A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.

In the present paper we demonstrate that the concept of a list code is from a mathematical point of view a more canonical notion than the classical code concept (list size one) in that it allows a unified treatment of various coding problems. In particular we determine for small list sizes the capacities of arbitrarily varying channels.

Stationary processes which are defined on the points of a square lattice and are Markovian in various senses are considered. It is shown that a certain assumption of linearity of regression forces the spectral distribution to be of a certain explicit form, and that given this form Gaussian processes of this kind are easily constructed. Certain non-Gaussian processes satisfying the various Markovian properties are also constructed and the difference from nearest-neighbour systems emphasized. It is conjectured, but not proved, that the assumption of linearity of regression also implies Gaussianity.

Heuristic methods yield simple limiting results for the behaviour of input-output systems under conditions of heavy traffic. It is assumed that the input is time-dependent, while the output rate depends entirely on the content of the system. It is shown that the content, when suitably normalized and scaled, behaves like a (non-stationary) Ornstein-Uhlenbeck process.

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

Traffic noise depends significantly on statistical properties of highway flow. The noise heard by an off-highway observer is the sum of the contributions of all the vehicles on the highway. We assume that vehicle spacings on a single-lane infinite straight road form a Markov renewal process (MRP) with N states or vehicle types. The noise impact is then a two-sided filtered MRP. We give explicit formulae for the mean and variance in the case N = 2 and exponential headways. Jewell's (1965) study (cars vs. trucks in the southbound curb lane of U.S. 40) gives parameters of the MPR for a numerical example. Even with only a modest degree of truck clustering and variability in vehicle spacings and noise emission, the variation in noise level is much greater than usually predicted. The coefficient of variation of arithmetic noise intensity is greater than unity at distances from the roadway less than 700 feet. The logarithmic noise level parameters L50 and L10 usually computed in engineering practice may be in error by 2 to 3 decibels if these sources of variability are ignored.

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

The paper begins with an iterated logarithm law of classical Hartman-Wintner form for stationary martingales. This is then used to obtain iterated logarithm results giving information on rates of convergence of estimators of the parameters in a stationary autoregressive process. In the case of an autoregression of small order, detailed rate results for each autocorrelation and for the estimators of all parameters can be obtained. A rate result for the convergence of the sample mean is given in the general case.

An identity, of the type of Green's equation, is deduced for the loss function of a stopping process. This yields a set of relations determining the optimal (minimal loss) stopping boundary, which do not require simultaneous determination of the loss function in the stopping region. A no-overshoot approximation is invoked, but a bound on the magnitude of the terms neglected is obtained by appeal to a general version of Chernoff's tangency condition at an optimal boundary.

We will consider the optimal search for a target whose motion is a Markov process. The classical detection law leads to the use of multiplicative functionals and the search is equivalent to the termination of the Markov process with a termination density. A general condition for the optimality is derived and for Markov processes in n-dimensional Euclidean space with continuous transition functions we derive a simple necessary condition which generalizes the result of Hellman (1972).

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij. Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

Examples of non-stationary random set processes are presented. The reconstruction of a pattern, generated by such a process, is studied; a particular arrangement of the sample-points is considered.

The use of a branching process argument in complex queueing situations often leads to a discussion of a non-linear matrix integral equation of Volterra type. By the use of a fixed point theorem we show these equations have a unique solution.

The paper considers the dam problem with Markovian inputs, with special reference to the serial correlation coefficient of the input process. An input model is proposed which by giving particular values to the parameters makes the stationary distribution of the inputs one of the standard discrete distributions. The probabilities of first emptiness before overflow are first obtained by using the Markovian analogue of Wald's Identity. From these, the stationary distributions of the dam content are obtained by a duality argument. Both, finite and infinite dams are considered.

An iterated logarithm type theorem is obtained for maxima of nonstationary discrete-parameter Gaussian processes.

The mean square and almost sure convergence of W(t) = e–αtZ(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration.

A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.

The transient behaviour of the waiting time process in GI/M/S is studied algebraically by means of a two-dimensional Markovian process {(vn, ln} , where the variables vn denote the times of full occupation immediately after the arrival instants Tn and where ln = max {0,– 1 + number of idle servers at (Tn + 0)}.

Let {Xn, n ≧ 1} be i.i.d. and Yn = max {X1,…, Xn}. Xj is a record value of {Xn} if Yj > Yj–1 The record value times are Ln, n ≧ 1 and inter-record times are Δn, n ≧ 1. The known limiting behavior of {Ln} and {Δn} is close to that of a non-homogeneous Poisson process and an explanation of this is obtained by embedding {Yn} in a suitable extremal process which jumps according to a non-homogeneous Poisson process.