\gamma$, there is a minimum sample size $N(P, \gamma, K)$ such that the condition $P_F\lbrack 1 - F(X_{N - K}) \leqq P\rbrack \geqq \gamma$ is met only if $N \geqq N(P, \gamma, K)$. Thus, in cases where sampling is very expensive and stringent requirements are made on the tolerance limit (small $P$ and large $\gamma$) or where the statistician is presented with a sample of size $N < N(P, \gamma, 0)$ without the possibility of obtaining additional observations, the only recourse has been to use parametric tolerance limits with the distinct possibility that the limit obtained is meaningless. In this paper we propose a compromise scheme whereby upper tolerance limits of the form $X_{N - K - j} + b(X_{N - K} - X_{N - K - j})$ are constructed which are valid for all $N \geqq 2$ and all $0 < P, \gamma < 1$. These limits satisfy (0) for the class, $\mathscr{F}_U$, of absolutely continuous distribution functions for which $\varphi = -\log (1 - F)$ is convex and, thus, earn the title log-convex (L.C.) tolerance limits. The class $\mathscr{F}_U$ has been studied by Barlow, Marshall and Proschan in [1] and was shown to contain most of the distributions commonly used as models in statistics including the normal and exponential. The distributions in this class are called increasing hazard rate distributions in reliability theory due to the fact that the instantaneous hazard rate $f/(1 - F)$, is non-decreasing on the support of the probability density $f$. We will elaborate on the properties of this class in Section 2. Lower $(P, \gamma)$ tolerance limits of the form $L_N = X_{K + j + 1} - b(X_{K + j + 1} - X_{K + 1})$, which satisfy $P_F\lbrack F(L_N) \leqq P\rbrack \geqq \gamma$ for the class $\mathscr{F}_L$ of absolutely continuous distributions for which $\psi = -\log F$ is convex, will be obtained in Section 3. It is thus of interest to note (in Section 2) that the important class of distributions possessing densities which are Polya frequency functions of order 2 $(PF_2)$ is contained in $\mathscr{F}_L \cap \mathscr{F}_U$. Hence both upper and lower tolerance limits are available for members of this class. Tables of the $b$-factors needed for both upper and lower tolerance limits are given in Section 4 for $j = 1$ and all combinations of $P = .500, .250, .100, .050$ and $\gamma = .90, .95, .99$. The sample sizes range from $N = 2$ to the smallest value of $N$ for which the usual non-parametric tolerance limit can be used. At this point the L.C. tolerance limits are equal to the non-parametric ones. No questions of optimality are considered here; however, some comparisons of the L.C. tolerance limits with the usual normal and exponential tolerance limits are made by means of Monte Carlo sampling in Section 5. Some extensions of the theory will also be considered in Section 5.

The basic problem considered here involves $k$ experimental categories. The experimenter must decide none of the $k$ categories is better than the control or decide a certain category is better. For this problem a fixed sample size procedure $\delta^\ast_m$ is given. With a definite loss function and a cost $c > 0$ per observation, $\delta^\ast_m$ and other fixed sample size procedures are compared in a certain asymptotic sense as $c \rightarrow 0$. In particular, $\delta^\ast_m$ is shown to be an optimal fixed sample size procedure in this asymptotic sense. By appealing to asymptotic results the procedure $\delta^\ast_m$ is compared with sequentially designed procedures.

Exact power for normal location alternatives is obtained for the bivariate sign tests of Blumen [1] and Hodges [5]. A recursive scheme, used in conjunction with a computer, permits comparison of the two tests for sample sizes $n = 8(1)12$. Efficiency values relative to Hotelling's bivariate $T^2$ test are also obtained for the test of Hodges. Slight power differences are noted for the sign tests along with surprisingly high power when compared with the $T^2$.

This paper is a continuation of [8], and considers the sequential compound decision problem for the case where the component decisions are of the simple versus simple hypothesis testing type, and thus can be stated in terms of testing whether $\theta = 0$ or $\theta = 1$. The loss for the compound decision is taken to be the average of the losses in the component decisions, and the risk for the compound decision is defined correspondingly. Let $R( )$ denote the Bayes envelope function of the component problem. In [8] two sequences of compound decision rules $\{T^\ast_n\}$ and $\{\hat T_n\}$ are exhibited, such that for $n$ sufficiently large, the risk incurred by $\hat{T}_n$ never exceeds $R(\vartheta_n) + \epsilon$ where $\vartheta_n$ is the average of the true $\theta$-values in the $n$ first components, and this holds uniformly in all possible sequences of $\theta$'s: for $T^\ast_n$ a corresponding statement is valid provided $R( )$ is differentiable for all $0 \leqq \eta \leqq 1$. Here we prove that for any sequence of $\theta$-values, the difference between the loss incurred by $\hat{T}_n$ and $R(\vartheta_n)$ converges to zero in probability, and under the differentiability assumption a corresponding statement holding with probability one is proved for $T^\ast_n$. Numerical data is provided to indicate the rate of convergence.

This paper describes how data from a multinomial distribution, and in particular data in the form of a contingency table, may be studied by using a prior distribution of the parameters and expressing the results in the form of a posterior distribution, or some aspects thereof, of the parameters. The analysis used must depend on the prior distribution and the form described here only applies to a certain type of prior knowledge but, for reasons given below, it is believed that this type is of frequent occurrence. The binomial situation is first considered and the results obtained there suggest a general result for the multinomial distribution, which is then established. A few remarks on Bayesian analysis in general enable the result to be applied, first to certain multinomial problems and then, with the aid of another general result, to contingency tables. The method used there has close connections with the Analysis of Variance and these connections are examined, particularly with a view to simplifying the analysis of contingency tables involving three or more factors.

After discussing the sequential decision problem in the general case and its Bayes solution as given by Wald [11], LeCam [8] and others, in Section 2, we give the integral equation satisfied by the Bayes risk. In Section 3 we specialise to Markov dependent observations and in the last section we give an illustrative example.

A generalization of Tukey's polykays [3], [4], and [5], was made by Hooke [1] in reference to sampling from a two-way array or population. These generalized polykays were christened "bipolykays." Smith [2] also developed these functions of degree two independently. Working with certain structural patterns in the analysis of variance, a set of functions denoted by $\Sigma$'s were introduced by Wilk and Kempthorne [6] and formally defined and extended to include all "balanced" population structures by Zyskind [7]. Because of certain "nice" properties of symmetric means of degree two, the $\Sigma$ expansions were found to be relatively simple and could be defined for a large class of structures. In contrast, the work on the extension of the polykays was limited only to sampling from a two-way population structure though polykays of higher degrees were also considered. Zyskind [8] recognized the equivalence of Hooke's bipolykays of degree two and a certain subset of the $\Sigma$'s, and conjectured the equivalence of appropriately extended polykays with the whole set of $\Sigma$'s for all balanced structures. In this paper an extension of the bipolykays of degree two to "$n$-way-polykays" (henceforth referred to as generalized polykays) is made to encompass all balanced structures (as defined in [7]). (Since this paper was submitted, general definitions of polykays and symmetric means of all degrees have been formulated, properties of these developed, and the basic results applied to obtaining variances and co-variances of estimates of components of variation in certain two and three-factor balanced structures.) The equivalence of these generalized polykays and the $\Sigma$'s is then shown.

Let $x_1, x_2, x_3, x_4$ be jointly distributed with a quadrivariate normal distribution with mean 0, and correlation matrix $\{\rho_{ij}\}$. The orthant probability, i.e. the probability that all the $x_i$'s will be simultaneously positive, is not, in general, given by a closed expression; but is easily computed in a special class of cases, here called orthoscheme probabilities. It is explicitly shown how the general orthant probability can be expressed as a linear combination of six orthoscheme probabilities. Orthoscheme probabilities have been tabulated by the author and instructions for the use of this table [1] are given. In addition, an abridged table is appended.

A lemma is given first which provides an easy method of expressing the product of an $s$th order Vandermonde type determinant and the $k$th and $l$th $(k, l \geqq 0)$ powers of the $r$th and $h$th $(r, h \leqq s)$ elementary symmetric functions (esf's) respectively as a linear compound of determinants. The lemma extends itself readily to the product of powers of any number of esf's up to the $s$th. Using this lemma and some reduction formulae for certain special types of Vandermonde type determinants, a second lemma has been proved to show that certain formulas for the moments of esf's in $s$ non-null characteristic roots $\lambda_i(0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_s < \infty)$ of a matrix can be easily derived from corresponding formulas for the moments of corresponding esf's in $s$ non-null roots, $\theta_i, (0 < \theta_1 \leqq \cdots \leqq \theta_s < 1)$ of another matrix and vice versa. Illustrations are given explaining both lemmas.

This paper is a study of Doeblin Ratio limit laws, the weak and strong laws of large numbers, and the Central Limit theorem for Markov Renewal processes. A general definition of these processes is given in Section 2. The means and variances of random variables associated with recurrence times are computed in Section 4. When restricted to the special case of a Markov chain, certain of the results of Sections 5 and 6 strengthen known results.

The existence of the class of orthogonal projections which map an arbitrary $q$-variate weakly stationary stochastic process again into a $q$-variate process contained in the span of $p(\leqq q)$ of its component processes is established. Mimicking the definitions of the partial and multiple correlation coefficients (e.g., Anderson, 1958), these projections are used to define partial and multiple coefficients of coherence, thus providing the foundation for the multivariate covariance and correlation analyses for weakly stationary processes employed in special cases by Tick (1963) and Jenkins (1963). Some of the properties of the partial and multiple correlation coefficients are established for the corresponding coefficients of coherence. In particular, formulas are established for generating these parameters iteratively. When used for the sample coefficients of coherence, these formulas provide useful methods of defining and constructing estimates of the multiple and partial coefficients of coherence from the usual estimates of the ordinary coefficient of coherence. Results due to Goodman (1963) concerning the distributions of these estimators when the process is Gaussian are indicated.

If $\beta_1$ and $\beta_2$ are not identically zero $\sigma$-finite invariant measures for a measurable invertible ergodic transformation $S$ on a measure space, and $\beta_1(E) > 0$ implies $\beta_2(E) > 0$ for measurable sets $E$, then $\beta_2 = c\beta_1$ for some constant $c \neq 0$ ([4], p. 35). In this paper a corresponding result will be proved for stationary measures of a Markov process (Theorem 1). Theorem 1 is a generalization of the corollary of [6], p. 863. In that paper, the authors impose conditions ensuring that the shift transformation has no wandering sets of positive measure, and then they use Hopf's theorem. In Section 3, some new and known results are seen to follow readily from Theorem 1. The recurrence condition introduced by Harris [5] is discussed, and Theorem 1 is used to give a new proof of the uniqueness theorem of [5] independent of the existence of stationary measures, and generalizing the theorem to $\sigma$-fields which are not necessarily separable.

The first clear and precise statement of the axioms of qualitative probability was given by de Finetti ([1], Section 13). A more detailed treatment, based however on more complex axioms for conditional qualitative probability, was given later by Koopman [5]. De Finetti and Koopman derived a probability measure from a qualitative probability under the assumption that, for any integer $n$, there are $n$ mutually exclusive, equally probable events. L. J. Savage [6] has shown that this strong assumption is unnecessary. More precisely, he proves that if a qualitative probability is only fine and tight, then there is one and only one probability measure compatible with it. No property equivalent to countable additivity has been used as yet in the development of qualitative probability theory. However, since the concept of countable additivity is of such fundamental importance in measure theory, it is to be expected that an equivalent property would be of interest in qualitative probability theory, and that in particular it would simplify the proof of the existence of compatible probability measures. Such a property is introduced in this paper, under the name of monotone continuity. It is shown that, if a qualitative probability is atomless and monotonely continuous, then there is one and only one probability measure compatible with it, and this probability measure is countably additive. It is also proved that any fine and tight qualitative probability can be extended to a monotonely continuous qualitative probability, and therefore, contrary to what might be expected, there is no loss in generality if we consider only qualitative probabilities which are monotonely continuous. At the present time there is still a controversy over the interpretation which should be given to the word probability in the scientific and technical literature. Although the present writer subscribes to the opinion that this interpretation may be different in different contexts, in this paper we do not enter into this controversy. We simply remark that a qualitative probability, as a numerical one, may be interpreted either as an objective or as a subjective probability, and therefore the following axiomatic theory is compatible with both interpretations of probability.