We extend the Markov additive methodology developed in [Ann. Appl. Probab. 9 (1999) 110-145, Ann. Appl. Probab. 11 (2001) 596-607] to obtain the sharp asymptotics of the steady state probability of a queueing network when one of the nodes gets large. We focus on a new phenomenon we call a bridge. The bridge cases occur when the Markovian part of the twisted Markov additive process is one null recurrent or one transient, while the jitter cases treated in [Ann. Appl. Probab. 9 (1999) 110-145, Ann. Appl. Probab. 11 (2001) 596-607] occur when the Markovian part is (one) positive recurrent. The asymptotics of the steady state is an exponential times a polynomial term in the bridge case, but is purely exponential in the jitter case. We apply this theory to a modified, stable, two node Jackson network where server two helps server one when server two is idle. We derive the sharp asymptotics of the steady state distribution of the number of customers queued at each node as the number of customers queued at the server one grows large. In so doing we get an intuitive understanding of the companion paper [Ann. Appl. Probab. 15 (2005) 519-541] which gives a large deviation analysis of this problem using the flat boundary theory in the book by Shwartz and Weiss. Unlike the (unscaled) large deviation path of a Jackson network which jitters along the boundary, the unscaled large deviation path of the modified network tries to avoid the boundary where server two helps server one (and forms a bridge).
We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics…
Exact Asymptotics for the Exit Time Probabilities of Scalar Ornstein-Uhlenbeck Bridges
State Key Laboratory of Mechanics and Control for Aerospace Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics（南京航空航天大学航空航天结构力学与控制国家重点实验室航空学院）；School of Automation, Nanjing University of Science and Technology（南京理工大学自动化学院）；College of Mechanical and Electrical Engineering, Wenzhou University（温州大学机械电气工程学院）
The space-time distribution, $Q_A(x,dt dξ)$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtdξ)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt dξ)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). The results are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joini…
In this paper, we investigate exact tail asymptotics for the stationary distribution of a fluid model driven by the $M/M/c$ queue, which is a two-dimensional queueing system with a discrete phase and a continuous level. We extend the kernel method to study tail asymptotics of its stationary distribution, and a total of three types of exact tail asymptotics is identified from our study and reported in the paper.
This short communication considers an infinite-server system with overdispersed input. The objective is to identify the exact tail asymptotics of the number of customers present at a given point in time under a specific scaling of the model (which involves both the arrival rate and time). The proofs rely on a change-of-measure approach. The results obtained are illustrated by a series of examples.
Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term to walkers on the circle conditioned to start and end at the same position. For each return time $T<π^2$ we show that if the absolute value of the drift is less than a critical value then the expected total winding number is asymptotically zero. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected total winding is fi…
We derive in this article the asymptotic behavior as well as non-asymptotical estimates of tail of distribution for self-normalized sums of random variables (r.v.) under natural classical norming. We investigate also the case of non-standard random norming function and the tail asymptotic for the maximum distribution for self-normalized statistics. We do not suppose the independence or identical distributionness of considered random variables, but we assume the existence and sufficient smoothness of its density. We show also the exactness of our conditions imposed on the considered random vari…
Consider the problem of matching two independent i.i.d. samples of size $N$ from two distributions $P$ and $Q$ in $\mathbb{R}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead in this paper the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with $N$ atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show…
Tied-down renewal processes are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time $t$. We give an analytical derivation of the two-time correlation function for such processes in the Laplace space of all temporal variables. This yields the exact asymptotic expression of the correlation in the Porod regime of short separations between the two times and in the persistence regime of large separations. We also investigate other quantities, such as the backward and forward recurrence times, as well as the …
We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expr…
Exact Solutions to the Quantum Schrödinger Bridge Problem
Machine Learning Research, Pfizer Bristol-Myers Squibb, ML Research Cambridge（机器学习研究部，辉瑞百灵益泰，ML研究剑桥）；Pfizer Worldwide Research Development and Medical（辉瑞全球研发与医疗）；Machine Learning Research group, Cambridge, MA, USA（机器学习研究组，剑桥，马萨诸塞州，美国）
