On the functional limits for partial sums under stable law

For the partial sums (Sn) of independent random variables we define a stochastic process sn(t)≔(1/dn)∑k≤[nt](Sk/k−μ) and prove that (1/logN)∑n≤N(1/n)I{sn(t)≤x}→Gt(x) a.s. if and only if (1/logN)∑n≤N(1/n)P(sn(t)≤x)→Gt(x), for some sequence (dn) and distribution Gt. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an α-stable law with α∈(1,2].