×

A remark on almost sure convergence of weighted sums. (English) Zbl 0599.60031

As a generalization of a theorem of Y. S. Chow [Ann. Math. Stat. 37, 1482-1493 (1966; Zbl 0152.169)] it is shown by an elementary method that for i.i.d. r.v.’s \(X_ 1,...,X_ n\), with expectation zero and finite p-th absolute moment (p\(\geq 2)\) the weighted sums \[ \sum^{n}_{i=1}a_{n,i}X_ i/n^{1/p}(\sum^{n}_{i=1}a^ 2_{n,i})^{1/2} \] converge to zero a.s.

MSC:

60F15 Strong limit theorems
60G42 Martingales with discrete parameter

Citations:

Zbl 0152.169
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Chow, Y. S., Some convergence theorems for independent random variables, Ann. Math. Statist., 37, 1482-1492 (1966) · Zbl 0152.16905
[2] Kleffe, J.; Thrum, R., Inequalities for moments of quadratic forms with applications to a.s. convergence, MOS, Series Statist., 14, 211-216 (1983) · Zbl 0545.60027
[3] Neveu, J., Bases mathématiques du calcul de probalités (1964), Paris: Masson, Paris · Zbl 0137.11203
[4] Shirjajev, A. N., Verojatnost (1980), Moscow: Nauka, Moscow
[5] Stout, W. F., Almost sure convergence (1974), New York: Academic Press, New York · Zbl 0321.60022
[6] Révész, P., The laws of large numbers (1967), Budapest: Akadémia kiadó, Budapest · Zbl 0203.50403
[7] Whittle, P., Bounds for the moments of linear and quadratic forms in independent variables, Teorija Yerojatnost. i primenen, 5, 331-334 (1960) · Zbl 0101.12003
[8] Chen, Gui-Jing; Lai, T. L.; Wei, C. Z., Convergence system and strong consistency of least square estimates in regression models, J. Multivariate Anal., 11, 319-333 (1981) · Zbl 0471.62065
[9] Thrum, R.: Uniform almost sure convergence of Weighted sums and applications in regression models. Paper at the 5-th Pannonian Symposium on Mathematical Statistics. May 20-24, 1985, Visegrád, Hungary (1985) · Zbl 0668.60033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.