A tauberian theorem for random walk

LetX ₁,X ₂,... be independent random variables, all with the same distribution symmetric about 0; $$S_n = \sum\limits_{i = 1}^n {X_i } $$ It is shown that if for some fixed intervalI, constant 1<a≦2 and slowly varying functionM one has $$\sum\limits_{k = 1}^n {P\{ S_k \in I\} \sim \frac{{n^{1 - 1/\alpha } }}{{M(n)}}} (n \to \infty )$$ then theX _i belong to the domain of attraction of a symmetric stable law.