[Paper Review] Pointwise estimates for exceedance times of perpetuity sequences

Dariusz Buraczewski, Ewa Damek+1 more|arXiv (Cornell University)|Dec 10, 2015

Stochastic processes and statistical mechanics13 references3 citations

TL;DR

This paper establishes precise pointwise asymptotics for the probability that the exceedance time $\tau_u = \inf\{n: Y_n > u\}$ of a perpetuity sequence $Y_n = B_1 + A_1B_2 + \cdots + (A_1\cdots A_{n-1})B_n$ equals $\rho \log u$, for $\rho > 0$ and $u \to \infty$. By analyzing path behavior of $Y_n$, it refines earlier interval-based estimates and reveals sharp analogies and distinctions with random walk behavior.

ABSTRACT

We consider large exceedence probabilities of the perpetuity sequence $$Y_n = B_1 + A_1 B_2 + \cdots + (A_1\ldots A_{n-1})B_n, $$ where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbf R} ^+ imes {\mathbf R}$ and the exceedance times are defined as $ au_u = \inf\{n:\; Y_n>u\}$. Applying techniques based on analyzing path behavior of $Y_n$ we provide the asymptotics of the sequence $\P[ au_u = ho \log u]$, $ ho >0$, $u o \infty $. This improves essentially the results of \cite{BCDZ}, where we identified probabilities $\P[ au_u \in I_u]$, for some large intervals $I_u$ around $k_u$, with lengths growing at least as $\log\log u$. Remarkable analogies and differences to random walks \cite{Laley} are discussed.

Motivation & Objective

• To derive sharp pointwise asymptotics for the exceedance time $\tau_u = \inf\{n: Y_n > u\}$ of a perpetuity sequence $Y_n$ as $u \to \infty$.
• To improve upon prior interval-based estimates $\P[\tau_u \in I_u]$ by focusing on exact probabilities $\P[\tau_u = \rho \log u]$ for $\rho > 0$.
• To analyze the path behavior of $Y_n$ to understand the stochastic dynamics leading to large exceedance times.
• To explore and clarify analogies and differences between the exceedance behavior of perpetuities and classical random walks.

Proposed method

• Analyzing the stochastic path behavior of the perpetuity sequence $Y_n = \sum_{k=1}^n \left(\prod_{i=1}^{k-1} A_i\right) B_k$ with i.i.d. $(A_n, B_n)$ in $\mathbb{R}^+ \times \mathbb{R}$.
• Applying techniques rooted in the structure of products and sums of i.i.d. random variables to characterize the tail behavior of $Y_n$.
• Deriving asymptotic expressions for $\P[\tau_u = \rho \log u]$ by focusing on the critical scaling $\rho \log u$ as $u \to \infty$.
• Using path decomposition and conditioning on early and late contributions to the sum to isolate dominant contributions to exceedance.
• Comparing the results to known behaviors in random walks to highlight structural similarities and differences in exceedance mechanisms.
• Leveraging the independence and identical distribution of $(A_n, B_n)$ to derive exact asymptotic forms for point probabilities.

Experimental results

Research questions

• RQ1 What is the precise asymptotic behavior of $\P[\tau_u = \rho \log u]$ for $\rho > 0$ as $u \to \infty$?
• RQ2 How does the path behavior of the perpetuity sequence $Y_n$ influence the distribution of the exceedance time $\tau_u$?
• RQ3 In what ways do the exceedance time laws of perpetuities resemble or differ from those of random walks?
• RQ4 Can the interval-based estimates of $\P[\tau_u \in I_u]$ be sharpened to pointwise probabilities $\P[\tau_u = \rho \log u]$?
• RQ5 What role does the logarithmic scaling $\rho \log u$ play in the limiting distribution of $\tau_u$?

Key findings

• The paper establishes the exact asymptotic behavior of $\P[\tau_u = \rho \log u]$ as $u \to \infty$, providing pointwise estimates where earlier results were limited to intervals.
• The asymptotics are derived through detailed analysis of the path structure of the perpetuity sequence, particularly the interplay between products of $A_i$ and the additive contributions of $B_i$.
• The results show a sharp transition in the exceedance time distribution at the logarithmic scale $\rho \log u$, indicating a critical scaling regime.
• There are notable analogies with random walk behavior, particularly in the logarithmic scaling of exceedance times, but key differences arise due to the multiplicative structure of the perpetuity.
• The analysis improves upon prior work by \'BCDZ\' by replacing interval-based probabilities with exact point probabilities, significantly increasing precision.
• The method reveals that the dominant contribution to $Y_n$ at exceedance time $n \approx \rho \log u$ comes from a balance between the growth of the product terms and the sum of $B_i$ terms.