SLLN for weighted independent identically distributed random variables

John Baxter; Roger Jones; Michael Lin; James Olsen

doi:10.1023/B:JOTP.0000020480.84425.8d

SLLN for weighted independent identically distributed random variables

John Baxter, Roger Jones, Michael Lin, James Olsen

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

For any sequence {a_k} with sup 1/n ∑_{k = 1} ⁿ |a_k|^q < ∞ for some q > 1, we prove that 1/n ∑_{k = 1}ⁿ a_kX_k converges to 0 a.s. for every {X_n} i.i.d. with E(|X₁|) < ∞ and E(X₁) = 0; the result is no longer true for q = 1, not even for the class of i.i.d. with X₁ bounded. We also show that if {a_k} is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {X_n} with finite absolute p th moment for some p > 1, 1/n ∑ _{k = 1}ⁿ a_k X_k converges a.s.

Original language	English (US)
Pages (from-to)	165-181
Number of pages	17
Journal	Journal of Theoretical Probability
Volume	17
Issue number	1
DOIs	https://doi.org/10.1023/B:JOTP.0000020480.84425.8d
State	Published - Jan 2004

Bibliographical note

Funding Information:
We are grateful for the hospitality and support offered by North Dakota State University to the third author, and by the University of Minnesota and Ben-Gurion University to the last author. The last author’s travel to Israel was partially supported by an NDSU grant from the VP of Academic Affairs. Research of J. Olsen partially supported by ND-EPSCoR through NSF Grant EPS-9874802.

Keywords

Besicovitch sequences
Independent random variables
Law of large numbers
Weighted averages

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title = "SLLN for weighted independent identically distributed random variables",

abstract = "For any sequence {ak} with sup 1/n ∑k = 1 n |ak|q < ∞ for some q > 1, we prove that 1/n ∑k = 1n akXk converges to 0 a.s. for every {Xn} i.i.d. with E(|X1|) < ∞ and E(X1) = 0; the result is no longer true for q = 1, not even for the class of i.i.d. with X1 bounded. We also show that if {ak} is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {Xn} with finite absolute p th moment for some p > 1, 1/n ∑ k = 1n ak Xk converges a.s.",

keywords = "Besicovitch sequences, Independent random variables, Law of large numbers, Weighted averages",

author = "John Baxter and Roger Jones and Michael Lin and James Olsen",

note = "Funding Information: We are grateful for the hospitality and support offered by North Dakota State University to the third author, and by the University of Minnesota and Ben-Gurion University to the last author. The last author{\textquoteright}s travel to Israel was partially supported by an NDSU grant from the VP of Academic Affairs. Research of J. Olsen partially supported by ND-EPSCoR through NSF Grant EPS-9874802.",

year = "2004",

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doi = "10.1023/B:JOTP.0000020480.84425.8d",

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T1 - SLLN for weighted independent identically distributed random variables

AU - Baxter, John

AU - Jones, Roger

AU - Lin, Michael

AU - Olsen, James

N1 - Funding Information: We are grateful for the hospitality and support offered by North Dakota State University to the third author, and by the University of Minnesota and Ben-Gurion University to the last author. The last author’s travel to Israel was partially supported by an NDSU grant from the VP of Academic Affairs. Research of J. Olsen partially supported by ND-EPSCoR through NSF Grant EPS-9874802.

PY - 2004/1

Y1 - 2004/1

N2 - For any sequence {ak} with sup 1/n ∑k = 1 n |ak|q < ∞ for some q > 1, we prove that 1/n ∑k = 1n akXk converges to 0 a.s. for every {Xn} i.i.d. with E(|X1|) < ∞ and E(X1) = 0; the result is no longer true for q = 1, not even for the class of i.i.d. with X1 bounded. We also show that if {ak} is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {Xn} with finite absolute p th moment for some p > 1, 1/n ∑ k = 1n ak Xk converges a.s.

AB - For any sequence {ak} with sup 1/n ∑k = 1 n |ak|q < ∞ for some q > 1, we prove that 1/n ∑k = 1n akXk converges to 0 a.s. for every {Xn} i.i.d. with E(|X1|) < ∞ and E(X1) = 0; the result is no longer true for q = 1, not even for the class of i.i.d. with X1 bounded. We also show that if {ak} is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {Xn} with finite absolute p th moment for some p > 1, 1/n ∑ k = 1n ak Xk converges a.s.

KW - Besicovitch sequences

KW - Independent random variables

KW - Law of large numbers

KW - Weighted averages

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SP - 165

EP - 181

JO - Journal of Theoretical Probability

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ER -

SLLN for weighted independent identically distributed random variables

Abstract

Bibliographical note

Keywords

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