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On the asymptotic behaviour of the mantissa distributions of sums. (English) Zbl 0637.60032
Let X be a random variable. For \(X=0\) let us set \(| X| =M e\) k, K integer, \(1\leq M<e\), and for \(X=0\) set \(M=e\). M is called the mantissa of X. It is known that under very general conditions the mantissa of the product of random variables converges in law when the number of factors tends to infinity. More exactly, if \(X_ 1,X_ 2,...,X_ n\) are independent, identically distributed random variables and we denote by \(G_ n(x)\) the distribution function of Mant\(\prod^{n}_{i}X_ i\) it can be proved that \(\sup_{1\leq x\leq \ell}| G_ n(x)-\log x| \leq C w\quad n,\) where C,w are positive constants, \(w<1.\)
An analogous property does not hold for the mantissa of the sum. The author proved elsewhere [see Math. Nachr. 127, 7-20 (1986; Zbl 0607.60022)] that if we consider the arithmetic means \(H_ k(a_ n)\) defined by recurrence \(H_ 0(a_ n)=a_ n,\quad H_{k+1}(a_ n)=(1/n)\sum^{n}_{j=1}H_ k(a_ j)\) or the logarithmic means \(L(a_ n)=(1+\log n)^{-1}\sum^{n}_{j=1}a_ j/j\) analogous inequalities can be proved (under suitable hypotheses), i.e. if \(G_ n(x)\) is the distribution function of Mant\(\sum X_ i\) we have \[ \sup_{1\leq x\leq \ell}| H_ k(G_ n(x))-\log x| \leq A(n,k),\quad \sup_{1\leq x\leq \ell}| L(G_ n(x))-\log x| \leq B(n). \] In the present paper he gives some interesting improvements of the terms A and B, whose form is too complicated to be referred here, for which the convergence to zero appears in a more straight-forward way.
Reviewer: M.Cugiani

60F05 Central limit and other weak theorems