Suppose $X_1,X_2,\ldots$ are i.i.d. with $P(X_1=+1) = P(X_1=-1) = \frac 12.$ We know that $n^{-1/2}\sum_{i=1}^n X_i \stackrel{d}{\to} Z$ where $Z\sim\mathcal{N}(0,1).$
How steeply can a continuous function $f:\mathbb{R}\to\mathbb{R}$ be growing so that we also have the uniform integrability condition required to ascertain
$$\mathbb{E}f\left(n^{-1/2}\sum_{i=1}^n X_i\right)\to\mathbb{E}f(Z)~~?$$
a) Can $f$ grow as fast as $f(x) = |x|^p$ for any $p>0?$
b) Can $f$ grow as fast as $f(x) = e^{\lambda x}$ for any $\lambda>0?$
c) Can $f$ grow as fast as $f(x) = e^{\lambda x^2}$ for any $\lambda>0?$
I would also like to have estimates on the rate of convergence in each of these cases. Thanks!
We can show that for any integer $k$, the sequence $\left(\mathbb E\left[\left(\frac 1{\sqrt n}\sum_{i=1}^nX_i\right)^{2k}\right]\right)_{n\geqslant 1}$ is bounded, by expanding the sum. Therefore, we have the uniform integrability of the sequence $\left(f\left(n^{-1/2}\sum_{i=1}^n X_i\right)\right)_{n\geqslant 1}$ where $f(x)=\left|x\right|^p$ for any $p\gt 1$.
We can also show that for each $t$, the sequence $\left(\mathbb E\left[e^{tS_n/\sqrt n}\right]\right)_{n\geqslant 1}$ is bounded.
In the third case, we have to take $\lambda$ small enough, otherwise we can have $\mathbb E\left[f(Z)\right]=+\infty$, which would not be interesting.
For the rates of convergence, one can use Berry-Esseen bounds and express the moments in terms of tails.