A note on the asymptotic behaviour of sequences of generalized subgaussian random vectors

The aim of the present note is to define some new concepts related to φ-subgaussianity. More precisely, we introduce the concept of scalarly φ-subgaussian random vector and the concept of φ-subgaussian martingale sequence. The space of φ-subgaussian variables, denoted by Subφ(Ω) has been firstly considered in the book [2], and wider definitions are given in [4]. Among other properties, it is proved in [2] and [4] that the tail probability of a variable belonging to Subφ(Ω) verifies a useful inequality (Lemma (4.3) of [2] and Lemma 3.1 of [4]). In the present paper we prove that analogous inequalities hold in the multidimensional case (Prop. (2.6) and Theorem (2.7)). With these inequalities at our disposal, in section 3 we prove some results concerning the asymptotic behaviour of a sequence (X(n)) of IRd-valued scalarly φ- subgaussian random vectors (Theorems (3.1), (3.3), (3.5) and (3.8)). The interest of these results is in the fact that they are analogous to those concerning the first part of the classical law of iterated logarithm. In section 5 we consider the sequence (S(n)) of partial sums of the (X(n)); we prove an exponential bound concerning the tail probability of the norm of S(n) (Proposition (5.3)) and we use it in order to study the asymptotic behaviour of partial sums (Theorem (6.1)). In the last section we introduce the concept of φ-subgaussian martingale sequence and we prove a result concerning the tail probability of their modulus.