Processus empiriques des rapports de $m$-espacements uniformes disjoints (Non-overlapping uniform $m$-spacings-ratio empirical processes)

We consider an empirical process based upon ratio of selected pair of the non-overlapping $m$-spacings generated by independent samples of arbitrary sizes. As a main result, we show that when both samples are uniformly distributed on intervals of equal lengths, this empirical process converges to a mean-centered Brownian bridge of the form $\ \displaystyle (B\circ H_{m})_{C}(v)=B(H_{m}(v))-2(2m+1)C\ \frac{(2m-1)!}{m((m-1)!)^2}(v(1-v))^m\int_{0}^{1}B(H_{m}(s))\ud s,\ $ for $\ \displaystyle 0\leq v\leq 1,\ $ where $\ \displaystyle B(.)\ $ denotes a Brownian bridge, $\ \displaystyle H_{m},\ $ the distribution function of the Beta distribution with parameters $m$ and $m,\ $ and $C,\ $ a constant.