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Almost Sure Convergence of Delayed Renewal Processes
Journal of the London Mathematical Society, 1987Consider a renewal process \(\{N(t)\}_{t\geq 0}\) associated with a sequence \(\{X_ i\}_{i=1,2,...}\) of nonnegative and independent, identically distributed ''failure times''. Appropriate generalized moment conditions are presented which are necessary and sufficient for certain law of iterated logarithm type results on the ''delayed renewal process'',
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2017
This chapter gives the basic theory of almost sure convergence and Kolmogorov’s strong law of large numbers (1933) according to which the empirical mean of an iid sequence of integrable random variables converges almost surely to the probabilistic mean (the expectation).
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This chapter gives the basic theory of almost sure convergence and Kolmogorov’s strong law of large numbers (1933) according to which the empirical mean of an iid sequence of integrable random variables converges almost surely to the probabilistic mean (the expectation).
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2012
This chapter studies essentially Strong Laws of Large Numbers (SLLN) for associated variables and their applications to the characterization of asymptotics of statistical estimators under associated sampling. It is possible to prove SLLN under fairly general assumptions, but, in order to prove characterizations of convergence rates, a closer care on ...
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This chapter studies essentially Strong Laws of Large Numbers (SLLN) for associated variables and their applications to the characterization of asymptotics of statistical estimators under associated sampling. It is possible to prove SLLN under fairly general assumptions, but, in order to prove characterizations of convergence rates, a closer care on ...
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Almost sure convergence of sample range
Extremes, 2007An almost sure limit theorem is derived for the sample range statistics \( R_n=\max_{1\leq i\leq n} X_i-\min_{1\leq i\leq n} X_i \), where \(X_i\) are i.i.d. r.v.s. It states that if for some nonrandom \(\alpha_n\), \(\beta_n\) and some nondegenerate CDF \(G\) \(\zeta_n=\alpha_n(R_n-\beta_n)\) converges in distribution to \(G\) then at any point \(x ...
Zhongquan, Tan +2 more
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Convergence Almost Surely and in Probability
1989In this lesson we look at some theorems on convergence of sequences of numerical (extended real) valued functions. All will be used at some point later particularly in the lessons on integration. We keep a fixed [ΩSP].
Hung T. Nguyen, Gerald S. Rogers
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Almost sure convergence and bounded entropy
Israel Journal of Mathematics, 1988Let (\({\mathcal X},\mu)\) be a probability space, \(S_ n\) a sequence of operators on \(L^ 2(\mu)\), \(\| S_ n\| \leq 1\), \(T_ j\) a sequence of positive isometric operators satisfying \(T_ j(1)=1\), \(J^{-1}\sum_{j\leq J}T_ jf\to \int f d\mu\) \(\forall f\in L^ 1\) and \(T_ jS_ n=S_ nT_ j\).
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Convergence: Almost Surely and in Probability
1996For nets of maps defined on a single, fixed probability space (Ω, A, P), convergence almost surely and in probability are frequently used modes of stochastic convergence, stronger than weak convergence. In this section we consider their nonmeasurable extensions together with the concept of almost uniform convergence, which is equivalent to outer almost
Aad W. van der Vaart, Jon A. Wellner
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Almost sure convergence of branching processes
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1977A general theorem concerning the almost sure convergence of some nonhomogeneous Markov chains, whose conditional distributions satisfy a certain convergence condition, is given. This result applied to branching processes with infinite mean yields almost sure convergence for a large class of processes converging in distribution, as well as a ...
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On $(B,\rho )$-Amarts and Almost Sure Convergence
Theory of Probability & Its Applications, 1996The main aim of this paper is to generalize a conditional amart [\textit{D. Szynal} and the second author, Bull. Pol. Acad. Sci., Math. 34, 635-642 (1986; Zbl 0615.60027), the second author, Theory Probab. Appl. 36, No. 3, 637-639 (1991) and Teor. Veroyatn. Primen. 36, No. 3, 616-617 (1991; Zbl 0739.60036)] and a \(D_v\)-amart [\textit{I.
Kruk, L., Ziȩba, W.
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Almost Sure Convergence of Renewal Processes
2018Consider some renewal sequence, that is, a sequence of partial sums {Sn}n≥0 of independent identically distributed random variables {Xn}n≥1. Our aim in this chapter is to show that various functionals of partial sums and corresponding renewal processes are asymptotically equivalent if one considers them from the point of view of generalized renewal ...
Valeriĭ V. Buldygin +3 more
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