Acta Mathematica Academiae Scientiarum Hungaricae 67. (1995)

1995 / 1-2. szám - Manstavičius, E.: Functional approach in the divisor distribution problems

2 E. MANSTAVlClUS We say that Yk converges to Y Pn-almost surely (P„-a.s.) if for each e > О Thus, a compact set A C S such that, for each £ > 0 and each X £ A, and may be called a cluster set of the sequence {Yk} Pn-a.s. In what follows we denote the relations (1) and (2) by Instead of (2) sometimes we shall write Ykl — X (P„-a.s.) having in mind that k\ is a random increasing subsequence. The following lemma will be repeatedly used to deduce corollaries from the principal result. Lemma 1. Let (S,d) and (Si,e?i) be separable metric spaces and let f : S —* Si be a continuous map into Si. Then (3) in (S,d) implies the convergence No) => f(A) (Pn-a.s.) in the second space (S1? di ). The proof is evident (see [6], Lemma 137, or [21], Lemma 1.5.11). If Pn = P does not depend on n, due to monotonicity of the events in (1) and (2), one returns to the traditionally treated situation. Here it should be observed that the proposition Yk => A for a compact set A is equivalent to two assertions: the sequence Yk is relatively compact and A is the set of its limit points. Observe that in the concept of strong convergence introduced above we need the separability of the spaces to assure the measurability of the distances only. This condition is superfluous when Pn has a finite carrier, as in the case considered afterwards. So, at the cost of simplicity we have gained the desired property: we do not need a product space or some other construction of the fixed probabilistic space, common for all Yk I­In the present paper we consider sequences of functions related to the multiplicative structure of natural numbers. Let С = C[0,1] be the Banach lim limsupPn( max d(Yk,Y) ^ e) = 0 r n—кх> x^k^n (1) lirri limsupPn( max d(Yk,A) ^ e) = 0 % n—юо х'йк'^п (2) lim liminfPn( min d(Yk,X) < e) =1 x-»oo n—t-oo x<k<n (3) Yk=> A (Pn-a.s.). Acta Mathematica Hungarica 67, 1995

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