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A simple note on some empirical stochastic process as a tool in uniform L-statistics weak laws

In this paper, we are concerned with the stochastic process \begin{equation} β_{n}(q_{t},t)=β_{n}(t)=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}\left\{G_{t,n}(Y(t))-G_{t}(Y_{j}(t))\right\} q_{t}(Y_{j}(t)), \tag{A} \end{equation} where for $n\geq1$ and $T>0$, the sequences $\{Y_{1}(t),Y_{2}(t),...,Y_{n}(t),t\in [0,T]\}$ are independant observations of some real stochastic process ${Y(t),t\in [0,T]}$, for each $t \in [0,T]$, $G_{t}$ is the distribution function of $% Y(t)$ and $G_{t,n}$ is the empirical distribution function based on $% Y_{1}(t),Y_{2}(t),...,Y_{n}(t)$, and finally $q_{t}$ is a bounded real fonction defined on $\mathbb{R}$. This process appears when investigating some time-dependent L-Statistics which are expressed as a function of some functional empirical process and the process (A). Since the functional empirical process is widely investigated in the literature, the process reveals itself as an important key for L-Statistics laws. In this paper, we state an extended study of this process, give complete calculations of the first moments, the covariance function and find conditions for asymptotic tightness.