On the rate of almost sure convergence of Dümbgen's change-point estimators

Consider a triangular array of rowwise independent random elements with values in a measurable space. Suppose there exist [theta]n [set membership, variant]Tn={in-1: 1 [less-than-or-equals, slant]i[less-than-or-equals, slant]n-1} such that X1n,...,Xn,n[theta]n have distribution Pn and Xn,n[theta]n+1,..., Xnn have distribution Qn[not equal to]Pn, where Pn, Qn and [theta]n are unknown. We investigate a large class of change-point estimators n due to Dümbgen. Dümbgen proved that n - [theta]n = Op([gamma]2nn-1), where the sequence ([gamma]n) measures the 'distance' between Pn and Qn. We show that with probability one.