Kamil Marcin Kosinski

Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let

On the functional limits for partial sums under stable law

\{Y^{(a)}_{n}:n\ge1\}

Logarithmic asymptotics for multidimensional extremes under nonlinear scalings

be a sequence of independent and identically distributed random variables and

Gaussian queues in light and heavy traffic

\{X^{(a)}_{t}:t\ge0\}

Queue lengths and workloads in polling systems

be a Lévy process such that

Gaussian queues in light and heavy-traffic

X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)}

On the infimum attained by a reflected Lévy process

,

On the infimum attained by a reflected Lvy process

\mathbb{E}X_{1}^{(a)}<0

L\'evy-driven polling systems

and

Extremes of multidimensional Gaussian processes

\mathbb{E}X_{1}^{(a)}\uparrow0