Let , be independent identically distributed -valued random vectors. Assuming that has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of we prove three functional limit theorems for the logarithm of convergent discounted perpetuities as . 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.
A. Iksanov and A. Marynych were supported by the National Research Foundation of Ukraine (project 2020.02/0014 ‘Asymptotic regimes of perturbed random walks: on the edge of modern and classical probability’).
We thank the anonymous referee for many useful suggestions which greatly improved the presentation of our results.
"Limit theorems for discounted convergent perpetuities II." Electron. J. Probab. 28 1 - 22, 2023. https://doi.org/10.1214/23-EJP907