On the strong law of large numbers and additive functions
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ON THE STRONG LAW OF LARGE NUMBERS AND ADDITIVE FUNCTIONS ´n Berkes1 , Wolfgang Mu ¨ller2 and Michel Weber3 Istva 1
Graz University of Technology, Institute of Statistics M¨ unzgrabenstrasse 11, A-8010 Graz, Austria E-mail: [email protected]
2
Graz University of Technology, Institute of Statistics M¨ unzgrabenstrasse 11, A-8010 Graz, Austria E-mail: [email protected] 3
IRMA, Universit´e Louis-Pasteur et C.N.R.S. 7 rue Ren´e Descartes, 67084 Strasbourg Cedex, France E-mail: [email protected] (Received September 24, 2010; Accepted October 29, 2010)
Dedicated to Endre Cs´ aki and P´ al R´ev´esz on the occasion of their 75th birthdays
Abstract Let f (n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f , we prove the following weighted strong law of large numbers: if X, X1 , X2 , . . . is any sequence of integrable i.i.d. random variables, then PN n=1 f (n)Xn lim = EX a.s. PN N →∞ n=1 f (n)
1. Introduction Let X, X1 , X2 , . . . be i.i.d. integrable random variables and f (n), n = 1, 2, . . . Pn a positive numerical sequence, F (n) = k=1 f (k). By a classical result of Jamison, Orey and Pruitt [5], under the condition 1 lim sup #{n : F (n) ≤ xf (n)} < ∞, x→∞ x
(1)
Mathematics subject classification numbers: 60F15,11A25. Key words and phrases: strong law of large numbers, weighted i.i.d. sums, strongly additive functions. 1 Research supported by FWF grant S9603-N23 and OTKA grants K 67961 and K 81928. 0031-5303/2011/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
2
¨ I. BERKES, W. MULLER and M. WEBER
we have the weighted strong law Pn k=1 f (k)Xk lim P = EX n n→∞ k=1 f (k)
a.s.
(2)
Conversely, if (2) holds for all i.i.d. sequences X, X1 , X2 , . . . with finite means, then (1) is valid. Note that condition (1) puts a restriction on the distribution of the weight sequence f (n) and not on the magnitude of the weights, as it happens, e.g., in central limit theory. In particular, (1) can fail even for bounded weight sequences f (n), see [5]. Condition (1) is generally difficult to check for irregular sequences f (n), and this leads to the question to study (2) for typical irregular sequences in number theory, for example, additive arithmetic functions. Let f (n), n = 1, 2, . . ., be a real-valued, strongly additive function, i.e., assume that f (mn) = f (m) + f (n)
for (m, n) = 1
(3)
and f (pα ) = f (p),
for p prime, α = 2, 3, . . .
It follows that f (n) =
X
(4)
f (p),
p|n
so that f is completely determined by its values taken over the primes. A typical example is ω(n), the number of different prime factors of n. Put An =
X f (p) , p
Bn =
p≤n
X |f (p)|2 . p
(5)
p≤n
In [1] we studied the weighted SLLN with coefficients f (n) and proved the following result (see Theorem 1.1 in [1]). Theorem 1. Assume that f ≥ 0 and Bp → ∞,
f (p) = o(Bp1/2 )
as p → ∞.
(6)
Then for any i.i.d. sequence X, X1 , X2 , . . . with finite means, the weighted strong law (2) holds. Condition (6) plays an important role in probabi
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