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Lösungen inhomogener stochastischer Fixpunktgleichungen
Get PDFGegenstand der Arbeit ist die Untersuchung der Lösungen von inhomogenen stochastischen Fixpunktgleichungen. Die Fixpunktgleichungen sind Verteilungsgleichungen, die unter anderem bei der Analyse von zufälligen Divide-and-Conquer-Algorithmen eine wichtige Rolle spielen. Unter einer Lösung von stochastischen Fixpunktgleichungen versteht man ein Wahrscheinlichkeitsmaß, das ein Fixpunkt unter der zur Fixpunktgleichung assoziierten Abbildung ist.
Für mehrere Arten von stochastischen inhomogenen Fixpunktgleichungen, nämlich des Summen-, Infimum- und Supremumtyps wird eine Darstellungen von analytischen Transformierten (z.B. der Laplace-Transformierten oder der Überlebensfunktion) der Lösungen angegeben, die auf der positiven Halbachse konzentiert sind. Mithilfe dieser Darstellung können dann Lösungsmengen charakterisiert werden.
Für Lösungen inhomogener Fixpunktgleichungen des Summentyps, deren Träger die gesamten reellen Zahlen sind, wird eine Darstellung der Fourier-Transformierten angegeben und in einigen Spezialfällen werden die Lösungen charakterisiert
Large deviation tail estimates and related limit laws for stochastic fixed point equations
Get PDFWe study the forward and backward recursions generated by a stochastic fixed point equation (SFPE) of the form , where , for both the stationary and explosive cases.In the stationary case (when {\bf E} [\log \: A] < 0), we present results concerning the precise tail asymptotics for the random variable satisfyingthis SFPE. In the explosive case (when {\bf E} [\log \: A] > 0), we establish a central limit theorem for the forward recursion generated by the SFPE,namely the process , where \{ (A_n,B_n,D_n): n \in \pintegers \} is an i.i.d.\ sequence of random variables.Next, we consider recursionswhere the driving sequence of vectors, \{(A_n, B_n, D_n): n \in \pintegers \}, is modulated by a Markov chain in general state space. We demonstrate an asymmetry between the forward and backward recursions and develop techniques for estimating the exceedance probability. In the process, we establish an interesting connection between the regularity properties of and the recurrence properties of an associated -shifted Markov chain. We illustrate these ideas with several examples
Multivariate Mixed Poisson Processes
Get PDFMultivariate mixed Poisson processes are special multivariate counting processes whose coordinates are, in general, dependent. The first part of this thesis is devoted to properties which multivariate counting processes may possess. Such properties are, for example, the Markov property, the multinomial property and regularity. With regard to regularity we study the properties of transition probabilities and intensities. The second part of this thesis restricts the class of all multivariate counting processes by additional assumptions leading to different types of multivariate mixed Poisson processes which, however, are connected with each other. Using a multivariate version of the Bernstein-Widder theorem, it is shown that multivariate mixed Poisson processes are characterized by the multinomial property. Furthermore, regularity of multivariate mixed Poisson processes and properties of their moments are studied in detail. Throughout this thesis, two types of stability of properties of multivariate counting processes are studied: It is shown that most properties of a multivariate counting process are stable under certain linear transformations including the selection of single coordinates and summation of all coordinates. It is also shown that the different types of multivariate mixed Poisson processes under consideration are in a certain sense stable in time
On the moments of certain first passage times for linear growth processes
No full textLet be a stochastic process adapted to the filtration and with increments X1, X2, ... Set and Ln = m1 + ... + mn for n [greater-or-equal, slanted] 1. Then we call a linear growth process (LGP) if 1. (1) [mu] [less-than-or-equals, slant] Ln/n [less-than-or-equals, slant] [nu] a.s.f.a. n [greater-or-equal, slanted] n0 and2. (2) Ln/n --> [theta] a.s., as n --> [infinity] for suitable [mu], [nu], [theta] > 0 and some integer n0 [greater-or-equal, slanted] 1. In the case where (2) holds uniformly on a subevent of probability 1, is called a uniform linear growth process (ULGP), and if (1) and (2) are satisfied with Ln/n replaced by mn in (1), then is called a strong linear growth process (SLGP). For b [greater-or-equal, slanted] 0 and positive, continuous functions f on [0, [infinity]) we examine the first passage times [tau] = [tau] (b) = inffn [greater-or-equal, slanted] 1: Sn > b {(n)} as to existence of the moments of [tau] and S[tau] and related asymptotics. We will show that many results which are valid in the i.i.d. case carry over to LGP's under quite weak additional assumptions. In the case where is a SLGP and f(·) [triple bond; length as m-dash] 1, we will furthermore provide uniform integrability of the moments of the excess over the boundary S[tau] - b by renewal theoretic methods. This yields an expansion for E[tau] up to terms of order O(1), as b --> [infinity], when (Sn - n[theta])n[greater-or-equal, slanted]0 constitutes a martingale. In the final section the results will be applied to several examples from applied probability.first passage times excess over the boundary linear growth process renewal theory random walk queueing theory birth and death process branching process
Nonnegativity of odd functional moments of positive random variables with decreasing density
No full textIn this note we give some results on the nonnegativity of odd functional moments of random variables with a decreasing density. More precisely, we prove by purely elementary arguments, Egf(X - EX) [greater-or-equal, slanted] 0 for suitable functions gf that satisfy gf(x) = -gf(-x) for all x [greater-or-equal, slanted] 0 and random variables X [greater-or-equal, slanted] 0 with a decreasing Lebesgue density on (0, [infinity]) or counting density on 0. The motivation came from a problem recently published in Statistica Neerlandica (Vol. 43, p. 66). We give a more specialized result in this paper.Functional moments Skewness Decreasing density
RANDOM WALKS WITH STOCHASTICALLY BOUNDED INCREMENTS: RENEWAL THEORY VIA FOURIER ANALYSIS
No full textapplication/pdfRandom walks ＄S_{N}=(S_{n})_{n￥geqq 0}＄ with stochastically bounded increments ＄X_{0},＄ ＄X_{1},＄ ＄￥cdots＄ have been introduced in [2], [3] as natural generalizations of those with i.i.d. increments. In this article we present Blackwell-type renewal theorems proved by means of Fourier analysis. In the special case of independent ＄X_{0},＄ ＄X_{1}＄ , ＄￥cdot＄ these results lead to generalizations of earlier ones in the literature, notably in [3] where proofs were based on coupling technique which is a purely probabilistic device. As a further application we prove Blackwell's renewal theorem for certain random walks with stationary 1-dependent increments that appear in Markov renewal theory as subsequences of Markov random walks
Superposed continuous renewal processes A Markov renewal approach
No full textAbstractLam and Lehoczky (1991) have recently given a number of extensions of classical renewal theorems to superpositions of p independent renewal processes. In this article we want to advertise an approach that more explicitly uses a Markov renewal theoretic framework and thus leads to a simplified derivation of their main results together with a number of new ones. Those include a Stone-type decomposition for the resulting Markov renewal measure and a number of convergence rate results which extend the corresponding results for single renewal processes
On the Markov renewal theorem
No full textLet (S, £) be a measurable space with countably generated [sigma]-field £ and (Mn, Xn)n[greater-or-equal, slanted]0 a Markov chain with state space S x and transition kernel :S x ( [circle times operator] )-->[0, 1]. Then (Mn,Sn)n[greater-or-equal, slanted]0, where Sn = X0+...+Xn for n[greater-or-equal, slanted]0, is called the associated Markov random walk. Markov renewal theory deals with the asymptotic behavior of suitable functionals of (Mn,Sn)n[greater-or-equal, slanted]0 like the Markov renewal measure [Sigma]n[greater-or-equal, slanted]0P((Mn,Sn)[epsilon]Ax (t+B)) as t-->[infinity] where A[epsilon] and B denotes a Borel subset of . It is shown that the Markov renewal theorem as well as a related ergodic theorem for semi-Markov processes hold true if only Harris recurrence of (Mn)n[greater-or-equal, slanted]0 is assumed. This was proved by purely analytical methods by Shurenkov [15] in the one-sided case where (x,Sx[0,[infinity])) = 1 for all x[epsilon]S. Our proof uses probabilistic arguments, notably the construction of regeneration epochs for (Mn)n[greater-or-equal, slanted]0 such that (Mn,Xn)n[greater-or-equal, slanted]0 is at least nearly regenerative and an extension of Blackwell's renewal theorem to certain random walks with stationary, 1-dependent increments.Markov renewal theory Markov random walk semi-Markov process Harris recurrence> (null) regeneration epochs Blackwell's renewal theorem random walks with stationary 1-dependent increments coupling
Convergence rates in the law of large numbers for martingales
No full textAbstractIn this paper we extend well-known results by Baum and Katz (1965) and others on the rate of convergence in the law of large numbers for sums of i.i.d. random variables to general zero-mean martingales SN. For 12 < α ⩽ 1, p>1α and f(x) = |x| (two-sided case) or = x+ or x− (one-sided case), it is e.g. shown that if, for some γ ϵ (1α, 2] and q>(pα − 1)(γα − 1), supn⩾1n−1∑j=1nE(|Xj|γ|S0,…Sn)q<∞ and an additional mixing condition holds in the one-sided case, then ∑n⩾1npα−2P(f(Sn)>εnα<∞for allε> 0 holds iff ∑n⩾1∑j⩾njpα−2P(f(Xn)>εjα)<∞for all ε>0, X1, X2, … being the increments of SN. The latter condition reduces to the well-known moment condition Ef(X1)p<∞, if X1, X2, … are i.i.d. Our results also extend recent ones by Irle (1985, 1987)
On the moments of certain first passage times for linear growth processes
No full textAbstractLet SN be a stochastic process adapted to the filtration FN and with increments X1, X2, … Set mn = E(Xn|Fn−1) and Ln = m1 + … + mn for n ⩾ 1. Then we call SN a linear growth process (LGP) if 1.(1) μ ⩽ Ln/n ⩽ ν a.s.f.a. n ⩾ n0 and2.(2) Ln/n → θ a.s., as n → ∞
for suitable μ, ν, θ > 0 and some integer n0 ⩾ 1. In the case where (2) holds uniformly on a subevent of probability 1, SN is called a uniform linear growth process (ULGP), and if (1) and (2) are satisfied with Ln/n replaced by mn in (1), then SN is called a strong linear growth process (SLGP). For b ⩾ 0 and positive, continuous functions f on [0, ∞) we examine the first passage times τ = τ (b) = inffn ⩾ 1: Sn > b {(n)} as to existence of the moments of τ and Sτ and related asymptotics. We will show that many results which are valid in the i.i.d. case carry over to LGP's under quite weak additional assumptions. In the case where SN is a SLGP and f(·) 1, we will furthermore provide uniform integrability of the moments of the excess over the boundary Sτ − b by renewal theoretic methods. This yields an expansion for Eτ up to terms of order O(1), as b → ∞, when (Sn − nθ)n⩾0 constitutes a martingale. In the final section the results will be applied to several examples from applied probability
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