On the Increase of Dispersion of Sums of Independent Random Variables

Let ξ ₁ ,ξ ₂ , ⋯ ,ξ ₙ be independent random variables, \[ Q_k \{ l \} = { }_x { P} \{ x ξ _k x + l \}, \]\[ Q(L) = { }_x { P} \{ {x ξ _1 + ⋯ + ξ _n x + L} \}, s = ∑k + 1^n {( {1 - Q_k (l_k )} )lk^2 } \] We prove: Theorem 1. If L > max l₂, then\[ Q(L) {{CL}}{{l√ s }} \], whereCis an absolute constant Special cases of this theorem correspond to the results of [1]–[4].