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 $h_p(t)=(1/z_p(t))p\log\log t$

Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process $X\equiv\{X(t):t\in\mathbb R\}$ with stationary increments and variance function $\sigma^2_X(\cdot)$, equipped with a deterministic drift $c>0$, reflected at 0: $$Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)).$$ We study the resulting stationary workload process $Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\}$ in the limiting regimes $c\to 0$ (heavy traffic) and $c\to\infty$ (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function $\delta(c)$ such that $Q^{(c)}_X(\delta(c)\cdot)/\sigma_X(\delta(c))$ converges to a non-trivial limit in $C[0,\infty)$

On the infimum attained by a reflected L\'evy process

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided L\'evy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of \prob{M(T_u)> u} (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios

Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals

Let {X(t),t a parts per thousand yen 0} be a centered Gaussian process and let gamma be a non-negative constant. In this paper we study the asymptotics of as , with an independent of X non-negative random variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes

Logarithmic Asymptotics For Probability Of Component-Wise Ruin In A Two-Dimensional Brownian Model

We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function P(u) for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where u is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of − ln P(u)/u as u tends to infinity, which depends essentially on the correlation ρ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem

Parisian ruin of self-similar Gaussian risk processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times