On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

Giuliano, Rita; Hadjikyriakou, Milto

doi:10.1007/s10959-020-01010-3

Cited by 7 publications

(6 citation statements)

References 27 publications

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“…In closing this introduction it is worth pointing out that the extensions of Theorem 1.2 obtained in the present work are in the spirit of weighted exact laws as the ones proved in [2], [5] and more recently in [12], [3] and [4]. We recall that an exact law is a convergence result for a sequence {Z n } n≥1 in which a suitable array of real numbers {c k,n } n≥1, k≤n ensures that…”

Section: Introductionsupporting

confidence: 56%

Convergence for weighted sums of Luroth type random variables

Giuliano¹,

Hadjikyriakou²

2020

Preprint

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In this work we prove an asymptotic result, that under some conditions on the involved distribution functions, is valid for any Oppenheim expansion, extending a classical result proven by W. Vervaat in 1972 for denominators of the Lüroth case. Furthermore, we study the convergence in distribution of weighted sums of a sequence of independent random variables. Although the result is of its own interest, in the present setting it is used to prove convergence in distribution of specific sequences of random variables generalizing known results obtained for Lüroth random variables.

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Section: Introductionsupporting

confidence: 56%

Convergence for weighted sums of Luroth type random variables

Giuliano¹,

Hadjikyriakou²

2020

Preprint

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“…That is certainly not fair for the house. This is why we need to study Exact Strong Laws, one recent paper on such limit theorems is [8]. Others also interested in the infinite mean case are [10], [11] and [13].…”

Section: Introductionmentioning

confidence: 99%

Unusual limit theorems for the difference of order statistics from a Pareto

Adler¹

2021

Bulletin of the Inst. of Math. Academia Sinica(N.S.)

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In this paper we establish various limit theorems for the difference in order statistics from a sample from the Pareto distribution. The underlying density is f (x) = x −2 I(x ≥ 1).We look at both fixed and slowly increasing samples sizes. For our strong and weak laws of large numbers the first moment will be infinite and for our central limit theorem the second moment will be infinite. These theorems are quite unusual since the usual moment conditions do not hold. In order to achieve these results we must attach weights to these random variables and find these appropriate weights and norming sequences in order to establish our results.

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“…The earlier research of laws of large numbers without finite means, one can refer to Adler [5,6]. For the one sided or Pareto-type laws with infinite means, we can refer to Adler [7]- [15], Matsumoto and Nakata [11], Nakata [12]- [14], Adler and Matula [15], Yang et al [16], Matula et al [17], Xu et al [18], Giuliano and Hadjikyriakou [19], Adler and Matula [20] and the references therein. For the laws of asymmetrical Cauchy random variables, we can refer to Adler [2] and Xu et al [18].…”

Section: Introductionmentioning

confidence: 99%