Log-classes for the increments of a multivariate emipirical process (Q1269940)

Let \(\xi_1,\xi_2,\dots\) be independent random vectors with common uniform distribution on the \(d\)-dimensional unit cube \(I^d\), and, for every integer \(n\geq 1\), let \(\alpha_n\) denote the normalized and centered empirical measure based on the first \(n\) of these random vectors. For a given sequence \((h_n)_{n\geq 1}\) of positive constants, set \(C(x)=z+[0,x_1h_n]\times\dots\times[0,x_dh_n], x=(x_1,\dots,x_d), z\in I^d\). Under the condition \[ 0<\liminf_{n\to\infty}| \log h_n| /\log n \leq \limsup_{n\to\infty}| \log h_n| /\log n < 1/d \] two Strassen-type functional limit theorems for \(\alpha_n(C(x))\) are obtained.