On Functional Versions of the Arc-Sine Law

Let X ₁,X ₂,… be a sequence of random variables. Let S _k =X ₁+???+X _k and assume that S _k /b _k converges in distribution for some numerical sequence (b _k ). We study the weak convergence of the random processes {Λ _n (z), z∈?}, where

$\Lambda_{n}(z)=\frac{1}{n}\sum_{k=1}^{n}I\left\{\frac{S_{k}}{b_{k}}\leq z\right\}.$

We consider the same problem when the normalized partial sums S _k /b _k are replaced by other functionals of the sequence (X _n ). In particular, we investigate the case of sample extremes in detail.