Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σk=0∞pksk and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk, and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.

This note is concerned with integral representations of some transition probabilities of countable state space Markov chains embedded in birth and death processes, of the serial covariances of functions defined on such chains when stationary, and finally the properties of the spectral measure of stationary processes with monotonically decreasing or completely monotonic serial covariances.

1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).)

We consider a branching process for which the offspring distribution has the generating function f(t) and mean f'(1) = m < 1. The probability that a line descended from an individual still survives in generation n is asymptotically of the form cmn. A method is derived whereby good bounds for c may be obtained. This method makes use of the first three moments of the distribution of offspring.

In this note we consider an age-dependent branching process in which the life-spans of sister cells are correlated as well as the numbers of offspring of sister cells, but otherwise cells live and reproduce independently. One might surmise that in many populations there would be some positive correlation among siblings due to similar characteristics inherited from parents. Data of Powell (1955) seem to corroborate this conjecture. Powell's data indicate that in certain bacterial populations the life-spans of sister cells have significant positive correlations while the correlation between the life-spans of mother and daughter cells is negligible. More recently positive correlations have been observed in cell populations among the life-spans of first cousins and also among second cousins (Kubitschek (1967)). However, the model presented in this note permits correlations only among siblings.

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t1, t2 ɛ T, t1 〈 t2; the random variable X(t2) – X(t1) is called the increment of the process X(t) over the interval [t1, t2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

Let two points be taken at random in an n-dimensional convex body K, and let σ be the line joining them. The distribution of σ is found and compared with other distributions for random secants of K. More generally, if r + 1 ≦ npoints are taken in K, the distribution of the r-dimensional affine subspace containing them is computed. The results find application to the n-dimensional case of a problem of Sylvester.

In a previous paper, [1] it was shown that a solution, f(x) will exist for the optimal inventory equation (where f(y − z) = f(0), y < z) provided:

1. 1. g(x) ≧ 0, x ≧ 0;

2. 2. 0 < a < 1;

3. 3. h(x) is monotonically nondecreasing, h(0) = 0;

4. 4. F is a distribution function on [0, ∞);

   (In [1], 1–4 were denoted collectively as (A).)

   and either

5. 5a. g(x) is continuous for all x ≧ 0;

6. 5b. limx→∞g(x) = ∞;

7. 5c h(x) is continuous for all x > 0 (Theorem 2 of [1]);

   or

8. 6. g(x) is uniformly continuous for all x ≧ 0 (Theorem 3 of [1]).

A recursive procedure leading to the solution of triangular systems of differential-difference equations is presented. It is shown that the equations for special families of multi-dimensional pure birth processes and pure death processes can be represented as such a system. The theory is used to resolve a difficult problem of stochastic cross-infection among groups. Additional applications to problems in epidemic theory are discussed.

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S0= 0, Sn = Σ i=1nXi, n ≧ 1, and Mn = max0 ≦ k ≦ nSk. In the case where the Xi are such that Σ1∞n−1Pr(Sn > 0) < ∞, we have limn→∞Mn = M which is finite with probability one, while in the case where Σ1∞n−1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1∞n−1Pr(Sn < 0) < ∞, Σ1∞n−1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.

This paper is a continuation of [1]. The techniques of [1] are used to get specific information about the distribution of the total progeny in a branching process. This distribution is also related to one which arises in a random walk problem.

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

Several mechanisms under which the randomness of straight line paths through convex bodies can arise are described. Some general results are given relating four of these mechanisms, and the corresponding distributions of the lengths of the straight line paths are found for the circle, the rectangle, and the cube.

Suppose Q1⋆, … Qn⋆ are independent, identically distributed random variables, each with probability density function f(x), cumulative distribution function F(x), where F(1) – F(0) = 1, f(x) is continuous in the open interval (0, 1) and continuous on the right at x = 0 and on the left at x = 1, and there exists a positive C such that f(x) > C for all x in (0, l). f(0) is defined as f(0+), f(1) is defined as f(1–).

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates.

Items arrive at a processing plant at a Poisson rate λ. At time T, all items are dispatched from the system. An intermediate dispatch time is to be chosen to minimize the total wait of all items. It is shown that if the dispatch time must be chosen at time 0 then T/2 not only minimizes the expected total wait but it also maximizes the probability that the total wait is less than a for every a > 0. If the intermediate dispatch time is allowed to be a (random) stopping time, then it is shown that the policy which dispatches at time t iff N(t) > λ(T – t) is optimal, where N(t) denotes the number of items present at time t. The distribution of the optimal dispatch time and the optimal expected total wait are determined. A generalization to the case of a non-homogeneous Poisson process, a time lag, and batch arrivals is given. Finally, the case where the process goes on indefinitely and any number of dispatches are allowed (at a cost K per dispatch) is considered.

This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t < ∞) by the equation where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (< K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) > K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.

It is well known that any stochastically continuous real valued stochastic process with independent increments defined on a compact time interval can be decomposed into a sum of independent processes, one of which is Gaussian with continuous sample paths, and the remainder of which have sample paths which are continuous except at a finite number of points with the discontinuities occurring at Poisson time points. The purpose of this note is to announce a proof of the above theorem in the case where the process takes values in an abelian group G. The detailed proof will appear elsewhere. The basic ideas of the proof in the case when G is finite dimensional Euclidean space are contained in Chapter VI of Gikhman and Skorohod (1965).