The functional law of the iterated logarithm for the empirical process based on sample means (Q5939316)

For a scheme of i.i.d. random variables (mean \(\mu\), variance \(\sigma^2>0\)) \(\{X_{i,j}\mid i,j \geq 1\}\) the empirical distributions \(\Phi_{N,n}\) of \((Z_{1,N}, \cdots , Z_{N,n})\) where \(Z_{i,n}=n^{-1/2}\sum_{j=1}^n (X_{i,j}-\mu)/\sigma\) are shown to satisfy a functional law of the iterated logarithm.