[Paper Review] A conditional limit theorem for random walks under extreme deviation

Michel Broniatowski, Zhansheng Cao|arXiv (Cornell University)|Jun 29, 2012

Stochastic processes and statistical mechanics6 references3 citations

TL;DR

This paper establishes a conditional limit theorem for i.i.d. random walks under extreme deviation, showing that when the sum $ S_n/n = a_n $ with $ a_n \to \infty $, the conditional distribution of $ X_1 $ converges in variation norm to the tilted distribution at $ a_n $, under light-tailed assumptions with regularity. The result extends the classical Gibbs conditioning principle to the extreme deviation regime, providing a local approximation of the conditional density via second-order expansions.

ABSTRACT

This paper explores a conditional Gibbs theorem for a random walkinduced by i.i.d. (X_{1},..,X_{n}) conditioned on an extreme deviation of its sum (S_{1}^{n}=na_{n}) or (S_{1}^{n}>na_{n}) where a_{n} ightarrow\infty. It is proved that when the summands have light tails with some additional regulatity property, then the asymptotic conditional distribution of X_{1} can be approximated in variation norm by the tilted distribution at point a_{n}, extending therefore the classical LDP case.

Motivation & Objective

• To investigate the asymptotic behavior of the conditional distribution of $ X_1 $ given extreme deviations of the sum $ S_n/n = a_n $, where $ a_n \to \infty $.
• To extend the classical Gibbs conditional principle, valid for constant $ a > \mathbb{E}[X_1] $, to the case of growing $ a_n $.
• To establish convergence in total variation norm between the conditional distribution and the tilted distribution at $ a_n $, under light-tailed and regularity conditions.
• To provide a local approximation of the conditional density using second-order Edgeworth-type expansions.

Proposed method

• Condition the i.i.d. sequence $ (X_1, \dots, X_n) $ on the event $ S_n/n = a_n $ with $ a_n \to \infty $, using the local conditioning principle.
• Use the tilted distribution $ \pi^a $ with parameter $ t $ such that $ m(t) = a $, where $ m(t) = \frac{d}{dt} \log \Phi(t) $, and $ \Phi(t) $ is the moment generating function.
• Apply asymptotic analysis techniques for regularly varying functions and Laplace's method to approximate the conditional density near the mode of the likelihood.
• Derive second-order expansions of the conditional density using the third cumulant $ \mu_3(t) $ and variance $ s^2(t) $, showing that higher-order terms vanish in the limit.
• Establish convergence in variation norm by bounding the $ L^1 $-distance between the true conditional density and the tilted density.
• Use regular variation theory for the density tail $ p(x) \sim x^{-\beta}L(x) $, with $ \beta > 1 $, to analyze the behavior of $ \mu_3(t) $ and $ s^2(t) $ as $ t \to \infty $.

Experimental results

Research questions

• RQ1 Does the conditional distribution of $ X_1 $ given $ S_n/n = a_n $ with $ a_n \to \infty $ converge to the tilted distribution at $ a_n $?
• RQ2 How does the convergence rate in variation norm behave under extreme deviation compared to the classical large deviation principle?
• RQ3 Can second-order Edgeworth-type corrections improve the approximation of the conditional density in the extreme deviation regime?
• RQ4 What role do the regularity and tail behavior (e.g., regularly varying tails) of the underlying distribution play in the convergence of the conditional law?
• RQ5 Under what conditions does the third cumulant $ \mu_3(t) $ become negligible relative to the variance, ensuring the validity of the tilted approximation?

Key findings

• The conditional distribution of $ X_1 $ given $ S_n/n = a_n $ converges in total variation norm to the tilted distribution $ \pi^{a_n} $ as $ n \to \infty $, under light-tailed and regularity conditions.
• For $ h \in R_\beta $ with $ \beta > 1 $, the third cumulant $ \mu_3(t) \sim \frac{M_6 - 3}{2} \psi''(t) $, and $ \mu_3(t)/s^3(t) \to 0 $, implying higher-order terms vanish.
• For $ h \in R_\infty $, the ratio $ \mu_3(t)/s^3(t) \to 0 $ under condition $ \psi(t)\epsilon(t)/t \to 0 $, ensuring the validity of the Gaussian approximation.
• The variance $ s^2(t) \sim \psi'(t) $, and the second derivative of the cumulant generating function satisfies $ h''(\hat{x}) \sim \beta(\beta-1)\psi(t)^{\beta-2}l_0(\psi(t)) $, which controls the curvature of the likelihood.
• The $ L^1 $-distance between the conditional density and the tilted density converges to zero, confirming convergence in variation norm.
• The result generalizes the Gibbs principle from constant deviations to divergent $ a_n $, showing that all summands $ X_i $ concentrate near $ a_n $ under extreme deviation, with the conditional law approximated by $ \pi^{a_n} $.