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  • 1
    Language: English
    In: Queueing Systems, 2011, Vol.68(3), pp.261-266
    Description: We discuss the problem of establishing an upper bound for the distribution tail of the stationary waiting time D in the GI / GI /1 FCFS queue.
    Keywords: FCFS single server queue ; Stationary waiting time ; Heavy tails ; Large deviations ; Long tailed distribution ; Subexponential distribution ; Integrated tail distribution ; Accuracy of approximation ; Lower and upper bounds
    ISSN: 0257-0130
    E-ISSN: 1572-9443
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  • 2
    Language: English
    In: Stochastic Processes and their Applications, April 2018, Vol.128(4), pp.1316-1332
    Description: We study subexponential tail asymptotics for the distribution of the maximum of a process with negative drift for the entire range of . We consider compound renewal processes with linear drift and Lévy processes. For both processes we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes with drift particularly includes the Cramér–Lundberg renewal risk process.
    Keywords: Lévy Process ; Compound Renewal Process ; Distribution Tails ; Heavy Tails ; Long-Tailed Distributions ; Subexponential Distributions ; Random Walk ; Mathematics
    ISSN: 0304-4149
    E-ISSN: 1879-209X
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  • 3
    Description: It is shown how a natural representation of perpetuities as asymptotically homogeneous in space Markov chains allows to prove various asymptotic tail results for stable perpetuities and limit theorems for unstable ones. Some of these results are new while others essentially improve moment conditions known in the literature. Both subexponential and Cram\'er's cases are considered.
    Keywords: Mathematics - Probability ; 60h25, 60j05, 60f10, 60f05
    Source: Cornell University
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  • 4
    Description: We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t〉0$. We consider compound renewal processes with linear drift and L\'evy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cram\'er-Lundberg risk process.
    Keywords: Mathematics - Probability ; 60f10, 60g51, 60k05
    Source: Cornell University
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  • 5
    Language: English
    In: Mathematics of operations research, 2012, Vol.37(2), pp. 201-218
    ISSN: 0364765x
    Source: Deutsche Zentralbibliothek für Wirtschaftswissenschaften
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  • 6
    Language: English
    In: Mathematics of Operations Research, 2012, Vol.37(2), p.201(18)
    Description: We present upper and lower bounds for the tail distribution of the stationary waiting time D in the stable GI / GI / s first-come first-served (FCFS) queue. These bounds depend on the value of the traffic load ρ which is the ratio of mean service and mean interarrival times. For service times with intermediate regularly varying tail distribution the bounds are exact up to a constant, and we are able to establish a “principle of s − k big jumps” in this case (here k is the integer part of ρ), which gives the most probable way for the stationary waiting time to be large. Another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time.
    Keywords: Probability Theory – Analysis
    ISSN: 0364-765X
    E-ISSN: 15265471
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  • 7
    Language: English
    In: Stochastic Processes and their Applications, August 2013, Vol.123(8), pp.3027-3051
    Description: We consider a positive recurrent Markov chain on with asymptotically zero drift which behaves like at infinity; this model was first considered by Lamperti. We are interested in tail asymptotics for the stationary measure. Our analysis is based on construction of a harmonic function which turns out to be regularly varying at infinity. This harmonic function allows us to perform non-exponential change of measure. Under this new measure Markov chain is transient with drift like at infinity and we compute the asymptotics for its Green function. Applying further the inverse transform of measure we deduce a power-like asymptotic behaviour of the stationary tail distribution. Such a heavy-tailed stationary measure happens even if the jumps of the chain are bounded. This model provides an example where possibly bounded input distributions produce non-exponential output.
    Keywords: Markov Chain ; Invariant Distribution ; Lamperti Problem ; Asymptotically Zero Drift ; Test (Lyapunov) Function ; Regularly Varying Tail Behaviour ; Convergence to [Formula Omitted]-Distribution ; Renewal Function ; Harmonic Function ; Non-Exponential Change of Measure ; Martingale Technique ; Mathematics
    ISSN: 0304-4149
    E-ISSN: 1879-209X
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  • 8
    Language: English
    In: Extremes, 2015, Vol.18(3), pp.315-347
    Description: For a centered d -dimensional Gaussian random vector ξ = ( ξ 1 , … , ξ d ) and a homogeneous function h : ℝ d → ℝ we derive asymptotic expansions for the tail of the Gaussian chaos h ( ξ ) given the function h is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of h ( ξ ) and its density at infinity and then discuss possible extensions for some general ξ with polar representation.
    Keywords: Wiener chaos ; Polynomial chaos ; Gaussian chaos ; Multidimensional normal distribution ; Subexponential distribution ; Determinant of a random matrix ; Gaussian orthogonal ensemble ; Diameter of random Gaussian clouds ; Max-domain of attraction
    ISSN: 1386-1999
    E-ISSN: 1572-915X
    Source: Springer Science & Business Media B.V.
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  • 9
    Language: English
    In: Journal of Theoretical Probability, 2008, Vol.21(1), pp.234-245
    Description: We consider a time-homogeneous real-valued Markov chain  X n , n ≥0. Suppose that this chain is transient, that is, X n generates a σ -finite renewal measure. We prove the key renewal theorem under the condition that this chain has jumps that are asymptotically homogeneous at infinity and asymptotically positive drift.
    Keywords: Transient Markov chain ; Renewal kernel ; Renewal measure ; Key renewal theorem ; Green function
    ISSN: 0894-9840
    E-ISSN: 1572-9230
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  • 10
    Language: English
    In: Queueing Systems, Feb, 2014, Vol.76(2), p.111(2)
    Description: Byline: Larisa G. Afanasyeva (1), Dmitry Korshunov (2), Dmitry Shabanov (1) Author Affiliation: (1) Lomonosov Moscow State University, Moscow, Russia (2) Sobolev Institute of Mathematics, Novosibirsk , 630090, Russia Article History: Registration Date: 16/12/2013 Online Date: 11/01/2014
    Keywords: Economics / Management Science ; Operations Research/Decision Theory ; Computer Communication Networks ; Probability Theory and Stochastic Processes ; Production/Logistics/Supply Chain ; Systems Theory, Control ; Engineering;
    ISSN: 0257-0130
    E-ISSN: 15729443
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