Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X 1 , …, X N a random sample from a distribution F, let the process S δ N ( t ) be defined as\documentclass{article}\pagestyle{empty}\begin{document}$$ S_N^\Delta (t) = K_N^{ - 1} \sum\limits_{i = 1}^N {(c_i - \bar c})a_N (R_{x_i + \Delta d_i ,} t)\;\;0 \le t1,\;\Delta \in {\bf R}, $$\end{document}where K 2 N = σ N i=1 (c i − c̄) 2\documentclass{article}\pagestyle{empty}\begin{document}$$ a_N (i,t) = (i - Nt)I(Nt \le i < Nt + 1) + I(i \ge Nt + 1),\;i = \cdots ,N, $$\end{document}and R x i , + Δ d , is the rank of X i + Δd i , among X 1 + Δd 1 , …, X N + Δd N . The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c i and d i , S Δ N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C . The special case of two samples is considered.