Let $s, s^*:\mathbb{R}^+ \to \mathbb{R}^+$ ($0\in\mathbb{R}^+$) such that $0 \le s(x), s^*(x) \le x$ for every $x \in \mathbb{R}^+$ and $X, X_1, \ldots$ is iid with $\mathbb{E}X>0$. Let $U_{0,s}=1$ and $U_{n,s}=U_{n-1,s}+s(U_{n-1,s})X_n$ for $n\in \mathbb{N}$.

What can we say about asymptotic distribution of $U_{n,s}$? If $s$ is linear then we have log-normal distribution, but what can we say in general? I do not even know how to compute expected value and variance of $U_{n,s}$ so any information will be useful. My main goal is to compute (or estimate): $\lim_{n \to \infty} \frac{F^{-1}_{U_{n,s}}(\alpha)}{F^{-1}_{U_{n,s^*}}(\alpha)}$ (ratio of quantiles of order $\alpha$) for any $\alpha \in (0,1)$ when $s$ and $s^*$ are given.

I will be grateful for any tips, literature or any idea which may help me dealing with such objects..


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