A law of the iterated logarithm for arithmetic functions

Berkes, István; Weber, Michel

doi:10.1090/S0002-9939-06-08557-1

A law of the iterated logarithm for arithmetic functions
HTML articles powered by AMS MathViewer

by István Berkes and Michel Weber PDF

Proc. Amer. Math. Soc. 135 (2007), 1223-1232 Request permission

Abstract:

Let $X,X_1,X_2,\ldots$ be a sequence of centered iid random variables. Let $f(n)$ be a strongly additive arithmetic function such that $\sum _{p < n}\tfrac {f^2(p)}{p}\to \infty$ and put $A_n= \sum _{p < n}\tfrac {f(p)}{p}$. If $\mathbf {E} X^2 <\infty$ and $f$ satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm: \[ \limsup _{N\to \infty }{\sum _{n=1}^N f(n) X_n \over A_N \sqrt {2 N \log \log N}}\buildrel {a.s.}\over {=} \|X\|_2.\] We also prove the validity of the corresponding weighted strong law of large numbers in $L^1$.

References

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
P. D. T. A. Elliott, Probabilistic number theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 239, Springer-Verlag, New York-Berlin, 1979. Mean-value theorems. MR 551361
P. Erdös and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738–742. MR 2374, DOI 10.2307/2371483
Evan Fisher, A Skorohod representation and an invariance principle for sums of weighted i.i.d. random variables, Rocky Mountain J. Math. 22 (1992), no. 1, 169–179. MR 1159950, DOI 10.1216/rmjm/1181072802
H. Halberstam, On the distribution of additive number-theoretic functions, J. London Math. Soc. 30 (1955), 43–53. MR 66406, DOI 10.1112/jlms/s1-30.1.43
Benton Jamison, Steven Orey, and William Pruitt, Convergence of weighted averages of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 40–44. MR 182044, DOI 10.1007/BF00535481
J. Kubilius, Probabilistic methods in the theory of numbers, Translations of Mathematical Monographs, Vol. 11, American Mathematical Society, Providence, R.I., 1964. MR 0160745
Walter Philipp, Mixing sequences of random variables and probablistic number theory, Memoirs of the American Mathematical Society, No. 114, American Mathematical Society, Providence, R.I., 1971. MR 0437481
Harold N. Shapiro, Distribution functions of additive arithmetic functions, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 426–430. MR 79609, DOI 10.1073/pnas.42.7.426
Michel Weber, An ergodic theorem of arithmetic type, Tatra Mt. Math. Publ. 31 (2005), 123–129. MR 2208793
Weber M., [2004] An LIL of arithmetical type, preprint.
Michel Weber, An arithmetical property of Rademacher sums, Indag. Math. (N.S.) 15 (2004), no. 1, 133–149. MR 2061474, DOI 10.1016/S0019-3577(04)90011-0
Michel Weber, On the order of magnitude of the divisor function, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 377–382. MR 2214359, DOI 10.1007/s10114-005-0679-1
Michel Weber, Divisors, spin sums and the functional equation of the zeta-Riemann function, Period. Math. Hungar. 51 (2005), no. 1, 119–131. MR 2180637, DOI 10.1007/s10998-005-0024-6