The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

Open Access, Copyright CERN, for the benefit of the CLICdp Collaboration. Article funded by SCOAP 3 .

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Let X i , i = 1. .. n be a sequence of positive i. i. d. random variables. Define R n := E X 2 1 + X 2 2 +. .. + X 2 n (X 1 + X 2 +. .. + X n) 2. Let φ(s) = Ee −sX. We give an explicit representation of R n in terms of φ and with the help of the Karamata theory of functions of regular variation we study the asymptotic behaviour of R n for large n. Résumé Soient X i , i = 1. .. n une suite de variables aléatoires positiveséquidistribuées, R n := E X 2 1 + X 2 2 +. .. + X 2 n (X 1 + X 2 +. .. + X n) 2 , et φ(s) = Ee −sX. Nous donnons une expression explicite de R nà l'aide de φ et grâceà la théorie de Karamata des fonctionsà variation régulière nous décrivons le comportement asymptotique de R n pour les grandes valeurs de n.