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Many physical problems are described by stochastic processes with stationary increments. We present a general description of such processes. In particular we give an expression of a process in terms of its increments and we show that there are two classes of processes: diffusion and asymptotically stationary. Moreover, we show that the nth increments are given by a linear filtering of an arbitrary stationary process. STOCHASTIC PROCESSES; STATIONARY; DIFFUSION; LINEAR FILTERING; CORRELATION THEORY Stochastic processes (s.p.) are very often used as statistical models to represent random physical phenomena. In most cases the assumption of stationarity is introduced, either for mathematical simplifications or for physical reasons. Nevertheless there are some situations in which the observed phenomenon is nonstationary, even if it is driven by a stationary source. The best example is the phase ((t) of an oscillator the instantaneous frequency F(t) of which is stationary. Thus we have

The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.

An algorithm is presented for computing the unique stationary distribution of an infinite regular stochastic matrix of a structural form subsuming both upper-Hessenberg and generalized renewal matrices of this kind. Convergence is elementwise, monotone from above, from information within finite truncations, of increasing order.

The Monte-Carlo method makes use of sequences of “random numbers” which are often defined by purely arithmetical properties. That suggests the interpretation of the Monte-Carlo method in analytical, nonstatistical terms. Random numbers are replaced by “uniformly distributed sequences”. A short study of such sequences and their realizations is made. Similarly, it happens that random functions can be replaced by nonrandom functions, defined by properties of “temporal mean values,” like Mf = lim T→∞ 1 2T ∫ −T −T f(t) dt . Functions for which Mf and M f (t) f(t + τ) (the correlation function of f) exist are called stationary functions. A study is made of the correlation function γ(τ), which is a positive-definite function, the Fourier transform of a positive bounded measure. The class of stationary functions contains, essentially, almost periodic functions and pseudorandom functions, for which limτ → ∞ γ(τ) = 0. Processes are given for the construction of pseudorandom functions, involving uniformly distributed sequences. Through convolution by an integrable function a stationary function f is transformed into a stationary function of the same category, but possibly more regular (continuous, differentiable,…) than f. The class of all stationary functions does not have a good algebraic structure, but can be embedded in a Banach space, the Marcinkiewicz space, and contains linear subspaces. The most important of these subspaces is generated by the translation of a given stationary function; in this space a harmonic analysis is possible. Some final remarks are made about the “asymptotic measure,” i.e., the distribution of the values of a stationary function, and the effect of a change of scale. In this paper, only some elementary proofs are given. In the appendices, the reader will find the proofs of those essential theorems which are not contained in the main text. Nevertheless the proofs of a number of useful theorems are too long and too technical to be developed here. References are given, in which the reader will find all explanations he may wish. This paper will be followed by a second one, in which the theory of stationary functions will be applied to the construction of turbulent solutions of Navier-Stokes equations.

The first-excursion probability of a stationary narrow-band Gaussian process with mean zero has been studied. Within the framework of point process approach, series approximations derived from the theory of random points and approximations based on the maximum entropy principle have been developed. With the aid of numerical examples, merits of the approximations proposed previously as well as of those developed in this paper have been compared. The results indicate that the maximum entropy principle has not produced satisfactory approximations but the approximation based on nonapproaching random points is found to be the best among all the approximations proposed herein. A conclusion drawn from the present and the previous studies is that the point process approach produces a number of useful approximations for the first-excursion probability, particularly those based on the concepts of the Markov process, the clump-size, and the nonapproaching random points.

The coefficient of coherence is defined for bivariate weakly stationary stochastic processes which have spectral distributions dominated by a fixed Lebesgue-Stieltjes measure. This quantity is shown to possess two of the important properties which make the ordinary correlation coefficient a desirable measure of linear regression for pairs of random variables. This provides a justification for the already common use of the coefficient of coherence as a measure of linear-regression for pairs of stationarily correlated, weakly stationary time series.

In Moran's theory of reservoirs, the system is subject to a stationary sequence of stochastically fluctuating but mutually independent inflows, with a draft policy governed by current levels. The theory gives the resulting distribution of levels, including the probability of spilling and of emptying. An example in engineering terms has recently been published by Langbein. The present paper extends this analysis to the case where the inflows exhibit “persistence”, with correlation pattern assumed to be Markovian. After an initial section summarising the probability concepts required, the method for correlated inflows is illustrated by a simple example which approximately parallels Langbein's independent-inflows example. Explicit expressions are obtained for the distribution of levels, in terms of the correlation coefficient ϱ between consecutive inflows. For a symmetrical input distribution, of standard deviation σ, with constant draft equal to mean input, a typical result is that the probability of spilling is 1 {2 + 2 3 H(1 − ϱ)} for a reservoir of capacity Hσ. A final section discusses modifications to allow for seasonal inflow effects, losses due to seepage and evaporation, and other elaborations.

This study concerns the relative efficiency of the least-squares estimator of the mean of a covariance stationary sequence that has a constant mean and an autoregressive representation of order p. Our results augment those in [2]. The results in Section 1 include for any positive integer p the variance of the LSE, the BLUE in terms of autoregressive coefficients, the variance of the BLUE and the range of applications of Ylvisaker's theorem [7] regarding a lower bound on relative efficiency. For second-order schemes it also includes bounds on parameter values for relative efficiency >0.80. Graphical results for the first-order scheme in Section 2 add to the detailed study made in [2] and imply that the relative desirability of the LSE should be based on variance as well as relative efficiency considerations.

The spreading of excitation in a random chain was considered as a Markov process of the first order. The stationary probability distribution of the excitation as a functional of the probability distribution of distances in the chain has been found.

The set of stationary probability vectors arising when the transition probabilities of an n-state Markov chain are perturbed, is defined. They are shown to lie in a convex cone bounded by at most $3n$ hyperplanes. There is an infinite number of transition matrices which will yield a given stationary probability vector. The effect of the same perturbation of this set of matrices is examined and several relations are established. It is shown that the largest perturbations in the stationary probability vector tend to occur for diagonally dominant matrices and some particular results are given for this case.

It is shown in this paper that if a continuous stationary stochastic process is given on the unit interval, a stationary extension of the given process exists on the whole line.

This paper is concerned with the spectral analysis of wide sense stationary time series which possess a spectral density function and whose fourth moment functions satisfy an integrability condition (which includes Gaussian processes). Consistent estimates are obtained for the spectral density function as well as for the spectral distribution function and a general class of spectral averages. Optimum consistent estimates are chosen on the basis of criteria involving the notions of order of consistency and asymptotic variance. The problem of interpolating the estimated spectral density, so that only a finite number of quantities need be computed to determine the entire graph, is also discussed. Both continuous and discrete time series are treated.

A formula is given for the stationary value of a function of any number of variables in terms of the values of the function and its differential coefficients at any point in the neighbourhood of the stationary position.

Summary Gnedenko's classical work [1] on the limit of the distribution of the maximum of a sequence of independent random variables is extended to the distribution of the maximum of a sequence of random variables defined on a finite Markov chain.

Nonequilibrium stationary thermodynamic systems are described by stationary random Markov processes with discrete time. For given macroscopic conditions, the condition of maximum path entropy determines the process.

Let $\{X_n, n = 0, \pm 1, \cdots\}$ be a real valued discrete parameter stationary stochastic process on a probability space $(\Omega, \mathscr{F}, P);$ for each $n = 1, 2, \cdots$, let $Z_n = \max (X_1, \cdots, X_n)$. We shall find general conditions under which the random variable $Z_n$ has a limiting distribution function (d.f.) as $n \rightarrow \infty$; that is, there exist sequences $\{a_n\}$ and $\{b_n\}, a_n > 0$, and a proper nondegenerate d.f. $\Phi(x)$ such that \begin{equation*}\tag{1.1}\lim_{n \rightarrow \infty} P\{Z_n \leqq a_nx + b_n\} = \Phi(x)\end{equation*} for each $x$ in the continuity set of $\Phi(x)$. The simplest type of stationary sequence $\{X_n\}$ is one in which the random variables are mutually independent with some common d.f. $F(x)$. In this case, $Z_n$ has the d.f. $F^n(x)$ and (1.1) becomes \begin{equation*}\tag{1.2}\lim_{n \rightarrow \infty} F^n(a_nx + b_n) = \Phi(x).\end{equation*} It is well known that in (1.2) $\Phi(x)$ is of one of exactly three types; necessary and sufficient conditions on $F$ for the validity of (1.2) are also known [9]. The three types are usually called extreme value d.f.'s [10]. Theorem 2.1 gives the limiting d.f. of $Z_n$ in a stationary sequence satisfying a certain condition on the upper tail of the conditional d.f. of $X_1$, given the "past" of sequence: the limiting d.f. is a simple mixture of extreme value d.f.'s of a single type. These are the same kind of d.f.'s found by us [3] to be the limiting d.f.'s of maxima in sequences of exchangeable random variables. The conditions of Theorem 2.1 are specialized to exchangeable and Markov sequences, and Theorem 2.2 extends the methods of Theorem 2.1 to general (not necessarily stationary) Markov sequences. It is shown that stationary Gaussian sequences, except for the trivial case of independent, identically distributed Gaussian random variables, do not obey the requirements of the hypothesis of Theorem 2.1: hence, Sections 3, 4, and 5 are devoted to a detailed study of the maximum in a stationary Gaussian sequence. Theorem 3.1 provides conditions on the rate of convergence of the covariance sequence to 0 which are sufficient for $Z_n$ to have the same extreme value limiting d.f. as in the case of independence, namely, $\exp (-e^{-x})$. The relation of these conditions to the spectral d.f. of the process is also discussed. A weaker condition on the covariance sequence ensures the "relative stability in probability" of $Z_n$ (Theorem 4.1). Theorem 5.1 describes the behavior of $Z_n$ when the spectrum has a discrete component with "not too many large jumps" and a "smooth" continuous component: when properly normalized, $Z_n$ converges in probability to a random variable representing the maximum of the process corresponding to the discrete spectral component. A special case was given by us in [2]. We now summarize some known results used in the sequel. The extreme value d.f.'s are continuous, so that (1.2) holds for all $x$; furthermore, this holds if and only if it holds for all $x$ satisfying $0 < \Phi(x) < 1.$ (1.2) implies that for all such $x$ $0 < F^n (a_nx + b_n) < 1,\quad\text{for all large} n,$ and \begin{equation*}\tag{1.3}\lim_{n \rightarrow \infty} F(a_nx + b_n) = 1.\end{equation*} Let $x_\infty$ be the supremum of all real numbers $x'$ for which $F(x') < 1$; then, for all $x$ satisfying $0 < \Phi(x) < 1$, we have \begin{equation*}\tag{1.4}\lim_{n \rightarrow \infty} a_nx + b_n = x_\infty.\end{equation*} From (1.3), and the asymptotic relation $-\log F \sim (1 - F), F \rightarrow 1$, we see that (1.2) holds if and only if \begin{equation*}\tag{1.5}\lim_{n \rightarrow \infty} n\lbrack 1 - F (a_nx + b_n)\rbrack = -\log \Phi(x).\end{equation*}

Let $\{X(t), t = \cdots -1, 0, 1, \cdots\}$ be a $P$ dimensional zero mean stationary Gaussian time series, $X(t) = \begin{pmatrix}X_1(t)\\X_2(t)\\\vdots\\X_P(t)\end{pmatrix}$ we let $R(\tau) = EX(t)X' (t + \tau)$, where $R(\tau) = \{R_{ij}(\tau), i,j = 1, 2, \cdots P\}$, and $F(\omega) = (2\pi)^{-1} \sum^\infty_{\tau=-\infty}e^{-i\omega\tau}R(\tau)$. It is assumed that $\sum^P_{i,j=1} \sum^\infty_{\tau=-\infty} |\tau| |R_{ij}(\tau)| < \infty$, and hence $F(\omega)$ exists and the elements possess bounded derivatives. It is further assumed that $F(\omega)$ is strictly positive definite, all $\omega$. Knowledge of $F(\omega)$ serves to specify the process. $F(\omega)$, and $S$, the covariance matrix of $x = \begin{pmatrix}x_1 \\ x_2\ \\ vdots\\x_P\end{pmatrix}$, a Normal $(0, S)$ random vector are known to enjoy many analogous properties. (See [7].) To cite two examples, the hypothesis that $X_i(s)$ is independent of $X_j(t)$ for $i \neq j = 1, 2, \cdots P$, any $s, t$, is equivalent to the hypothesis that $F(\omega)$ is diagonal, all $\omega$, while the hypothesis that $x_i$ is independent of $x_j$, for $i \neq j = 1, 2, \cdots P$ is equivalent to the hypothesis that $S$ is diagonal. The conditional expectation of $x_1$, given $x_2, \cdots x_P$ is \begin{equation*}E(x_1\mid x_2, \cdots x_P) = S_{12}S^{-1}_{22}\begin{pmatrix}x_2 \\ \vdots \\ x_P\end{pmatrix}, S = \bigg(\begin{array}{c|c} S_{11} & S_{12} \\ \hline S_{21} & S_{22}\end{array} \bigg)\end{equation*}. The corresponding regression problem for stationary Gaussian time series goes as follows. If \begin{equation*}E\{X_1(t)\mid X_2(s), \cdots X_P(s), s = \cdots -1, 0, 1, \cdots\} = \sum^P_{j=2} \sum^\infty_{s=-\infty} b_j(t - s)X_j(s)\end{equation*} then $B(\omega)$, defined by $B(\omega) = (B_2(\omega), \cdots B_P(\omega)), B_j(\omega) = \sum^\infty_{s=-\infty} b_j(s)e^{i\omega s}$ satisfies \begin{equation*}B(\omega) = F_{12}(\omega)F_{22}^{-1}(\omega), \quad F(\omega) = \bigg(\begin{array}{c|c}f_{11}(\omega) & F_{12}(\omega) \\ \hline F_{21}(\omega) & F_{22}(\omega)\end{array} \bigg).\end{equation*} It is interesting to ask how well these and similar analogies carry over to sampling theory and hypothesis testing. Goodman [3] gave a heuristic argument to support the conclusion that $\hat{F}_X(\omega_k)$, a suitably formed estimate of the spectral density matrix $F(\omega_k)$ has the complex Wishart distribution. The question is met here by the following results. Firstly if $\hat{F}_X(\omega_l), l = 1, 2, \cdots M$ are estimates of the spectral density matrix, each consisting of averages of $(2n + 1)$ periodograms based on a record of length $T$, with the $\omega_l$ equally spaced and $(2n + 1)M \leqq \frac{1}{2} T$, then it is possible to construct, on the same sample space as $X(t), M$ independent complex Wishart matrices $\hat{F}{\bar{\bar{X}}}(\omega_l), l = 1, 2, \cdots M$ such that $\{\hat{F}_X(\omega_l), l = 1, 2, \cdots M\}$ converge simultaneously in mean square to $\{\hat{F}_{\bar{\bar{X}}}(\omega_l), l = 1, 2,\cdots M\}$, as $n, M$ get large. Secondly, it is legitimate to use the natural analogies from multivariate analysis to test hypotheses about time series. One example is presented, as follows. The likelihood ratio test statistic for testing $S$ diagonal is $|\hat{S}|/\mathbf{\prod}^P_{i=1} \hat{s}_{ii}$ where $\hat{S} = \{\hat{s}_{ij}$ is the sample covariance matrix. The analogous statistic $\psi$ for testing $X_i(s), X_j(t)$ independent, $i,j 1 = 2, \cdots P$ from a record of length $T$ is $\psi = \prod^M_{l=1} \lbrack|\hat{F}_X(\omega_l)|/\prod^P_{i=1} \hat{f}_{ii}(\omega_l)\rbrack$ where $\hat{F}_X(\omega_l) = \{\hat{f}_{ij}(\omega_l)\}$ are the sample spectral density matrices as above. Letting ${\bar{dbar{\psi}}} = \prod^M_{l=1} \lbrack|\hat{F}_{\bar{\bar{x}}}(\omega_l)|/\prod^P_{i=1} \hat{h}_{ii}(\omega_l)\rbrack$ where $\hat{F}_{\bar{\bar{x}}}(\omega_l) = \{\hat{h}_{ij}(\omega_l)\}$ are the independent complex Wishart matrices referred to above, we show $EC_{n,M} |\log \psi - \log {\bar{\bar{\psi}}} \rightarrow 0$ for large $n, M$, where $C_{n,M}$ are chosen to make the result non-trival. The method of proof applies to any statistic which is a product over $l$ of sufficiently smooth functions of the entries of $\hat{F}_X(\omega_l)$. Applications to estimation and testing in the regression problem will appear elsewhere [8]. The distribution theory of functions of complex Wishart matrices has been well investigated by a number of authors [3] [5] [6], and hence can be easily applied here to statistics like ${\bar{\bar{\psi}}}$. The results above are shown for $P = 2$, it is clear that the proofs extended to any (fixed) finite $P$. The proofs proceed as follows, via a theorem which has somewhat more general application. For each $T$, let $X$ be the $2 \times T$ random matrix $X = \binom{X_1}{X_2} = \begin{pmatrix}X_1(1), \cdots, X_1(T)\\X_2(1), \cdots, X_2(T)\end{pmatrix}$ and let the $2T \times 2T$ covariance matrix $\Sigma$ be given by $\Sigma = \begin{pmatrix}\sum_{11} \sum_{12} \\ \sum_{21} \sum_{22}\end{pmatrix}$ where $\Sigma_{ij} = EX_i'X_j. \{\hat{F}_X(\omega_l)\}$, the sample spectral density matrices described above based on a record of length $T$, are each of the form $\hat{F}_X(\omega_l) = T^{-1}XQX'$ where $Q$ is a $T \times T$ circulant matrix with largest eigenvalue $ = T(2n + 1)^{-1} \leqq \frac{1}{2}M <

This paper presents a general approach to the derivation of series expansions of second-order wide-sense stationary mean-square continuous random process valid over an infinite-time interval. The coefficients of the expansion are orthogonal and convergence is in the mean-square sense. The method of derivation is based on the integral representation of such processes. It covers both the periodic and the aperiodic cases. A constructive procedure is presented to obtain an explicit expansion for a given spectral distribution.

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables, and write $Z_n$ for the maximum of $X_1, X_2, \cdots X_n$. Then there are two well known theorems concerning the limiting behaviour of the distribution of $Z_n$. (See, for example, Gumbel [7].) Firstly, if for some sequence of pairs of numbers $a_n, b_n$, the quantities $a^{-1}_n(Z_n - b_n)$ have a non-degenerate limiting distribution as $n \rightarrow \infty$, then this limit must take one of three forms. Secondly, if $c_n = c_n(\xi)$ is defined by $P\lbrack X > c_n\rbrack \leqq \xi/n \leqq P\lbrack X \geqq c_n\rbrack$, then $P\lbrack Z_n \leqq c_n\rbrack$ tends to $e^{-\xi}$ as $n \rightarrow \infty$. Suppose now that we drop the assumption of independence of the $X_j$, and require instead that the sequence $\{X_n\}$ be a stationary stochastic process: then it might be expected that similar results will hold, at least if $X_i$ and $X_j$ are nearly independent when $|i - j|$ is large. In Section 2 it will be shown that, if the process $\{X_n\}$ is uniformly mixing, then the only possible non-degenerate limit laws of $a^{-1}_n(Z_n - b_n)$ are just those that occur in the case of independence, and that the only possible limit laws of $P\lbrack Z_n \leqq c_n\rbrack$ are of the form $e^{-k\xi}, k$ being some positive constant not greater than one. The uniform mixing property is rather strong at first sight. It is however clear that some restriction is necessary, at least for example to ergodic processes, and the parallel which exists to a certain extent with normed sums of random variables suggests that uniform mixing hypotheses may be appropriate. (Cf. Rosenblatt [9]). In the independent case converse results hold, as we have already observed for the second problem, giving necessary and sufficient conditions for the existence of the limits. We give some results concerning this problem in Section 3, but they are not altogether satisfactory. Berman ([2] and especially [3]) has investigated the same problem, under somewhat different conditions. His results for the Gaussian case in [3], however, include those which can be obtained from Lemmas 1 and 2 of the present paper, and in consequence we do not reproduce them here. The second problem mentioned above was considered by Watson [10] for $m$-dependent stationary processes, and his paper did in fact suggest the present investigations. His results are contained in Section 3. Certain results were announced without proof by Chibisov [4], but these appear not to overlap our results.

Introduction. Consider a sequence of random variables I Xk: k = 1, 2, * * * I obeying the law of large numbers, i.e., there exists a constant c such that for every e>0 the sequence of probabilities P { n-1 k_ Xk-ci >e} =Pn(e) converges to zero as n-* oo. The object of the present paper is to study the relationships among an exponential convergence rate (i.e., Pn(e) = Q(pn) for p =p(e) < 1), the existence of the individual moment generating functions and the stochastic structure of the sequence { Xk }. Papers containing related studies (e.g., [1; 3; 5]) have treated the case of independent random variables and demonstrated exponential convergence under the hypothesis that the moment generating functions exist. The present paper studies the extent to which an exponential convergence rate implies the existence of the moment generating function and conversely. In particular, satisfactory necessary and sufficient conditions (Theorem 2) are found for exponential convergence of sequences of independent (not necessarily identically distributed) random variables. In the first section it is proved that an exponential rate of convergence for aniy stationary sequence necessarily implies the existence of the moment generating function of the random variables. Conversely an example is constructed showing that restrictions on the size of the variables cannot be sufficient to insure exponential convergence for the general stationary sequence. The terms size, smallness, etc., as used throughout the paper, refer essentially to the tail probabilities. The case of independence is treated in ?2 where it is shown that the existence of all the moment generating functions on the same interval is necessary and that a growth restriction on the product of the first n of these generating functions is necessary and sufficient. In ?3 it is shown that the existence of the moment generating functions is necessary for exponential convergence of the averages of a function of the variables of a Markov sequence having stationary transition probabilities. If the process satisfies Doeblin's condition then this condition is prov,ed sufficient. ?4 contains an example showing that the existence of moment generating functions

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.

Considerations on stochastic models frequently involve sums of dependent random variables (rv's). In many such cases, it is worthwhile to know if asymptotic normality holds. If so, inference might be put on a nonparametric basis, or the asymptotic properties of a test might become more easily evaluated for certain alternatives. Of particular interest, for example, is the question of when a weakly stationary sequence of rv's possesses the central limit property, by which is meant that the sum $\sum^n_1 X_i$, suitably normed, is asymptotically normal in distribution. The feeling of many experimenters that the normal approximation is valid in situations "where a stationary process has been observed during a time interval long compared to time lags for which correlation is appreciable" has been discussed by Grenander and Rosenblatt ([10]; 181). (See Section 5 for definitions of stationarity.) The general class of sequences $\{X_i\}_{-\infty}^\infty$ considered in this paper is that whose members satisfy the variance condition \begin{equation*}\tag{1.1}\operatorname{Var} (\sum^{a+n}_{a+1} X_i) \sim nA^2\text{uniformly in} a (n \rightarrow \infty) (A^2 > 0).\end{equation*} Included in this class are the weakly stationary sequences for which the covariances $r_j$ have convergent sum $\sum_1^\infty r_j$. A familiar example is a sequence of mutually orthogonal rv's having common mean and common variance. As a mathematical convenience, it shall be assumed (without loss of generality) that the sequences $\{X_i\}$ under consideration satisfy $E(X_i) \equiv 0$, for the sequences $\{X_i\}$ and $\{X_i - E(X_i)\}$ are interchangeable as far as concerns the question of asymptotic normality under the assumption (1.1). As a practical convenience, it shall be assumed for each sequence $\{X_i\}$ that the absolute central moments $E|X_i - E(X_i)|^\nu$ are bounded uniformly in $i$ for some $\nu > 2$ ($\nu$ may depend upon the sequence). When (1.1) holds, this is a mild additional restriction and a typical criterion for verifying a Lindeberg restriction ([15]; 295). We shall therefore confine attention to sequences $\{X_i\}$ which satisfy the following basic assumptions (A): \begin{equation*}\tag{A1}E(X_i) \equiv 0,\end{equation*}\begin{equation*}\tag{A2}E(T_a^2) \sim A^2 \text{uniformly in} a (n \rightarrow \infty) (A^2 > 0),\end{equation*}\begin{equation*}\tag{A3}E|X_i|^{2+\delta} \leqq M (\text{for some} \delta > 0 \text{and} M < \infty),\end{equation*} where $T_a$ denotes the normed sum $n^{-\frac{1}{2}} \sum^{a+n}_{a+1} X_i$. Note that the formulations of (A2) and (A3) presuppose (A1). We shall say, under assumptions (A), that a sequence $\{ X_i\}$ has the central limit property (clp), or that $T_1$ is asymptotically normal (with mean zero and variance $A^2$), if \begin{equation*}\tag{1.2}P\{(nA^2)^{-\frac{1}{2}}\sum^n_1 X_i \leqq z\} \rightarrow (2\pi)^{-\frac{1}{2}} \int^z_{-\infty} e^{-\frac{1}{2}t{}^2}dt\quad (n \rightarrow \infty).\end{equation*} The assumptions (A) do not in general suffice for (1.2) to hold. (The reader is referred to Grenander and Rosenblatt ([10]; 180) for examples in which (1.2) does not hold under assumptions (A), one case being a certain strictly stationary sequence of uncorrelated rv's, another case being a certain bounded sequence of uncorrelated rv's.) It is well known, however, that in the case of independent $X_i$'s the assumptions (A) suffice for (1.2) to hold. It is desirable to know in what ways the assumption of independence may be relaxed, retaining assumptions (A), without sacrificing (1.2). Investigators have weakened considerably the moment requirements (A2) and (A3) while retaining strong restrictions on the dependence. However, in many situations of practical interest, assumptions (A) hold but neither strong dependence restrictions nor strong stationarity restrictions seem to apply. Thus it is important to have theorems which take advantage of assumptions (A) when they hold, in order to utilize conclusion (1.2) without recourse to severe additional assumptions. A basic theorem in this regard is offered in Section 4. It is unfortunate that the additional assumptions required, while relatively mild, are not particularly amenable to verification, with present theory. This difficulty is alleviated somewhat by the strong intuitive appeal of the conditions. The variety of ways in which the assumption of independence may be relaxed in itself poses a problem. It is difficult to compare the results of sundry investigations in central limit theory because of the ad hoc nature of the suppositions made in each instance. In Section 2 we explore the relationships among certain alternative dependence restrictions, some introduced in the present paper and some already in the literature. Conditions involving the moments of sums $\sum^{a+n}_{a+1} X_i$ are treated in detail in Section 3. The central limit theorems available for sums of dependent rv's embrace diverse areas of application. The results of Bernstein [2] and Loeve [14], [15] have limited applicability within the class of sequences satisfying assumptions (A). A result that is apropos is one of Hoeffding and Robbins [11] for $m$-dependent sequences (defined in Section 2). In addition to assumptions (A1) and (A3), their theorem requires that, defining $A_a^2 = E(X^2_{a+m}) + 2 \sum^m_1E(X_{a+m-j}X_{a+m},$ \begin{equation*}\tag{H}\lim_{n\rightarrow\infty} n^{-1}\sum^n_{i = 1} A^2_{a+i} = A^2 \text{exists uniformly in} a (n \rightarrow \infty).\end{equation*} Now it can be shown easily that conditions (A2) and (H) are equivalent in the case of an $m$-dependent sequence satisfying (A1) and (A3). Therefore, a formulation relevant to assumptions (A) is THEOREM 1.1 (Hoeffding-Robbins). If $\{ X_i\}$ is an $m$-dependent sequence satisfying assumptions (A), then it has the central limit property. In the case of a weakly stationary (with mean zero, say) $m$-dependent sequence, the assumptions of the theorem are satisfied except for (A3), which then is a mild additional restriction. For applications in which the existence of moments is not presupposed, e.g., strictly stationary sequences, Theorem 1.1 has been extended by Diananda [6], [7], [8] and Orey [16] in a series of results reducing the moment requirements while retaining the assumption of $m$-dependence. In the present paper the interest is in extensions relaxing the $m$-dependence assumption. A result of Ibragimov [12] in this regard implies THEOREM 1.2 (Ibragimov). If $\{ X_i\}$ is a strictly stationary sequence satisfying assumptions (A) and regularity condition (I), then it has the central limit property. (Condition (I) is defined in Section 2.) Other extensions under condition (I) but not involving stationarity assumptions are Corollary 4.1.3 and Theorem 7.2 below. See also Rosenblatt [17]. Other extensions for strictly stationary sequences, further reducing the dependence restrictions, appear in [12] and [13] and Sections 5 and 6 below. Section 2 is devoted to dependence restrictions. The restrictions (2.1), (2.2) and (2.3), later utilized in Theorem 4.1, are introduced and shown to be closely related to assumptions (A). Although conditional expectations are involved in (2.2) and (2.3), the restrictions are easily interpreted. It is found, under assumptions (A), that if (2.3) is sufficiently stringent, then (2.1) holds in a stringent form (Theorem 2.1). A link between regularity assumptions formulated in terms of joint probability distributions and those involving conditional expectations is established by Theorem 2.2 and corollaries. Implications of condition (I) are given in Theorem 2.3. Section 3 is devoted to the particular dependence restriction (2.1). Theorem 3.1 gives, under assumptions (A), a condition necessary and sufficient for (2.1) to hold in the most stringent form, (3.1). The remaining sections deal largely with central limit theorems. Section 4 obtains the basic result and its general implications. Sections 5, 6 and 7 exhibit particular results for weakly stationary sequences, sequences of martingale differences and bounded sequences. NOTATION AND CONVENTIONS. We shall denote by $\{ X_i\}^\infty_{-\infty}$ a sequence of rv's defined on a probability space. Let $\mathscr{M}_a ^b$ denote the $\sigma$-algebra generated by events of the form $\{(X_{i_1},\cdots, X_{i_k}) \varepsilon E\}$, where $a - 1 < i_1 < \cdots < i_k < b + 1$ and $E$ is a $k$-dimensional Borel set. We shall denote by $\mathscr{P}_a$ the $\sigma$-algebra $\mathscr{M}^a_{-\infty}$ of "past" events, i.e., generated by the rv's $\{ X_a, X_{a-1},\cdots\}$. Conditional expectation given a subfield $\mathscr{B}$ will be represented by $E(\cdot\mid\mathscr{B}),$ which is to be regarded as a function measurable ($\mathscr{B}$). All expectations will be assumed finite whenever expressed.

Introductory. In the theory of random processes we may distinguish between ordinary processes and point processes. The former are concerned with a quantity, say x (t), which varies with time t, the latter with events, incidences, which may be represented as points along the time axis. For both categories, the stationary process is of great importance, i. e., the special case in which the probability structure is independent of absolute time. Several examples of stationary processes of the ordinary type have been examined in detail (see e. g. H. Wold 1). The literature on stationary point processes, on the other hand, has exclusively been concerned with the two simplest cases, viz. the Poisson process and the slightly more general process arising in renewal theory (see e. g. J. Doob 3).

“Linear threshold element”, is the generic term for a device, natural or artificial, of the input-output type which (1) has a finite numberk of input lines and one output line; (2) forms a linear combinationa1x1+a2x2+...+akxk of the inputk-tuple (x1,x2,...,xk) and compares it with a threshold quantity θ; and (3) yields one of two outputs depending on whether the linear combination does or does not exceed the threshold θ. Such elements find wide use as means for realizing linear discriminant functions and as analogs for certain features characterizing neurons. Adaptive classification of inputs into one of two categories, as designated by the two outputs, can sometimes be achieved by the suitable adjustment, based on experience, of the parametersa1,a2,...,ak and θ. One such adjustment rule having the characteristics of a steepest descent algorithm is here investigated. Established is a theorem giving the necessary and sufficient conditions that the mean values of the adjusted parameters converge to a solution of the equations required for minimizing the mean square error relative to a quantity associated with classification performance when the element is exposed to a stationary sequence of independent inputs.

This paper is concerned with everywhere local behaviour of certain classes of random processes which have stationary Gaussian increments. It is shown that for two classes of processes almost all the sample functions have the following property. The supremum of the increments in the neighbourhood of a point is everywhere of larger order than the standard deviation. For a third class of processes it is shown that the supremum is at least of the same order as the standard deviation.

It is shown how the Karhunen-Loeve (K-L) series representation for a finite sample of a discrete random sequence, stationary to the second order, may be further decomposed into a pair of series by utilizing certain symmetry properties of the covariance matrix of the sequence. The theory is applied to the particular example of a first-order Markov sequence, the series representation of which has not so far been reported in the literature. The generalization to the case of continuous random functions on a finite interval is similar and is therefore only briefly described.

Sufficient conditions are given for the optimal control of Markov processes when the control policy is stationary and the process possesses a stationary distribution. The costs are unbounded and additive, and may or may not be discounted. Applications to Semi-Markov processes are included, and the results for random walks are related to the author's previous papers on diffusion processes.

Power series f(z) =⌆ aizi are considered, where the sequence {ai} forms a homogeneous random process. If the sequence is exchangeable and the variance of the marginal distributions exists, it is proved that r, the random radius of convergence of f(z), takes the values 0 and 1. If the sequence is a second order stationary time series then r=1 with probability 1. If {ai} is a regular denumerable Markov chain, it can be proved that r=c≲=1 with probability 1, but both c=0 and c=1 can arise. A number of criteria are given for deciding the value of c in this situation.

N stationary, ergodic and independent sequences of events are considered. The interval probability density function and the expectation density function of the pooled sequence are expressed in terms of the interval probability density function and the expectation density function of the individual sequences. The case of two pooled sequences is treated in detail and applied to data on thalamic neuron discharge patterns. It is shown that the pooled sequence approaches a Poisson process if N tends to infinity.

0. Summary. Second-order stationary random sequences, especially if Gaussian, may be of statistical interest because their covariances may, under certain conditions, be consistently estimated with finite realizations of the sequences. It is shown that there is a class of non-stationary random sequences, namely sequences of orthogonal random variables with zero means and variances ft which form a uniformly almost periodic sequence, which are of statistical interest, at least in the Gaussian case, in the following sense. ft admits a generalized Fourier series expansion, and the Fourier coefficients ys of this expansion can be consistently estimated with finite realizations of the sequences. In certain situations, the nonconstant variance sequence ft may be directly estimated. The sequences of interest may be converted, via Fourier transformations, into second-order stationary random functions, and the Fourier coefficients yYs, of the expansion of ft, are shown to form a sequence of stationary covariances. A multiplicative representation is given for the nonstationary sequences considered. I. PRELIMINARIES

The analysis deals with Markov processes with stationary measure that is not finite. Various results on processes with finite stationary measure are generalized and few added. (Author)