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Summability methods and almost sure convergence. (English) Zbl 0551.60037
The authors investigate almost sure convergence for sequences of i.i.d. random variables under different methods of summability. We say \(s_ n\to s\) (P), if \(\sum^{\infty}_{j=0}s_ jP(S_ n=j)\to s\) as \(n\to \infty\), where \(S_ n:=\xi_ 1+...+\xi_ n\), and \(\xi_ 1,\xi_ 2,..\). are integer-valued independent random variables. The summability method (P) is called a random-walk method. This method is related to the family of circle-methods C, defined as follows: \(s_ n\to s\) (C), if \(\sum^{\infty}_{j=0}s_ jc_ j(n)\to s\) for given weights \(c_ j(n)\). For instance: \(c_ j(n):=\sqrt{(2\pi n)^{-1}a}\) \(\exp \{- \frac{1}{2}a(j-n)^ 2/n\}\) gives the Valiron methods \(V_ a\), and \(c_ j(n):=e^{-n}n^ j/j!\) gives the Borel method B. The paper contains among others, the following
Theorem. For \(X,X_ 0,X_ 1,..\). i.i.d. the following are equivalent:
(1) Var X\(<\infty\), \(EX=m.\)
(2) \(X_ n\to m\) a.s. (P), for P some (any) random walk.
(3) \(X_ n\to m\) a.s. \((V_ a)\), for some (all) \(a>0.\)
(4) \(X_ n\to m\) a.s. (C), for some (any) circle-method.
Reviewer: Z.Jurek

MSC:
60F15 Strong limit theorems
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[1] Acosta, A.de, Kuelbs, J.: Limit theorems for moving averages of random vectors. Z. Wahrscheinlichkeitstheorie verw. Gebiete 64, 67-123 (1983) · Zbl 0498.60008 · doi:10.1007/BF00532594
[2] Azlarov, T.A., Meredov, B.: Some estimates in the limit theorem for the Abel summability of random variables (Russian). Izv. Akad. Nauk Uz.SSR Ser. Fiz.-Mat. Nauk 5, 7-15 (1977) · Zbl 0389.60007
[3] Bikelis, A., Jasjunas, G.: Limit theorems in the metric of the spaces L 1, l 1 (Russian). Litovsk. Mat. Sb. 7, 195-218 (1967) · Zbl 0189.18004
[4] Bingham, N.H.: Tauberian theorems and the central limit theorem. Ann. Probability 9, 221-231 (1981) · Zbl 0459.60021 · doi:10.1214/aop/1176994464
[5] Bingham, N.H.: On Euler and Borel summability. J. London Math. Soc. (2) 29, 141-146 (1984) · Zbl 0528.40006 · doi:10.1112/jlms/s2-29.1.141
[6] Bingham, N.H.: On Valiron and circle convergence. Math. Z. 186, 273-286 (1984) · Zbl 0538.60030 · doi:10.1007/BF01161809
[7] Bingham, N.H.: On Tauberian theorems in probability. Nieuw Arch. Wiskunde (To appear) · Zbl 0619.40003
[8] Bingham, N.H.: Tauberian theorems for summability methods of random-walk type. J. London Math. Soc. (To appear) · Zbl 0518.40002
[9] Bingham, N.H., Goldie, C.M.: On one-sided Tauberian conditions. Analysis. 3, 159-188 (1983) · Zbl 0497.40003
[10] Chow, Y.S.: Delayed sums and Borel summability of independent identically distributed random variables. Bull. Inst. Math. Acad. Sinica 1, 207-220 (1973) · Zbl 0296.60014
[11] Chow, Y.S., Lai, T.L.: Limiting behaviour of weighted sums of independent random variables. Ann. Probability 1, 810-824 (1973) · Zbl 0303.60025 · doi:10.1214/aop/1176996847
[12] Cs?rg?, M., R?v?sz, P.: Strong approximations in probability and statistics. New York: Academic Press 1981
[13] Embrechts, P., Maejima, M.: The central limit theorem for summability methods of i.i.d. random variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete · Zbl 0535.60018
[14] Feller, W.: An introduction to probability theory and its applications, Volume 2, second ed. New York: Wiley 1971 · Zbl 0219.60003
[15] Gerber, H.: The discounted central limit theorem and its Berry-Esseen analogue. Ann. Math. Statist. 42, 389-392 (1971) · Zbl 0224.60012 · doi:10.1214/aoms/1177693529
[16] Hardy, G.H.: Divergent series. Oxford: University Press 1949 · Zbl 0032.05801
[17] Hardy, G.H., Riesz, M.: The general theory of Dirichlet series. Cambridge: University Press 1952 · JFM 45.0387.03
[18] Hyslop, J.M.: The generalisation of a theorem on Borel summability. Proc. London Math. Soc. 41, 243-256 (1936) · Zbl 0015.01504 · doi:10.1112/plms/s2-41.4.243
[19] Ibragimov, I.A., Linnik, Y.V.: Independent and stationary sequences of random variables. Groningen: Wolters-Noordhoff 1971 · Zbl 0219.60027
[20] Lai, T.L.: On Strassen-type laws of the iterated logarithm for delayed averages of the Wiener process. Bull. Inst. Math. Acad. Sinica 1, 29-39 (1973) · Zbl 0261.60030
[21] Lai, T.-L.: Limit theorems for delayed sums. Ann. Probability 2, 432-440 (1974) · Zbl 0305.60009 · doi:10.1214/aop/1176996658
[22] Lai, T.-L.: Summability methods for independent identically distributed random variables. Proc. Amer. Math. Soc. 45, 253-261 (1974) · Zbl 0339.60048
[23] 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. Probability 10, 320-335 (1982) · Zbl 0485.60031 · doi:10.1214/aop/1176993860
[24] Meyer-K?nig, W.: Untersuchungen ?ber einige verwandte Limitierungsverfahren. Math. Z. 52, 257-304 (1949) · Zbl 0041.18403 · doi:10.1007/BF02230694
[25] Petrov, V.V.: Sums of independent random variables. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0322.60043
[26] Pruitt, W.E.: Summability of independent random variables. J. Math. Mech. 15, 769-776 (1966) · Zbl 0158.36403
[27] Stout, W.F.: Almost sure convergence. New York: Academic Press 1974 · Zbl 0321.60022
[28] Zeller, K., Beekmann, W.: Theorie der Limitierungsverfahren. Berlin, Heidelberg, New York: Springer 1970 · Zbl 0199.11301
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