The Central limit theorem refers to the weak convergence (or convergence in distribution) of a properly normalised sum of iid random variables to an appropriate normal distribution. My question is, is the local limit theorem (which refers to the largest difference between the density of this normalised sum's distribution and the normal density, converging to zero) equivalent to convergence in the probability of said normalised sum to the normal distribution?
I know that the local limit theorem is stronger than the central limit theorem (in that local implies central) but I am unsure that the statement of the local limit theorem is strong enough to imply convergence in probability. Is it stronger, weaker or equivalent? Perhaps not related at all? Any help on the intuition would be greatly appreciated.
Let $X_j$ be i.i.d. variables of mean zero and variance 1. Write $S_n=X_1+\ldots +X_n$. The Central limit Theorem (CLT) states that the law $\mu_n$ of $n^{-1/2}S_n$ converges in distribution to a standard normal law $\mu$.
By the Skorokhod representation theorem, This is equivalent to the existence of variables $Y_n$ with law $\mu_n$ and a normal variable $Z$ such that $Y_n \to Z$ a.s.
It is easy to infer that it is also equivalent to the existence of variables $Y_n$ with law $\mu_n$ and a normal variable $Z$ such that $Y_n \to Z$ in probability.
Thelocal CLT is a finer result, and it separates into two cases: When the $X_i$ take values in a lattice and when they have a density. Both cases are most often (but not always!) proved via Fourier analysis. In contrast, for the CLT many other methods are available, such as Lindeberg coupling.
See the survey [2] for some proofs of local central limit theorems without Fourier analysis and a historical discussion.
[1] https://en.wikipedia.org/wiki/Skorokhod%27s_representation_theorem
[2] http://jirss.irstat.ir/article-1-127-en.pdf