I'm reading the proof of the Berry-Esseen Theorem from Varadhan's notes and I'm having trouble with a couple of steps. The proof starts at the top of page 97.
Confusion 1: On page 98 while estimating $\lambda[a,\infty)-\mu[a,\infty)$, there is the assumption that $\mu$ can be represented as $fdm$ with $m$ being the Lebesgue measure and $f$ being a bounded function -
Therefore, if we assume that $\mu$ has a density bounded by $C$.
I am ok with this since these computations are applied to the case when $\mu$ is the Gaussian measure. However, in the next line they seem to be assuming the same thing about $\lambda$ -
Since we get a similar bound in the other direction as well,
$\lambda$ is intended to be the pushforward-measure for $\frac{X_1 + \dots + X_n}{\sqrt{n}}$ where $X_i$ are iid with mean $0$ variance $1$ and with $(2+\alpha)$ moment finite.
Is the bounded density assumption on $\lambda$ being used? If so, how is it valid?
Confusion 2: At the bottom of page 98, they have the expansion for the c.f. $\hat{\lambda}_n$ as $$ \hat{\lambda_n}(t) = \left(1 - \frac{t^2}{2n} + O\left(\frac{1}{n^{1+\alpha/2}}\right)\right)^n $$
How do I get the rate of convergence of this to $\exp(-\frac{t^2}{2}$) as $$ \left| \hat{\lambda}_n(t) - e^{-t^2/2}\right| \leq C\frac{|t|^{\alpha+2}}{n^\alpha} \ \ \ \text{ for }\ \ \ |t| < n^{\frac{\alpha}{2+\alpha}}? $$ In the central limit theorem, I think one can justify taking logs and appealing to L'Hopital's rule to show convergence.
How do I effectivise the argument in this case?
Thanks for reading.
The bounded density assumption on $\mu$ suffices. The "similar bound" Varadhan is referring to is $$\mu[a,\infty)-\lambda[a,\infty) \le 2Ch+\mu[a+2h,\infty)-\lambda[a,\infty) \le \int f_{a+h,h} \, d(\mu-\lambda) \,.$$
It is important that in the first line of (3.37) two suprema are compared- the inequality need not hold for each $a$.
Regarding your confusion 2: You are missing a factor $|t|^{2+\alpha}$ inside the $O( \cdot)$. After taking logs and using the Taylor expansion of $\log$, I get the bound $\frac{|t|^{2+\alpha}}{n^{\alpha/2}}$ so the power in the denominator is different than in the notes. Correcting this will lead to a weaker final estimate (which is not sharp anyway, see [1])
[1] Katz, Melvin L. Note on the Berry-Esseen Theorem. Ann. Math. Statist. 34 (1963), no. 3, 1107--1108. https://projecteuclid.org/download/pdf_1/euclid.aoms/1177704037