On the LIL for Self-Normalized Sums of IID Random Variables

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Abstract

Let X, Xi, i ∈ ℕ, be i.i.d. random variables and let, for each n ∈ ℕ, Sn = ∑ni=1Xi and V2n = ∑ni=1X2i. It is shown that lim supn → ∞ | Sn | (Vn √ log log n) < ∞ a.s. whenever the sequence of self-normalized sums Sn/Vn is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if lim supt → ∞ [t| 𝔼X I(|X| ≤ t)|\𝔼X2I(|X| ≤ t)] < ∞.

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