An Erd\"os--R\'ev\'esz type law of the iterated logarithm for order statistics of a stationary Gaussian process

Let $\{X(t):t\in\mathbb R_+\}$ be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, $\mathbb E X(t) = 0$, $\mathbb E X^2(t) = 1$ and correlation function satisfying (i) $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha})$ as $t\to 0$ for some $0\le\alpha\le 2, C>0$, (ii) $\sup_{t\ge s}|r(t)|0$ and (iii) $r(t) = O(t^{-\lambda})$ as $t\to\infty$ for some $\lambda>0$. For any $n\ge 1$, consider $n$ mutually independent copies of $X$ and denote by $\{X_{r:n}(t):t\ge 0\}$ the $r$th smallest order statistics process, $1\le r\le n$. We provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(\mathscr E_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text{i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,1]}X_{r:n}(s)>f_p(t))=\mathscr C(t\log^{1-p} t)^{-1}$, $\mathscr C>0$, $\mathbb P(\mathscr E_{f_p})= 1_{\{p\ge 0\}}$. Consequently, with $\xi_p (t) = \sup\{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}$, for $p\ge 0$, $\lim_{t\to\infty}\xi_p(t)=\infty$ and $\limsup_{t\to\infty}(\xi_p(t)-t)=0$ a.s.. Complementary, we prove an Erd\"os-R\'ev\'esz type law of the iterated logarithm lower bound on $\xi_p(t)$, i.e., $\liminf_{t\to\infty}(\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$, $\liminf_{t\to\infty}\log(\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\in(0,1]$, where $h_p(t)=(1/z_p(t))p\log\log t$

Note on maximally entangled Eisert-Lewenstein-Wilkens quantum games

Maximally entangled Eisert-Lewenstein-Wilkens games are analyzed. For a general class of gate operators defined in the previous papers of the first author the general conditions are derived which allow to determine the form of gate operators leading to maximally entangled games. The construction becomes particularly simple provided one does distinguish between games differing by relabelling of strategies. Some examples are presented.Comment: 20 pages, no figures, appendix added, references added, concluding remarks extende

Geometric properties of semitube domains

In the paper we study the geometry of semitube domains in $\mathbb C^2$. In particular, we extend the result of Burgu\'es and Dwilewicz for semitube domains dropping out the smoothness assumption. We also prove various properties of non-smooth pseudoconvex semitube domains obtaining among others a relation between pseudoconvexity of a semitube domain and the number of connected components of its vertical slices. Finally, we present an example showing that there is a non-convex domain in $\mathbb C^n$ such that its image under arbitrary isometry is pseudoconvex.Comment: 6 page