Let {B H (t) : t ≥ 0} be a fractional Brownian motion with Hurst parameter H ∈ 1 2 , 1 . For the storage processas u → ∞. This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.
This paper considers extreme values attained by a centered, multidimensional
Gaussian process $X(t)= (X_1(t),\ldots,X_n(t))$ minus drift
$d(t)=(d_1(t),\ldots,d_n(t))$, on an arbitrary set $T$. Under mild regularity
conditions, we establish the asymptotics of \[\log\mathbb P\left(\exists{t\in
T}:\bigcap_{i=1}^n\left\{X_i(t)-d_i(t)>q_iu\right\}\right),\] for positive
thresholds $q_i>0$, $i=1,\ldots,n$, and $u\to\infty$. Our findings generalize
and extend previously known results for the single-dimensional and
two-dimensional cases. A number of examples illustrate the theory
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 {Y (a) n : n ≥ 1} be a sequence of independent and identically distributed random variables and {X (a) t : t ≥ 0} be a Lévy process such that X (a)→ R, for some random variable R and some function (·). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.
This paper considers a Lévy-driven queue (i.e., a Lévy 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 Lévy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of P(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.
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