The asymptotic behaviour of a superadditive bisexual Galton–Watson branching process is studied. Sufficient conditions for the almost sure and L¹ convergence of the suitably normed process are given. Finally, a first approach to the study of the L¹ convergence for a superadditive bisexual Galton–Watson branching process under the Z log⁺Z condition is considered.

The topic of this paper is Yanev's class of controlled branching processes with a random control function. We investigate the L2-convergence of the suitably normed process to a non-degenerate limit. Necessary and sufficient conditions for this convergence are established under certain constraints on the means and variances of the control variables.

In this work, an approach to the Bayesian estimation in a bisexual Galton-Watson process is considered. First we study an important parametric case assuming offspring distribution belonging to the bivariate series power family of distributions and then, we continue to investigate the nonparametric case. In both situations, Bayes estimators under weighted squared error loss function, for means, variances and covariance of the off spring distribution are obtained. For the superadditive case, the Bayes estimation of the asymptotic growth rate is also considered. Illustrative examples are given.

In this paper the L²-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.

In this paper a bisexual Galton–Watson branching process allowing the immigration of mating units is introduced and its limit behaviour is investigated. For the supercritical case, i.e., asymptotic growth rate greater than one, necessary and sufficient conditions for the almost sure and L 1 convergence of the suitably normed underlying sequences are given. *Research supported by the Consejerı´a de Educación y Juventud de la Junta de Extremadura and the Fondo Social Europeo, grant IPR98A023.

The limit behaviour of a controlled branching process with random control function is investigated. A necessary condition and a sufficient condition for the geometric growth of such a process are established by considering the L¹-convergence. Finally, taking into account the classical X log⁺X criterion in branching processes, a necessary and sufficient condition is provided.