The Brunk–Prokhorov strong law of large numbers for fields of martingale differences taking values in a Banach space

Son, Ta Cong; Thang, Dang Hung

doi:10.1016/j.spl.2013.04.023

Cited by 4 publications

(4 citation statements)

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“…Smythe's strong law of large numbers for multiple sums [15] implies that ( 16) holds for a stationary completely random measure under the same assumption on the logarithmic moment of S(I) as in Theorem 3.1. If also E S(A) = 0 for all Borel A, Corollary 2.2 yields that, if (17) holds if b −α n converges. By Corollary 2.5, (17) also holds if S(A) is (2q)-integrable with integer q ≥ 1 and ( 14) is satisfied, which is then the Brunk-Prohorov theorem for stationary completely random measures.…”

Section: Stationary Random Measuresmentioning

confidence: 98%

“…For this reason, Theorem 2.3 can be called the Brunk-Prokhorov theorem for multiple sums. Other generalizations of the Brunk-Prokhorov theorem are obtained in [12] and [17].…”

Section: Brunk-prohorov Theorem For Multiple Sumsmentioning

confidence: 98%

“…10.1.VI]. Write E (y 2 |x) for the second moment of y sampled from P (•|x) (assuming this moment is finite) and denote by Λ the intensity measure of the Poisson process {x i , i ≥ 1}.The strong law of large numbers of the type(17) follows from the strong law of large numbers for the partial sums of the discrete random field Z n = S(C n ), where C…”

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confidence: 99%

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Moment conditions in strong laws of large numbers for multiple sums and random measures

Klesov

Molchanov

2017

Statistics & Probability Letters

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The validity of the strong law of large numbers for multiple sums S n of independent identically distributed random variables Z k , k ≤ n, with r-dimensional indices is equivalent to the integrability of |Z|(log + |Z|) r−1 , where Z is the generic summand. We consider the strong law of large numbers for more general normalizations, without assuming that the summands Z k are identically distributed, and prove a multiple sum generalization of the Brunk-Prohorov strong law of large numbers. In the case of identical finite moments of order 2q with integer q ≥ 1, we show that the strong law of large numbers holds with the normalization (n 1 · · · n r ) 1/2 (log n 1 · · · log n r ) 1/(2q)+ε for any ε > 0.The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.

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Section: Stationary Random Measuresmentioning

confidence: 98%

“…For this reason, Theorem 2.3 can be called the Brunk-Prokhorov theorem for multiple sums. Other generalizations of the Brunk-Prokhorov theorem are obtained in [12] and [17].…”

Section: Brunk-prohorov Theorem For Multiple Sumsmentioning

confidence: 98%