Summability methods for independent identically distributed random variables

Lai, Tze Leung

doi:10.1090/S0002-9939-1974-0356194-4

Summability methods for independent identically distributed random variables
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Proc. Amer. Math. Soc. 45 (1974), 253-261 Request permission

Abstract:

In this paper, we present certain theorems concerning the Cesaro $(C,\alpha )$, Abel $(A)$, Euler $(E,q)$ and Borel $(B)$ summability of $\Sigma {Y_i}$, where ${Y_i} = {X_i} - {X_{i - 1}},{X_0} = 0$ and ${X_1},{X_2}, \cdots$ are i.i.d. random variables. While the Kolmogorov strong law of large numbers and the Hartman-Wintner law of the iterated logarithm are related to $(C,1)$ summability and involve the finiteness of, respectively, the first and second moments of ${X_1}$, their analogues for Euler and Borel summability involve different moment conditions, and the analogues for $(C,\alpha )$ and Abel summability remain essentially the same.

References

Y. S. Chow, Delayed sums and Borel summability of independent, identically distributed random variables, Bull. Inst. Math. Acad. Sinica 1 (1973), no. 2, 207–220. MR 343357
Y. S. Chow and T. L. Lai, Limiting behavior of weighted sums of independent random variables, Ann. Probability 1 (1973), 810–824. MR 353426, DOI 10.1214/aop/1176996847
William Feller, An extension of the law of the iterated logarithm to variables without variance, J. Math. Mech. 18 (1968/1969), 343–355. MR 0233399, DOI 10.1512/iumj.1969.18.18027
V. F. Gapoškin, The law of the iterated logarithm for Abel’s and Cesàro’s methods of summation, Teor. Verojatnost. i Primenen 10 (1965), 449–459 (Russian, with English summary). MR 0195133
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
Philip Hartman and Aurel Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), 169–176. MR 3497, DOI 10.2307/2371287
Tze Leung Lai, Limit theorems for delayed sums, Ann. Probability 2 (1974), 432–440. MR 356193, DOI 10.1214/aop/1176996658
Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 175194, DOI 10.1007/BF00534910
R. J. Tomkins, A generalization of Kolmogorov’s law of the iterated logarithm, Proc. Amer. Math. Soc. 32 (1972), 268–274. MR 292142, DOI 10.1090/S0002-9939-1972-0292142-1