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Limit theorems for discounted convergent perpetuities. II. (English) Zbl 1508.60040

Summary: Let \(({\xi_1},{\eta_1}), ({\xi_2},{\eta_2}),\dots\) be independent identically distributed \({\mathbb{R}^2} \)-valued random vectors. 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 theorems for the logarithm of convergent discounted perpetuities \(\sum_{k\ge 0} \text{e}^{\xi_1+\dots +\xi_k-ak } \eta_{k+1}\) as \(a\to 0+\). 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 part I of this paper [A. Iksanov et al., Electron. J. Probab. 26, Paper No. 131, 25 p. (2021; Zbl 1483.60049)], which focused on limit theorems for a different type of convergent discounted perpetuities.

MSC:

60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Citations:

Zbl 1483.60049
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References:

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