Sample Path Large Deviations Research Articles

Sample path large deviations for super-Brownian motion

We derive two large deviation principles of Freidlin-Wentzell type for rescaled super-Brownian motion. For one of the appearing rate functions an integral representation is given and interpreted as ‘Kakutani-Hellinger energy’. As a tool we develop estimates for the Laplace functionals of (historical) super-Brownian motion and certain maximal inequalities. Also it is shown that the Holder norm of index α<1/2 of the processt↦〈f, X t 〉 possesses some finite exponential moments provided the functionf is smooth.

Sample path large deviations for super-Brownian motion

Sample path large deviations for super-Brownian motion

Sample path large deviations and intree networks

Using the contraction principle, in this paper we derive a set of closure properties for sample path large deviations. These properties include sum, reduction, composition and reflection mapping. Using these properties, we show that the exponential decay rates of the steady state queue length distributions in an intree network with routing can be derived by a set of recursive equations. The solution of this set of equations is related to the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. We also prove a conditional limit theorem that illustrates how a queue builds up in an intree network.

A Problem on Large Deviations in a Space of Trajectories

Previous article Next article A Problem on Large Deviations in a Space of TrajectoriesI. F. PinelisI. F. Pinelishttps://doi.org/10.1137/1126006PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Godovanchuk, Probabilities of large deviations for sums of independent random variables attracted to a stable law, Theory Prob. Appl., 23 (1978), 602–608 0422.60020 LinkGoogle Scholar[2] A. A. Borovkov, Analysis of large deviations in boundary-value problems with arbitrary boundaries, Sibirsk. Mat. Ž., 5 (1964), 253–289 29:645 0166.14303 A. A. Borovkov, Analysis of large deviations in boundary-value problems with arbitrary boundaries. II, Sibirsk. Mat. Ž., 5 (1964), 750–767 29:4117 0202.48502 Google Scholar[3] A. A. Borovkov, Boundary-value problems for random walks and large deviations in function spaces, Theory Prob. Appl., 12 (1967), 575–595 0178.20004 LinkGoogle Scholar[4] A. A. Mogul'skii, Large deviations for trajectories of multi-dimensional random walks, Theory Prob. Appl., 21 (1976), 300–315 0366.60031 LinkGoogle Scholar[5] A. V. Nagaev, Limit theorems that take into account large deviations when Cramér's condition is violated, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk, 13 (1969), 17–22, (In Russian.) 43:8108 0226.60043 Google Scholar[6] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– 48:1287 0219.60027 Google Scholar[7] D. Kh. Fuk and , S. V. Nagaev, Probability inequalities for sums of independent random variables, Theory Prob. Appl., 16 (1971), 643–660 0259.60024 LinkGoogle Scholar[8] A. V. Nagaev, doctoral dissertation, Tashkent, 1970. (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Hidden regular variation for point processes and the single/multiple large point heuristicThe Annals of Applied Probability, Vol. 32, No. 1 Cross Ref On the Nonuniform Berry–Esseen Bound Cross Ref Tail asymptotics for delay in a half-loaded GI/GI/2 queue with heavy-tailed job sizes21 June 2015 | Queueing Systems, Vol. 81, No. 4 Cross Ref An asymptotically Gaussian bound on the Rademacher tailsElectronic Journal of Probability, Vol. 17, No. none Cross Ref Large deviations for random walks under subexponentiality: The big-jump domainThe Annals of Probability, Vol. 36, No. 5 Cross Ref An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions11 September 2008 | Siberian Mathematical Journal, Vol. 49, No. 4 Cross Ref Integro-local and integral theorems for sums of random variables with semiexponential distributionsSiberian Mathematical Journal, Vol. 47, No. 6 Cross Ref Large deviations of the first passage time for a random walk with semiexponentially distributed jumpsSiberian Mathematical Journal, Vol. 47, No. 6 Cross Ref On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution TailsA. A. Borovkov and K. A. Borovkov25 July 2006 | Theory of Probability & Its Applications, Vol. 46, No. 2AbstractPDF (215 KB)Large-Deviation Probabilities for Maxima of Sums of Independent Random Variables with Negative Mean and Subexponential DistributionD. A. Korshunov25 July 2006 | Theory of Probability & Its Applications, Vol. 46, No. 2AbstractPDF (155 KB)Exact Asymptotics for Large Deviation Probabilities, with Applications Cross Ref Sample path large deviations in finer topologies4 April 2007 | Stochastics and Stochastic Reports, Vol. 67, No. 3-4 Cross Ref The time until the final zero crossing of random sums with application to nonparametric bandit theoryApplied Mathematics and Computation, Vol. 63, No. 2-3 Cross Ref Probabilities of Large Deviations on the Whole AxisL. V. Rozovskii17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 1AbstractPDF (1937 KB)Sharp Exponential Inequalities for the Martingales in the 2-Smooth Banach Spaces and Applications to “Scalarizing” Decoupling Cross Ref Large Deviations of Sums of Independent Random Variables without Several Maximal SummandsV. V. Vinogradov and V. V. Godovan’chuk17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 3AbstractPDF (383 KB)Probabilities of Large Deviations of Sums of Independent Random Variables with Common Distribution Function in the Domain of Attraction of the Normal LawL. V. Rozovskii17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 4AbstractPDF (1377 KB) Volume 26, Issue 1| 1981Theory of Probability & Its Applications History Submitted:05 February 1979Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126006Article page range:pp. 69-84ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces

Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces