where D ∈ (0 , 1) and L$_{γ}$ is slowly varying at infinity. Also, we assume that The above set of assumptions we will call collectively LMSV model . The tail behavior of the sequence { X$_{j}$, j ≥ 1 } can be related to the tail behavior of { ε$_{j}$, j ≥ 1 } or { σ ( Y$_{j}$ ) , j ≥ 1 } or both. Here, we will specifically assume that for some p ∈ (0 , 1]; Under these conditions, it follows by Breiman’s Lemma (see (Resnick, 2007, Proposition 7.5)) that i.e. the process { X$_{j}$, j ≥ 1 } inherits the tail behavior from { ε$_{j}$, j ≥ 1 } . We would like to point out here that in the literature the usage of the term LMSV often presupposes that the sequences { Y$_{j}$, j ≥ 1 } and { ε$_{j}$, j ≥ 1 } are independent. In this paper, we will consider a more general model: instead of claiming mutual independence of { Y$_{j}$, j ≥ 1 } and { ε$_{j}$, j ≥ 1 } , we only assume that { ( η$_{j}$, ε$_{j}$ ) , j ≥ 1 } is a sequence of independent random vectors. Especially, this implies that for a fixed index j the random variables ε$_{j}$ and Y$_{j}$ are independent, while Y$_{j}$ may depend on { ε$_{i}$, i < j } . In many cases, this version of the LMSV model is referred to as LMSV with leverage. 4 • the random variables { ε$_{j}$, j ≥ 1 } have a marginal distribution with regularly varying right tail, i.e. ¯ F$_{ε}$ ( x ) := P ( ε$_{1}$ > x ) = x − $^{α}$L ( x ) for some α > 0 and a slowly varying function L , such that the following tail balance condition holds: • { ( ε$_{j}$, η$_{j}$ ) , j ≥ 1 } is a sequence of i.i.d. vectors. • we have E [ σ α + $^{δ}$( Y$_{1}$ ) ] < ∞ for some δ > 0. lim x →∞ P ( ε$_{1}$ > x ) P ( | ε$_{1}$ | > x ) = p = 1 − lim x →∞ P ( ε$_{1}$ < − x ) P ( | ε$_{1}$ | > x ) P ( X$_{1}$ > x ) ∼ E [ σ $^{α}$( Y$_{1}$ )] P ( ε$_{1}$ > x ) , (4)