2014/12/12 by Chengxiu Ling, Ling, Chengxiu
Economics, Econometrics and Finance · Mathematics · #60G15 #60G70 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60G15 #msc:60G70
paper · pdf · doi:10.48550/arxiv.1412.3934
openalex publication_date 2014/12/12 · arxiv created 2014/12/15 · arxiv updated 2014/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)=\mathbb P\Cr(u)≠ϕ\. Instead of stationary processes dealt with by Dȩbicki 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 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.