[Paper Review] Large Deviation Probabilities for Sums of Random Variables with Heavy or Subexponential Tails

Daren B. H. Cline, Tailen Hsing|arXiv (Cornell University)|Nov 29, 2022

Probability and Risk Models19 citations

TL;DR

This paper establishes conditions under which the large deviation probability of sums of i.i.d. random variables with heavy or subexponential tails is asymptotically equivalent to $ n(1-F(s)) $, extending prior results by linking this behavior to the slow variation of $ -\log(1-F(x)) $. The key contribution is a unified treatment via truncation and Markov's inequality, proving that subexponential distributions satisfy $ \lim_{n\to\infty}\sup_{s\geq t_n}\left|\frac{P(S_n>s)}{n(1-F(s))}-1\right|=0 $ for appropriately chosen $ t_n $, thereby characterizing subexponentiality through large deviation behavior.

ABSTRACT

Let $S_n$ be the sum of independent random variables with distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$ \lim_{n o\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1 ight| =0 $$ are given. These conditions extend and strengthen a series of previous results. Additionally, a connection with subexponential distributions is demonstrated. That is, $F$ is subexponential if and only if the condition above holds for some $t_n$ and $$ \lim_{t o\infty}{1-F(t+x)\over 1-F(t)} = 1 \quad ext{for each real $x$.}$$

Motivation & Objective

• To extend and strengthen existing results on large deviation probabilities for sums of i.i.d. random variables with heavy or subexponential tails.
• To establish conditions under which $ \lim_{n\to\infty}\sup_{s\geq t_n}\left|\frac{P(S_n>s)}{n(1-F(s))}-1\right|=0 $, ensuring asymptotic equivalence between sum and maximum tail probabilities.
• To unify the treatment of subexponential distributions whose tails satisfy $ -\log(1-F(x)) $ being slowly varying, including lognormal and Cauchy-like distributions.
• To demonstrate that $ F $ is subexponential if and only if the above limit condition holds for some sequence $ t_n $, linking subexponentiality to large deviation behavior.

Proposed method

• The authors use a truncation argument combined with Markov’s inequality to analyze the tail behavior of sums $ S_n $, focusing on the survival function $ \overline{F}(x) = 1 - F(x) $.
• They analyze the asymptotic behavior of $ P(S_n > s) $ by decomposing the sum into truncated and tail components, bounding the contribution of the truncated part using moment conditions.
• The method relies on the assumption that $ -\log(1-F(x)) $ is slowly varying as $ x \to \infty $, which characterizes the heaviest-tailed subexponential distributions.
• Key inequalities involve bounding $ \mathbb{E}[e^{\lambda X} \mathbf{1}_{\{X \leq x\}}] $ and using the slowly varying property to control the exponential moments.
• The proof structure follows a hierarchy of lemmas establishing bounds on $ \mu_1(s) $, $ \mu_2(s) $, and $ \eta(s) $, which control centering and variance terms.
• The authors derive sufficient conditions on $ t_n $ such that $ \sup_{s \geq t_n} \left| \frac{P(S_n > s)}{n \overline{F}(s)} - 1 \right| \to 0 $, using the interplay between $ \psi(s) = -\log \overline{F}(s) $ and its derivative.

Experimental results

Research questions

• RQ1 Under what conditions does the large deviation probability $ P(S_n > s) $ satisfy $ \frac{P(S_n > s)}{n(1-F(s))} \to 1 $ as $ n \to \infty $ for $ s \geq t_n $?
• RQ2 How can the asymptotic equivalence between the tail of the sum and the tail of the maximum be characterized for subexponential distributions with slowly varying $ -\log(1-F(x)) $?
• RQ3 What is the role of the slowly varying function $ \psi(x) = -\log(1-F(x)) $ in determining the rate of convergence of large deviation probabilities?
• RQ4 Can a unified approach be developed for subexponential distributions with heavy tails, such as lognormal and Cauchy, using truncation and moment bounds?
• RQ5 Is subexponentiality equivalent to the condition $ \lim_{n\to\infty}\sup_{s\geq t_n}\left|\frac{P(S_n>s)}{n(1-F(s))}-1\right|=0 $ for some sequence $ t_n $?

Key findings

• The paper proves that $ \lim_{n\to\infty}\sup_{s\geq t_n}\left|\frac{P(S_n>s)}{n(1-F(s))}-1\right|=0 $ holds if $ -\log(1-F(x)) $ is slowly varying and $ t_n $ is chosen such that $ n(1-F(t_n)) \to 0 $, extending prior results.
• For subexponential distributions where $ \overline{F}(t+x)/\overline{F}(t) \to 1 $ for all real $ x $, the condition $ \limsup_{n\to\infty}\sup_{s\geq t_n}\frac{P(S_n>s)}{n\overline{F}(s)} \leq 1 $ characterizes subexponentiality.
• The authors show that $ F \in \mathcal{S} $ if and only if (1.5) holds and there exists a sequence $ t_n $ such that the above limit superior is bounded by 1.
• For distributions with regularly varying tails, such as $ P(|X| > x) \sim x^{-\alpha} $, the result holds when $ \alpha < 2 $, with appropriate centering and moment conditions.
• The method yields non-trivial extensions for lognormal and Cauchy-type tails, which were previously untreated under a unified framework.
• The paper demonstrates that the condition $ \lim_{n\to\infty}\sup_{s\geq t_n}\left|\frac{P(S_n>s)}{n(1-F(s))}-1\right|=0 $ is both necessary and sufficient for subexponentiality when combined with the stability condition $ \overline{F}(t+x)/\overline{F}(t) \to 1 $.