Expectation of Truncated Random Variables (2)

Question

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ and $\epsilon >0$ and define $c_n:=\frac{2\delta\sigma}{\epsilon\sqrt{\log\log n}}$ and $b_n:=c_n\sigma\sqrt{n}$. In addition, for each $n\geq 1$ and $1\leq j \leq n$ define the truncated random variables as follows. $$X'_{n, j}:= X_j\mathbb I_{\{|X_j| < 0.5b_n\}}.$$ I already has

$$ \left|\mathbb E X'_{n, j}\right| = \left|\mathbb E X_j\mathbb I_{\{|X_j| < 0.5b_n\}}\right| = \left|-\mathbb EX_j \mathbb I_{\{|X_j|\geq 0.5b_n\}}\right|\leq \mathbb E |X_j|\mathbb I_{\{|X_j|\geq 0.5b_n\}} \leq \frac{2}{b_n} \mathbb EX^2 \mathbb I_{\{|X_j|\geq 0.5b_n\}}. $$

Define $S'_n:=\sum_{j=1}^n X'_{n, j}$ and use above result one can get

$$ \left| \mathbb E S'_n \right| \leq n\left| \mathbb E X'_{n, j} \right| \leq \frac{2n}{b_n} \mathbb EX^2\mathbb 1_{\left\{ \left|X\right| \geq \frac{b_n}{2} \right\}} = o\left( \frac{n}{b_n} \right) = o\left( \sqrt{n\log\log n} \right), \mathrm{as} \ n \to \infty. $$

Then I read the following line: for the chosen $\delta$ above, there is $n_0$ such that for $n > n_0$,

$$ \left| \mathbb E S'_n \right| \leq \epsilon\delta\sqrt{n\log\log n}. $$

I do NOT understand this last line. How do we get it? And how do I interpret it? Moreover, how do we get the range for $0 <\delta < 0.5$? Thank you!

Answer 1

You know that $$ \left| \mathbb E S'_n \right| = o\left( \sqrt{n\log\log n} \right), \mathrm{as} \ n \to \infty. $$ By definition of $o(\ )$, this means that, for every $\alpha\gt0$, there exists some finite $N_\alpha$ such that, for every $n\geqslant N_\alpha$, $$ \left| \mathbb E S'_n \right| \leq \alpha\sqrt{n\log\log n}. $$ Choose $\alpha=\epsilon\delta$ and let $n_0=N_{\epsilon\delta}$.

Moreover, how do we get the range for $0 <\delta < 0.5$?

Not apparent from the parts you reproduced. Note that we do not "get the range", we impose this condition, which is probably used somewhere else in the proof.