Limiting Behavior of Weighted Sums of Independent Random Variables

In this paper, we study weighted sums ∑ⁿᵢ₌₁ cₙ₋ᵢ Xᵢ of i.i.d. zero-mean random variables X₁, X₂, ⋯, under the condition that the sequence (cₙ) is square summable. It is proved that such weighted sums are, with probability 1, of smaller order than n1/α (respectively log n, etc.) iff E|X₁|^α < ∞ (respectively Eet|X₁| < ∞ for all t < ∞, etc.). Certain analogs of the law of the iterated logarithm for such weighted sums are also obtained.