Limit theorems for discounted convergent perpetuities II

Let (& xi;1, & eta;1), (& xi;2, & eta;2), ... be independent identically distributed R2-valued random vec-tors. Assuming that & xi;1 has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of & eta;1 we prove three functional limit theo-rems for the logarithm of convergent discounted perpetuities Ek & GE;0 e & xi;1+...+& xi;k-ak & eta;k+1 as a & RARR; 0+. Also, we prove a law of the iterated logarithm which corresponds to one of the aforementioned functional limit theorems. The present paper continues a line of research initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused on limit theorems for a different type of convergent discounted perpetuities.

Automated summary

Based on three groups of assumptions on the distribution of & eta;1, this paper proves three functional limit theorems for the logarithm of convergent discounted perpetuities Ek & GE;0 e & xi;1+...+& xi;k-ak & eta;k+1, where (& xi;1, & eta;1), (& xi;2, & eta;2), ... are independent identically distributed R2-valued random vectors with & xi;1 having zero mean and finite variance. Additionally, a law of the iterated logarithm corresponding to one of the functional limit theorems is also proved. This paper continues the research initiated in the paper Iksanov, Nikitin, and Samoillenko (2022) on limit theorems for a different type of convergent discounted perpetuities.