Invariance principle under self-normalization for nonidentically distributed random variables
The authors consider the adaptive process \(V^{-1}_n\zeta_n\) of self-normalized partial sums \(S_k\) of independent random variables \(X_j\), defined by linear interpolation between the points \((V^2_k/V^2_n, S_k/V_n)\), \(k\leq n\), where \(V^2_k=\sum_{j\leq k}X^2_j\). They prove that if the \(X_k\)'s are symmetric, \(V^{-1}_n\zeta_n\) converges weakly to the Brownian motion \(W\) in each Hölder space supporting \(W\) if \(\max_{1\leq k\leq n}X^2_k/V^2_n@>P>>0\) as \(n\to\infty.\)
scientific article; zbMATH DE number 2012903