Consider the equation

where are linear positive continuous operators and f : Cloc(ℝ;ℝ) → Lloc(ℝ;ℝ) is a continuous operator satisfying the local Carathéodory conditions. Efficient conditions guaranteeing the existence of a global solution, which is bounded and non-negative in the neighbourhood of –∞, to the equation considered are established provided that ℓ0, ℓ1 and f are Volterra-type operators. The existence of a solution that is positive on the whole real line is discussed as well. Furthermore, the asymptotic properties of such solutions are studied in the neighbourhood of –∞. The results are applied to certain models appearing in the natural sciences.

This paper discusses the distribution of tumor size at detection derived within the framework of a new stochastic model of carcinogenesis. This distribution assumes a simple limiting form, with age at detection tending to infinity which is found to be a generalization of the distribution that arises in the length-biased sampling. Two versions of the model are considered with reference to spontaneous and induced carcinogenesis; both of them show similar asymptotic behavior. When the limiting distribution is applied to real data analysis its adequacy can be tested through testing the conditional independence of the size, V, and the age, A, at detection given A > t*, where the value of t* is to be estimated from the given sample. This is illustrated with an application to data on premenopausal breast cancer. The proposed distribution offers the prospect of the estimation of some biologically meaningful parameters descriptive of the temporal organization of tumor latency. An estimate of the model stability to the prior distribution of tumor size and some other stability results for the Bayes formula are given.

This paper is concerned with the estimation of the variance for the multitype Galton-Watson process X = {Xn = (Xn(1),…, Xn(p)); n ≧ 0}. Two estimators for the variance matrix are obtained and asymptotic results for the estimators are given. The first is a maximum likelihood estimator based upon knowledge of individual offspring sizes, the second estimator is based on parent-offspring type combination counts only. Estimators for the asymptotic variances of the Asmussen and Keiding estimator and Becker estimator are also proposed.

The problem initially considered is that of testing whether ρ = 0 in a model y(n) = x(n)+ η (n), x(n) = ρx(n – 1) + ξ (n) where only y(n) is observed and η (n), ξ (n) are white noise. This is equivalent to distinguishing between an ARMA (1, 1) model and white noise. The asymptotic distribution of the likelihood ratio criterion is derived. This is shown to be of an unusual form. This result is then used to discuss the asymptotic properties of Akaike's procedure for estimating (p, q) in an ARMA (p, q) model. If p0, q0 are the true values and p0 < P, q0< Q, when P, Q are the maximum values considered, then it is shown that, in a certain asymptotic sense, the procedure is sure to overestimate p0, q0. However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.

For a supercritical branching process x = {xn; n ≧ 0, x0 = 1} with random environments, define when xn > 0; and = 1 when xn = 0. When x is assumed to satisfy the standard regularity assumptions, under the non-extinction hypothesis, is a strongly consistent and asymptotically unbiased estimator for the criticality parameter π and is asymptotically normal. A strongly consistent estimator, is also proposed for the associated variance, σ2.

A random spatial series is a collection of random variables F(xi, · · ·, xn) depending on several spatial coordinates (x1, · · ·, xn). An attempt is made to construct a statistical second-order theory of such series when (x1, · · ·, xn) varies over a regular cartesian lattice. Using the properties of the linear (Hilbert) space associated with the series, the concepts of innovation and purely non-deterministic (p.n.d.) series are introduced. For a p.n.d. series F(x1, · · ·, xn) a unilateral representation is obtained in terms of a white innovations series Z(y1, · · ·, yn) where The representatation is specialized to the homogeneous case and we discuss spectral conditions for p.n.d. The familiar time-series condition ∫ log f(λ) dλ > –∞ on the spectral density f is necessary but not sufficient. A sufficient condition is stated. Motivated by the p.n.d. unilateral representation results we define unilateral arma (autoregressive-moving average) spatial series models. Stability and invertibility conditions are formulated in terms of the location of zero sets of polynomials relative to the unit polydisc in Cn, and a rigorous shift operator formalism is established. For autoregressive spatial series a Yule–Walker-type matrix equation is formulated and it is shown how this can be used to obtain estimates of the autoregressive parameters. It is demonstrated that under mild conditions the estimates are consistent and asymptotically normal.

In this article, we consider maximum likelihood estimators of the initial probabilities and the mean of a supercritical Galton-Watson process; we find for these estimators, as well as for the Lotka-Nagaev estimator of the mean, the asymptotic distributions and deduce confidence intervals. As these results hold even if the independence between individuals of the same generation is not satisfied, application to living populations may be considered.

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.