数学物理学报(英文版) ›› 2000, Vol. 20 ›› Issue (4): 533-541.
孔繁超, 唐启鹤
KONG Fan-Chao, TANG Qi-He
摘要:
Let Xn, n 1, be a sequence of independent random variables satisfying
P(Xn = 0) = 1 − P(Xn = an) = 1 − 1/pn, where an, n 1, is a sequence of real
numbers, and pn is the nth prime,set FN(x) = P
NP
n=1
Xn x
. The authors investigate a
conjecture of Erd¨os in probabilistic number theory and show that in order for the sequence
FN to be weakly convergent, it is both sufficient and necessary that there exist three
numbers x0 and x1 < x2 such that limsup
N!1
(FN(x2) − FN(x1)) > 0 holds, and L0 =
lim
N!1
FN(x0) exists. Moreover, the authors point out that they can also obtain the same
result in the weakened case of liminf
n!1
P(Xn = 0) > 0.
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