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Tail estimates for the Brownian excursion area and other Brownian areas  [PDF]
Svante Janson,Guy Louchard
Mathematics , 2007,
Abstract: Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.
Exact L_2-small ball asymptotics of Gaussian processes and the spectrum of boundary value problems with "non-separated" boundary conditions  [PDF]
A. I. Nazarov
Mathematics , 2007,
Abstract: We sharpen a classical result on the spectral asymptotics of the boundary value problems for self-adjoint ordinary differential operator. Using this result we obtain the exact $L_2$-small ball asymptotics for a new class of zero mean Gaussian processes. This class includes, in particular, integrated generalized Slepian process, integrated centered Wiener process and integrated centered Brownian bridge.
From the Lifshitz tail to the quenched survival asymptotics in the trapping problem  [PDF]
Ryoki Fukushima
Mathematics , 2009,
Abstract: The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion in the Poissonian obstacles. As an application, we derive the quenched asymptotics for the Brownian motion in traps distributed according to a random perturbation of the lattice.
Tail Asymptotics of Deflated Risks  [PDF]
E. Hashorva,C. Ling,Z. Peng
Mathematics , 2013,
Abstract: Random deflated risk models have been considered in recent literatures. In this paper, we investigate second-order tail behavior of the deflated risk X=RS under the assumptions of second-order regular variation on the survival functions of the risk R and the deflator S. Our findings are applied to approximation of Value at Risk, estimation of small tail probability under random deflation and tail asymptotics of aggregated deflated risk
Beta-gamma tail asymptotics  [PDF]
Jim Pitman,Miklos Z. Racz
Mathematics , 2015,
Abstract: We compute the tail asymptotics of the product of a beta random variable and a generalized gamma random variable which are independent and have general parameters. A special case of these asymptotics were proved and used in a recent work of Bubeck, Mossel, and R\'acz in order to determine the tail asymptotics of the maximum degree of the preferential attachment tree. The proof presented here is simpler and highlights why these asymptotics hold.
Tail Asymptotics of Supremum of Certain Gaussian Processes over Threshold Dependent Random Intervals  [PDF]
Krzysztof D?bicki,Enkelejd Hashorva,Lanpeng Ji
Mathematics , 2013,
Abstract: Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ 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.
Tail asymptotics for busy periods  [PDF]
Ken R. Duffy,Sean P. Meyn
Mathematics , 2011, DOI: 10.1214/13-SSY098
Abstract: The busy period for a queue is cast as the area swept under the random walk until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the large-deviations asymptotics of $B$ is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs: I) The scaled probability of a large busy period has the asymptote, for any $b>0$, \lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P(B\geq bn) = -K\sqrt{b}, \hbox{where} \quad K = 2 \sqrt{-\int_0^{\lambda^*} \Lambda(\theta) d\theta}, \quad \hbox{with $\lambda^*=\sup\{\theta:\Lambda(\theta)\leq0\}$,} and with $\Lambda$ denoting the scaled cumulant generating function of the increments process. II) The most likely path to a large swept area is found to be a simple rescaling of the path on $[0,1]$ given by, [\psi^*(t) = -\Lambda(\lambda^*(1-t))/\lambda^*.] In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have very different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero. These results partially answer an open problem of Kulick and Palmowski regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics $(\lambda^*, K)$ based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.
Tail Asymptotics for Discrete Event Systems  [PDF]
Marc Lelarge
Mathematics , 2007,
Abstract: In the context of communication networks, the framework of stochastic event graphs allows a modeling of control mechanisms induced by the communication protocol and an analysis of its performances. We concentrate on the logarithmic tail asymptotics of the stationary response time for a class of networks that admit a representation as (max,plus)-linear systems in a random medium. We are able to derive analytic results when the distribution of the holding times are light-tailed. We show that the lack of independence may lead in dimension bigger than one to non-trivial effects in the asymptotics of the sojourn time. We also study in detail a simple queueing network with multipath routing.
Brownian Bridge and Self-Avoiding Random Walk  [PDF]
Yevgeniy Kovchegov
Mathematics , 2002,
Abstract: We establish the Brownian bridge asymptotics for a scaled self-avoiding walk conditioned on arriving to a far away point $n \vec{a}$ for $\vec{a}$ in $Z^d$, as $n$ increases to infinity.
Tail asymptotics for a random sign Lindley recursion  [PDF]
Maria Vlasiou,Zbigniew Palmowski
Mathematics , 2008,
Abstract: We investigate the tail behaviour of the steady state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queuing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.
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