The paper considers a particular family of set–valued continuous time stochastic processes modeling birth–and–growth processes. The proposed setting allows us to infer the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a logical consequence, different consistent set–valued estimators are studied for growth process. M...

Abstract. This paper concerns the structure of the space C of real valued cocycles for a flow (X,Zm). We show that C is always larger than the set of cocycles cohomologous to the linear maps if the flow has a free dense orbit. By considering appropriate dual spaces for C, we obtain the concept of an invariant cocycle integral. The extreme points of the set of invariant cocycle integrals paralle...

In this paper we present inequalities for integrals of functions that are the composition of nonnegative convex functions on an open convex set of a vector space R and vectorvalued functions in a weakly compact subset of a Banach vector space generated by m Lμ-spaces for 1 ≤ p < +∞ and inequalities when these vector-valued functions are in a weakly* compact subset of a Banach vector space gener...

We show that the general stable convergence results proved in Peccati and Taqqu (2007) for generalized adapted stochastic integrals can be used to obtain limit theorems for multiple stochastic integrals with respect to independently scattered random measures. Several applications are developed in a companion paper (see Peccati and Taqqu, 2008a), where we prove central limit results involving si...

Stochastic integration rules are derived for infinite integration intervals, generalizing rules developed by Siegel and O’Brien [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 169–181] for finite intervals. Then random orthogonal transformations of rules for integrals over the surface of the unit m-sphere are used to produce stochastic rules for these integrals. The two types of rules are combine...