I am reading Lehmann and Romano's Testing Statistical Hypotheses, 2008, and having difficulties on understanding Lemma 11.4.1.
The Lemma is stated as follows:
Suppose $X_{n,1}, ..., X_{n,n}$ are iid $F_{n}$ with $F_{n} \in \tilde{F}$, where $\tilde{F}$ satisfies: $$\lim_{\lambda \rightarrow \infty} \sup_{F \in \tilde{F}} E_{F} [\frac{|X - \mu(F)|^{2}}{\sigma^{2}(F)}] 1\{\frac{|X - \mu(F)|}{\sigma(F)} > \lambda \}] = 0$$ Let $\overline{X}_{n} = \sum_{i = 1}^{n} \frac{X_{n,i}}{n}$. Then, under $F_{n}$, $$\frac{ n^{\frac{1}{2}} [\overline{X}_{n} - \mu(F_{n})]}{\sigma(F_{n})} \xrightarrow[]{d} N(0, 1).$$
The strategy of the proof is to verify the Lindeberg Condition. However, I am confused by the following inequality in the proof:
But, for every $\lambda > 0$, $$\limsup_{n} E [{Y_{n,i}^{2}} 1\{ |Y_{n,i}| > \varepsilon n^{\frac{1}{2}} \}] \leq \limsup_{n} E [{Y_{n,i}^{2}} 1\{ |Y_{n,i}| > \lambda \}]$$
where $Y_{n,i} = \frac{[X_{n,i} - \mu(F_{n})]}{\sigma(F_{n})}$. Therefore, my question is, can anyone help me understand why the above inequality works? Thanks!
I think I figured it out. We can pick $n$ such that $n \geq (\frac{\lambda}{\varepsilon})^{2}$, then, the inequality holds immediately.