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  • Mathematics
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Year
  • 1
    Language: English
    In: Russian Mathematical Surveys, 2012, Vol.67(1), pp.93-165
    Description: Boolean functions are among the fundamental objects of discrete mathematics, especially in those of its subdisciplines which fall under mathematical logic and mathematical cybernetics. The language of Boolean functions is convenient for describing the operation of many discrete systems such as contact networks, Boolean circuits, branching programs, and some others. An important parameter of discrete systems of this kind is their complexity. This characteristic has been actively investigated starting from Shannon's works. There is a large body of scientific literature presenting many fundamental results. The purpose of this survey is to give an account of the main results over the last sixty years related to the complexity of computation (realization) of Boolean functions by contact networks, Boolean circuits, and Boolean circuits without branching. Bibliography: 165 titles.
    Keywords: Mathematics;
    ISSN: 0036-0279
    E-ISSN: 1468-4829
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  • 2
    Language: English
    In: Siberian Mathematical Journal, 2011, Vol.52(4), pp.655-664
    Description: We consider a Markov chain on ℝ + with asymptotically zero drift and finite second moments of jumps. We assume that the chain has invariant distribution. The paper is devoted to the existence and nonexistence of moments of invariant distribution. Our analysis is based on the technique of test functions.
    Keywords: stationary Markov chain ; asymptotically zero drift ; invariant distribution ; heavy-tailed distribution ; power moments ; Weibull-type moments ; test (Lyapunov) functions ; equilibrium identity
    ISSN: 0037-4466
    E-ISSN: 1573-9260
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  • 3
    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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  • 4
    Language: English
    In: Doklady Mathematics, Sept, 2013, Vol.88(2), p.566(3)
    Description: Byline: D. A. Korshunov (1), V. I. Piterbarg (2), E. Hashorva (3) Author Affiliation: (1) Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia (2) Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia (3) UNIL-Dorigny, 1015, Lausanne, Switzerland Article History: Registration Date: 23/10/2013 Received Date: 22/03/2013 Online Date: 26/10/2013 Article note: Original Russian Text (c) D.A. Korshunov, V.I. Piterbarg, E. Hashorva, 2013, published in Doklady Akademii Nauk, 2013, Vol. 452, No. 5, pp. 483--485. Presented by Academician A.N. Shiryaev March 6, 2013
    Keywords: Mathematics;
    ISSN: 1064-5624
    E-ISSN: 15318362
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  • 5
    Language: English
    In: Mathematical Notes, 2015, Vol.97(5), pp.878-891
    Description: The asymptotics of the multidimensional Laplace integral for the case in which the phase attains its minimum on an arbitrary smooth manifold is studied. Applications to the study of the asymptotics of the distribution of Gaussian and Weibullian random chaoses are considered.
    Keywords: Laplace asymptotic method ; Gaussian chaos ; Weibullian chaos ; Gelfand-Leray differential form ; random chaos
    ISSN: 0001-4346
    E-ISSN: 1573-8876
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  • 6
    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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  • 7
    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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  • 8
    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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  • 9
    Language: English
    In: Statistics and Probability Letters, 2011, Vol.81(9), pp.1419-1424
    Description: We consider a transient random walk on which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time .
    Keywords: Multidimensional Random Walk ; Transience ; First Return to the Origin ; Local Limit Theorem ; Defective Renewal Function ; Statistics ; Mathematics
    ISSN: 0167-7152
    E-ISSN: 1879-2103
    Source: ScienceDirect Journals (Elsevier)
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  • 10
    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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