Almost sure convergence of sample range

摘要

Let X_1,X_2,... be i.i.d. random variables. The sample range is R_n = max{X_i, 1 ≤ i ≤ n} - min{X_i, 1 ≤ i ≤ n}. If α_k (R_k - β_k) →~w G for anon-degenerate distribution G and some sequences (α_k),(β_k) then we have lim_(n→∞) 1/log n ∑_(k=1)~n 1/k I(α_k(R_k-β_k))≤ x) = G(x) and lim_(n→∞) 1/log n ∑_(k=1)~n 1/k f (α_k(R_k-β_k))=∫_(-∞)~∞ f(x)dG(x) almost surely for any continuity point x of G and for any bounded Lipschitz function f : R → R.