zbMATH — the first resource for mathematics

On a law of the iterated logarithm for sums mod 1 with application to Benford’s law. (English) Zbl 0619.60032
Let \(Z_ n\) be the sum mod 1 of n i.i.d. r.v. and let \(1_{[0,x)}(\cdot)\) be the indicator function of the interval [0,x). Then the sequence \(1_{[0,x)}(Z_ n)\) does not converge for any x. But if arithmetic means are applied then under suitable suppositions convergence with probability one is obtained for all x as is well-known.
In the present paper the rate of this convergence is shown to be of order \(n^{-1/2}(\log \log n)^{1/2}\) by using estimates of the remainder term in the CLT for m-dependent r.v.

60F15 Strong limit theorems
60F05 Central limit and other weak theorems
Full Text: DOI
[1] Barlow J.L., Bareiss, E.H.: On roundoff error distributions in floating point and logarithmic arithmetic. Computing 34, 325–347 (1985) · Zbl 0576.65035
[2] Benford F.: The law of anomalous numbers. Proc. Amer. Phil. Soc. 78, 552–572 (1938) · JFM 64.0555.03
[3] Bhattacharya R.N.: Speed of convergence of the n-fold convolution of a probability measure on a compact group. Z. Wahrscheinlichkeitstheor. Verw. Geb. 25, 1–10 (1972) · Zbl 0247.60008
[4] Chung K.L.: A course in probability theory. New York-London: Academic Press 1974 · Zbl 0345.60003
[5] Egorov V.A.: Some limit theorems for m-dependent random variables (Russian). Liet. mat. rink. 10, 51–59 (1970) · Zbl 0213.20201
[6] Heinrich L.: A method for the derivation of limit theorems for sums of m-dependent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 501–515 (1982) · Zbl 0492.60028
[7] Herrmann H.: Konvergenzgeschwindigkeit der Folge der Faltungspotenzen eines Wahrscheinlichkeitsmaßes auf einer kompakten topologischen Gruppe. Math. Nachr. 104, 49–59 (1981) · Zbl 0494.60009
[8] Holewijn P.J.: On the uniform distribution of random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 14, 89–92 (1969) · Zbl 0185.44601
[9] Kuipers L., Niederreiter H.: Uniform distribution of sequences. New York: Wiley 1974 · Zbl 0281.10001
[10] Loynes R.M.: Some results in the probability theory of asymptotic uniform distribution modulo 1. Z. Wahrscheinlichkeitstheor. Verw. Geb. 26, 33–41 (1973) · Zbl 0289.60017
[11] Petrov V.V.: Sums of independent random variables. Berlin Heidelberg New York: Springer 1975 · Zbl 0322.60043
[12] Philipp W.: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Ann. Probab. 5, 319–350 (1977) · Zbl 0362.60047
[13] Raimi R.A.: The first digit problem. Amer. Math. Mon. 83, 521–538 (1976) · Zbl 0349.60014
[14] Robbins H.: On the equidistribution of sums of independent random variables. Proc. Amer. Math. Soc. 4, 786–799 (1953) · Zbl 0053.26704
[15] Schatte P.: Zur Verteilung der Mantisse in der Gleitkommadarstellung einer Zufallsgröße. Zeitschr. Angew. Math. Mech. 83, 553–565 (1973) · Zbl 0267.60025
[16] Schatte P.: On the asymptotic uniform distribution of sums reduced mod 1. Math. Nachr. 115, 275–281 (1984) · Zbl 0557.60008
[17] Schatte P.: The asymptotic uniform distribution modulo 1 of cumulative processes. Optimization 16, 783–786 (1985) · Zbl 0583.60026
[18] Schatte P.: On the asymptotic uniform distribution of the n-fold convolution mod 1 of a lattice distribution. Math. Nachr. 128, 233–241 (1986) · Zbl 0617.60019
[19] Schatte P.: On the asymptotic behaviour of the mantissa distributions of sums. J. Inf. Process. Cybern. ElK. 23, 353–360 (1987) · Zbl 0637.60032
[20] Schatte P.: On the almost sure convergence of floating-point mantissas and Benford’s law. Math. Nachr. 135 (1988) · Zbl 0645.60038
[21] Schatte P.: On mantissa distributions in computing and Benford’s law. J. Inf. Process. Cybern. ElK (in press) · Zbl 0662.65040
[22] Schmidt V.: On the asymptotic uniform distribution of stochastic clearing processes. Optimization 17, 125–134 (1986) · Zbl 0592.60023
[23] Shergin V.V.: On the speed of convergence in the central limit theorem for m-dependent random variables (Russian). Teor. Verotn. Primen. 24, 781–794 (1979) · Zbl 0437.60018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.