## Afrika Statistika

- Afr. Stat.
- Volume 5, Number 1 (2010), 245-251.

### 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.

#### Article information

**Source**

Afr. Stat., Volume 5, Number 1 (2010), 245-251.

**Dates**

First available in Project Euclid: 1 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.as/1388545346

**Mathematical Reviews number (MathSciNet)**

MR2920301

**Zentralblatt MATH identifier**

1328.62288

**Subjects**

Primary: 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators 62G20: Asymptotic properties G2G05

**Keywords**

Empirical processes Order Statistics L-statistics

#### Citation

Samb Lô, Gane. A simple note on some empirical stochastic process as a tool in uniform L-statistics weak laws. Afr. Stat. 5 (2010), no. 1, 245--251. https://projecteuclid.org/euclid.as/1388545346