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

Thrum, Rolf

Probab. Theory Relat. Fields 75, 425-430 (1987).

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

Keywords:

weighted sums; almost sure convergence; least squares

Citations:

Zbl 0152.169

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