zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The strong law of large numbers for weighted averages under dependence assumptions. (English) Zbl 0857.60021

The authors prove strong laws of large numbers for weighted averages of dependent random variables, generalizing the classical work of B. Jamison, S. Orey and W. Pruitt [Z. Wahrscheinlichkeitstheorie Verw. Geb. 4, 40-44 (1965; Zbl 0141.16404)] for i.i.d. sequences. The dependence structure imposed is asymptotic quadrant sub-independent, requiring that

P(X i >s,X j >t)-P(X i >s)P(X j >t)q| i - j |α ij (s,t),

together with a similar condition on P(X i <s,X j <t). This condition generalizes the notion of asymptotic quadrant independence, introduced by T. Birkel [Stat. Probab. Lett. 7, No. 1, 17-20 (1988; Zbl 0661.60048)]. The authors also prove a Marcinkiewicz-Zygmund SLLN for weighted averages. The proofs make heavy use of unpublished results by the same authors.

60F15Strong limit theorems
[1]Birkel, T. (1992). Laws of large numbers under dependence assumptions.Statist. Prob. Lett. 14, 355–362. · Zbl 0925.60023 · doi:10.1016/0167-7152(92)90096-N
[2]Chandra, T. K. (1991). Extensions of Rajchman’s strong law of large numbers.Sankhyā, Ser. A 53, 118–121.
[3]Chandra, T. K., and Ghosal, S. (1993). Some elementary strong laws of large numbers: a review. Technical Report #22/93, Indian Statistical Institute.
[4]Chandra, T. K., and Ghosal, S. (1996). Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund.Acta Math. Hung. 72(3) (to appear).
[5]Etemadi, N. (1983). Stability of sums of weighted random variables.J. Multivariate Anal. 13, 361–365. · Zbl 0531.60034 · doi:10.1016/0047-259X(83)90032-5
[6]Hall, P., and Heyde, C. C. (1980).Martingale Limit Theory and Its Application., Academic Press, New York.
[7]Jamison, B., Orey, S., and Pruitt, W. E. (1965). Convergence of weighted averages of independent random variables.Z. Wahrsch. Verw. Gebiete 4, 40–44. · Zbl 0141.16404 · doi:10.1007/BF00535481
[8]McLeish, D. L. (1975). A maximal inequality and dependent strong laws.Ann. Prob. 3, 829–839. · Zbl 0353.60035 · doi:10.1214/aop/1176996269
[9]Pruitt, W. E. (1966). Summability of independent random variables.J. Math. Mech. 15, 769–776.
[10]Rothatgi, V. K. (1971). Convergence of weighted sums of independent random variables.Proc. Cambridge Phil. Soc. 69, 305–307. · doi:10.1017/S0305004100046685
[11]Rosalsky, A. (1987). Strong stability of normed sums of pairwise i.i.d. random variables.Bull. Inst. Math. Acad. Sinica 15, 203–219.