Extremes of order statistics of self-similar processes

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Let {Xᵢ(t),t≥0}, 1≤ i≤ n be independent copies of a random process {X(t), t≥0}. For a given positive constant u, define the set of rth conjunctions Cᵣ(u):={t∈[0,1]: Xᵣ:ₙ(t)>u} with Xᵣ:ₙ the rth largest order statistics of Xᵢ, 1≤ i≤ n. In numerical applications such as brain mapping and digital communication systems, of interest is the approximation of pᵣ(u)=P{Cᵣ(u)≠ϕ}. Instead of stationary processes dealt with by Dȩbicki et al. (2014), we consider in this paper X a self-similar R-valued process with P-continuous sample paths. By imposing the Albin's conditions directly on X, we establish an exact asymptotic expansion of pᵣ(u) as u tends to infinity. As a by-product we derive the asymptotic tail behaviour of the mean sojourn time of Xᵣ: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.