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

Abstract

In this paper, we are concerned with the stochastic process $$\beta_n (q_t, t) = \beta_n(t) = \frac{1}{\sqrt{n}}\sum_{j=1}^n \{G_{t,n}(Y(t)) - G_t(Y_j(t))\} q_t(Y_j(t)), \tag{A}$$ where for $n \geq 1$ and $T > 0$, the sequences $\{Y_1(t), Y_2(t), \cdots, Y_n(t), t \in [0, T]\}$ are independent 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 $Y_1(t), Y_2(t), \cdots, Y_n(t)$ and finally $q_t$ is a bounded real function 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.