Cesàro α-Integrability and Laws of Large Numbers-II

A STRONG LIMIT THEOREM FOR SEQUENCES OF BLOCKWISE AND PAIRWISE m-DEPENDENT RANDOM VARIABLES

“…Specifically, if sup n≥1 E|X n | r < ∞ for some r > 0, then there exists a random variable X with E|X| p < ∞ for all 0 < p < r such that P (|X n | > t) ≤ P (|X| > t), t ≥ 0, n ≥ 1 (The proviso that r > 1 in Lemma 3 of Wei and Taylor [27] is not needed as was pointed out by Adler, Rosalsky, and Taylor [1]). So that Theorem 2.1 reduces to Corollary 4 of Chandra and Goswami [3] when the {X n , n ≥ 1} are pairwise independent and sup n≥1 E|X n | r < ∞ for some r > 1.…”

“…Chandra and Goswami [3], Bose and Chandra [2] considered this problem under uniformly integrable condition. Hong and Hwang [12], and Czerebak-Mrozowicz, Klesov and Rychlik [6] extended the result of Choi and Sung to the multidimensional case.…”

Use of SAMC for Bayesian analysis of statistical models with intractable normalizing constants

“…Extension to the case s > 1 is trivial. To prove this theorem, we first introduce a definition of strongly residually Cesàro α-integrable and a lemma of strong law of large numbers (Chandra and Goswami, 2005). …”

Almost sure Cesàro and Euler summability of sequences of dependent random variables

“…There exists a vast literature on weakening the independence assumptions there. For references see, e.g., [27], [6], [28], compare also Remark 2.1 which concerns the case α = 2 in Theorem 2.1.…”

“…Thus by introducing the logarithmic factor in the mentioned conditions, one can avoid the additional conditions X n ≥ k < ∞ or sup EX n < ∞ in[6],[7, Theorem 1], and in[28, Remark 7]. The a.s. convergence assertion can be interpreted as an assertion on weighted means of…”