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Almost certain convergence in double arrays. (English) Zbl 0548.60028

For double arrays of constants \(\{a_{ni}\), \(1\leq i\leq k_ n,n\geq 1\}\) and i.i.d. random variables \(\{X,X_ i\), \(i\geq 1\}\), conditions are given under which the row sums \(\sum^{k_ n}_{i=1}a_{ni}X_ i\to^{a.c.}0\). Both cases \(k_ n\uparrow\infty \) and \(k_ n=\infty\) are treated. In general, the hypotheses involve a trade-off between moment requirements on X and the magnitude of the \(\{a_{ni}\}\). A Marcinkiewicz-Zygmund type strong law is obtained for the special case \(a_{ni}=a_ i/(\sum^{n}_{j=1}a^ p_ j)^{1/p}\), \(a_ i>0\), \(\sum^{n}_{1}a^ p_ j\to\infty \), \(0<p<2\).

MSC:

60F15 Strong limit theorems
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[1] Chow, Y. S.; Lai, T. L., Limiting behavior of weighted sums of independent random variables, Ann. Probab., 1, 810-824 (1973) · Zbl 0303.60025
[2] Chow, Y. S.; Teicher, H., Probability Theory: Independence, Interchangeability, Martingales (1978), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0399.60001
[3] Fernholz, L. T.; Teicher, H., Stability of Random Variables and Iterated Logarithm Laws for Martingales and Quadratic Forms, Ann. Probab., 8, 765-774 (1980) · Zbl 0442.60032
[4] Hanson, D. L.; Koopmans, L. H., On the convergence rate of the law of large numbers for linear combinations of independent random variables, Ann. Math. Statist., 36, 559-564 (1965) · Zbl 0132.38604
[5] Lai, T. L.; Wei, C. Z., A law of the iterated logarithm for double arrays of independent random variables with applications to regression and time series models, Ann. Probab., 10, 320-335 (1982) · Zbl 0485.60031
[6] Longnecker, M.; Serfling, R., General moment and probability inequalities for the maximum partial sum, Acta Math. Acad. Sci. Hung., 30, 129-133 (1977) · Zbl 0373.60066
[7] Pruitt, W. E., Summability of independent random variables, J. Math. Mech., 15, 769-776 (1966) · Zbl 0158.36403
[8] Serfling, R. J., Convergence properties of S_n under moment restrictions, Ann. Math. Statist., 41, 1235-1248 (1970) · Zbl 0302.60018
[9] Teicher, H., On the law of the iterated logarithm, Ann. Probab., 2, 714-728 (1974) · Zbl 0286.60013
[10] Teicher, H., Generalized Exponential Bounds, Iterated Logarithm and Strong Laws, Z. ahrscheinlichkeitstheorie Verw. Geb., 48, 293-307 (1979) · Zbl 0387.60034
[11] Teicher, H., Almost Certain Behavior of Row Sums of Double Arrays. Analytical Methods in Probability Theory, Oberwohlfach, Germany, Lecture Notes in Mathematics 861, 155-165 (1980), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York
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