Logarithmic tails of sums of products of positive random variables bounded by one

Abstract

In this paper, we show under weak assumptions that for R d = 1+M1+M1M2+. . ., where P(M ∈ [0, 1]) = 1 and Mi are independent copies of M , we have lnThe constant C is given explicitly and its value depends on the rate of convergence of ln P(M > 1 − 1/x). Random variable R satisfies the stochastic equation R d = 1 + M R with M and R independent, thus this result fits into the study of tails of iterated random equations, or more specifically, perpetuities.MSC 2010 subject classifications: Primary 60H25; secondary 60E99

On perpetuities with gamma-like tails

“…Decomposition (23) with τ = 1 is equivalent to ψ(r) = Ee rB ψ(rA). (27)Now we use(27) to obtainψ(r) = Ee rB ψ(rA) ≥ Ee rB ½ {A=−1} ψ(−r)which shows that ψ(−r) < ∞ whence Ee r|X| ≤ ψ(r) + ψ(−r) < ∞. This proves the ⇒ implication, the implication ⇐ being trivial.…”

“…Suppose now that P{A = 1} ∈ (0, 1). In order to check the second inequality in (25) we use once again (27) to infer…”

On perpetuities with light tails

“…Observe that if Q = 1 a.s. then h(x) = −x log P(M > 1 − 1/x), so we recover (1.6). Thus, we generalize the results of [11] and [21], but with new proofs, which are very different from those in [11] and [21]. Our proofs are based on a new formulation of the classical Tauberian theorems; see Section 2.4.…”

“…If the moment generating function of R is finite over all R, we will see that the dependence structure may have significant impact on the rate of convergence even for logarithmic tails; this can be observed in the following example (see also Example 5.2). thus, the results of [11] and [21] apply. For this example, the asymptotics of P(R > x) as x → ∞ are also known [30].…”

“…It turns out that such an approach, proposed in [11], gives the appropriate logarithmic asymptotics for constant Q; in [21] (with an earlier contribution in [16]), under some weak assumptions on the distribution of M near 1−, it was proved that…”