[Paper Review] Extremes of Order Statistics of Stationary Processes

Krzysztof Dȩbicki, Enkelejd Hashorva+2 more|arXiv (Cornell University)|Mar 28, 2014

Bayesian Methods and Mixture Models26 references4 citations

TL;DR

This paper derives exact asymptotic expansions for the probability that the rth order statistics process of n i.i.d. stationary processes exceeds a high threshold u over time interval [0,T], under Albin’s conditions. It establishes tail asymptotics for skew-Gaussian processes and a Gumbel limit theorem for minima of Gaussian processes, extending Li and Shao’s normal comparison lemma to extreme order statistics.

ABSTRACT

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \ldots , X_n(t), t\ge 0$. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's conditions on $X$, in this paper we obtain an exact asymptotic expansion of this probability as $u$ tends to infinity. Further, we establish the tail asymptotics of the supremum of a generalized skew-Gaussian process and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.

Motivation & Objective

• To derive exact asymptotic expansions for the probability that the rth order statistics process exceeds a high threshold u over [0,T].
• To extend Li and Shao’s normal comparison lemma to minimum and maximum of Gaussian random vectors.
• To establish tail asymptotics for the supremum of order statistics processes of skew-Gaussian processes.
• To prove a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes.
• To provide a theoretical foundation for applications in FMRI, brain mapping, and digital communications involving rare conjunction events.

Proposed method

• Imposes Albin’s conditions (Gumbel MDA, conditional limit, short-lasting exceedance) on the stationary process X.
• Uses a time-scale transformation and conditional limit theorems to analyze exceedance behavior of order statistics.
• Applies a generalized form of Li and Shao’s normal comparison lemma to bound differences in joint tail probabilities of Gaussian vectors.
• Derives exact asymptotic expansions for P(sup_{t∈[0,T]} X_{r:n}(t) > u) as u→∞.
• Establishes a Gumbel limit law for minima of stationary Gaussian processes via extreme value theory.
• Uses Slepian’s inequality and integral bounds to control tail differences in extreme order statistics.

Experimental results

Research questions

• RQ1 What is the exact asymptotic behavior of P(sup_{t∈[0,T]} X_{r:n}(t) > u) as u→∞ for stationary processes satisfying Albin’s conditions?
• RQ2 How do the tail properties of skew-Gaussian processes affect the supremum of their order statistics?
• RQ3 Can Li and Shao’s normal comparison lemma be extended to handle minimum and maximum order statistics of Gaussian vectors?
• RQ4 What is the limiting distribution of the minimum order statistics of stationary Gaussian processes at high thresholds?
• RQ5 How does the generalized Albin constant influence the asymptotic approximation of conjunction probabilities?

Key findings

• The paper derives an exact asymptotic expansion for P(C_{r,T,u} ≠ ∅) = P(sup_{t∈[0,T]} X_{r:n}(t) > u) as u→∞ under Albin’s conditions.
• The tail asymptotics of the supremum of order statistics processes of skew-Gaussian processes are established, extending results beyond Gaussian models.
• A Gumbel limit theorem is proven for the minimum order statistics of stationary Gaussian processes, confirming convergence to the Gumbel distribution.
• A new version of Li and Shao’s normal comparison lemma is derived for minimum and maximum order statistics of Gaussian vectors.
• The bound on the difference of joint tail probabilities involves a term proportional to n u^{-2(n-1)} exp(-n u^2 / (1 + ρ_{il})), showing exponential decay in n and u.
• The proof relies on a stochastic coupling via a family of Gaussian processes with varying correlation structures, enabling precise tail comparison.