Regular variation of series and random sums

Abstract

In this chapter, we will consider infinite weighted sums of i.i.d. regularly varying random vectors. We will provide conditions on the weights, possibly random and possibly dependent of the summands, to ensure the existence and the regular variation of the sum of the series. This will be the main result of the chapter, Theorem 4.1.2. We will apply it to series with deterministic weights and random sums. The results of this chapter will be useful to study certain models in Part III. Throughout this chapter, $\left| \cdot \right|$ denotes an arbitrary norm on $\mathbb {R}^d$ and $\left\| \cdot \right\|$ the associated matrix norm.

4.4 Bibliographical notes

The main result of this chapter, Theorem 4.1.2, is due to [HS08]. The case of a univariate series with deterministic coefficients has a long history. The conditions of Corollary 4.2.1 were first obtained in [MS00].

The models in Problem 4.2 and Problem 4.3 are considered in [Kul06] and [DR96] for instance. For Problem 4.8, see for instance the monograph [Mik09] on insurance mathematics. Problem 4.9 and extensions can be found in [RS08].

The converse to Corollary 4.2.4 which is obtained in Problem 4.10 can be extended to all $\alpha >0$; see [FGAMS06, Proposition 4.8].