Extremes of order statistics of self-similar processes

Abstract

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a random process $\{X(t), t\ge0\}$. For a given positive constant $u$, define the set of $r$th conjunctions $C_r(u):=\{t\in[0,1]: X_{r:n}(t)>u\}$ with $ X_{r:n}$ the $r$th largest order statistics of $X_i, 1\le i\le n$. In numerical applications such as brain mapping and digital communication systems, of interest is the approximation of $p_r(u)=\mathbb P\{C_r(u)\neq\phi\}$. Instead of stationary processes dealt with by D\c{e}bicki et al. (2014), we consider in this paper $X$ a self-similar $\mathbb R$-valued process with $P$-continuous sample paths. By imposing the Albin's conditions directly on $X$, we establish an exact asymptotic expansion of $p_r(u)$ as $u$ tends to infinity. As a by-product we derive the asymptotic tail behaviour of the mean sojourn time of $X_{r:n}$ over an increasing threshold. Finally, our findings are illustrated for the case that $X$ is a bi-fractional Brownian motion, a sub-fractional Brownian motion, and a generalized self-similar skew-Gaussian process.