Abstract
We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is \[ \varepsilon _{n}(f):={\mathbb{E} }\Big (f\Big (\frac 1{\sqrt n}\sum _{i=1}^{n}Z_{i}\Big )\Big )-{\mathbb{E} }\big (f(G)\big )\rightarrow 0 \] where $Z_{i}$, $i\in \mathbb{N} $, are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables $Z_{i}$, $i\in{\mathbb {N}} $, on hand is needed. Essentially, one needs that the law of $Z_{i}$ is locally lower bounded by the Lebesgue measure (Doeblin’s condition). This topic is also widely discussed in the literature. Our main contribution is to discuss convergence in distribution norms, that is to replace the test function $f$ by some derivative $\partial _{\alpha }f$ and to obtain upper bounds for $\varepsilon _{n}(\partial _{\alpha }f)$ in terms of the infinite norm of $f$. Some applications are also discussed: an invariance principle for the occupation time for random walks, small balls estimates and expected value of the number of roots of trigonometric polynomials with random coefficients.
Highlights
Our aim is to compare the law of Sn(Y ) with the law of Sn(G) where G = (Gk)1≤k≤n denote n standard independent Gaussian random variables
Notice that by (2.2) |σni,jk| ≤ 1 so we may assume without loss of generality that E(|Cn,kGk|p) ≤ Cp(Y ) for the standard normal random variables as well
[33] Prohorov considers a sequence of i.i.d. random variables Xn and proves that the convergence in the Central Limit Theorem (CLT) holds in total variation distance if and only if the following hypothesis holds: there exists n∗ such that the law of X1+...+Xn∗ has an absolute continuous component, that is X1+...+Xn∗ ∼ μ(dx)+p(x)dx
Summary
In his seminal paper [33] Prohorov proved that the convergence in total variation distance is equivalent to the fact that there exists r such that the law of Y1 + · · · + Yr has an absolutely continuous component This is “essentially” equivalent to the Doeblin’s condition that we present (see Remark 2.1): we assume that there exists r, ε > 0 and there exists yk ∈ Rm such that for every measurable set A ⊂ Br(yk). In the recent paper [24], the above result has been proved for general independent and identically distributed random variables Yk, k ∈ N, which are centered and with variance one. This is the main content of the forthcoming article [10] which follows the series [27, 3, 2] of articles dedicated to this task in the Gaussian case
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