82 research outputs found
On convergence to stationarity of fractional Brownian storage
With M(t) := sups2[0,t] A(s) − s denoting the running maximum of a fractional Brownian motion A(·)
with negative drift, this paper studies the rate of convergence of P(M(t) > x) to P(M > x). We define
two metrics that measure the distance between the (complementary) distribution functions P(M(t) > · )
and P(M > · ). Our main result states that both metrics roughly decay as exp(−#t2−2H), where # is the
decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was
computed recently [16]. The proofs extensively rely on application of the well-known large deviations
theorem for Gaussian processes. We also show that the identified relation between the decay of the
convergence metrics and busy-period asymptotics holds in other settings as well, most notably when
G¨artner-Ellis-type conditions are fulfilled
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 with stationary increments and variance
function , equipped with a deterministic drift ,
reflected at 0: We
study the resulting stationary workload process
in the limiting regimes (heavy
traffic) and (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 such that
converges to a non-trivial
limit in
Large deviations of infinite intersections of events in Gaussian processes
The large deviations principle for Gaussian measures in Banach space is given by the generalized Schilder's theorem. After assigning a norm ||f|| to paths f in the reproducing kernel Hilbert space of the underlying Gaussian process, the probability of an event A can be studied by minimizing the norm over all paths in A. The minimizing path f*, if it exists, is called the most probable path and it determines the corresponding exponential decay rate. The main objective of our paper is to identify the most probable path for the class of sets A that are such that the minimization is over a closed convex set in an infinite-dimensional Hilbert space. The `smoothness' (i.e., mean-square differentiability) of the Gaussian process involved has a crucial impact on the structure of the solution. Notably, as an example of a non-smooth process, we analyze the special case of fractional Brownian motion, and the set A consisting of paths f at or above the line t in [0,1]. For H>1/2, we prove that there is an s such that
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