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Abstract
We introduce a new, weak Cramér condition on the characteristic function of a random vector which does not only hold for all continuous distributions but also for discrete (non-lattice) ones in a generic sense. We then prove that the normalized sum of independent random vectors satisfying this new condition automatically verifies some small ball estimates and admits a valid Edgeworth expansion for the Kolmogorov metric. The latter results therefore extend the well known theory of Edgeworth expansion under the standard Cramér condition, to distributions that are purely discrete.
Highlights
If Xi are i.i.d. real random variables, centered with unit variance, Central Limit Theorem ensures that, if Sn := X1 + . . . + Xn, lim P √Sn ≤ x − Φ(x) = 0, n→+∞n where Φ denotes the cumulative distribution function of the standard Gaussian variable
C − Φ(x) ≤ √, n n and is very natural to ask if an higher order asymptotic expansion of the distribution function can be made explicit
Despite the richness of the class of random variables or vectors satisfying the weak Cramér condition, the latter is flexible enough to prove some fairly general results that are classical for continuous random variables but difficult to obtain as soon as the underlying variables have a discrete component
Summary
If the entries Xi admit a finite moment of order q ≥ 3 and satisfy the so-called Cramér condition, there exists explicit polynomials P1, . We introduce a new, weak Cramér condition on the characteristic function of a random vector which englobes the classical Cramér condition but is satisfied by discrete, non-lattice distributions, in a generic sense, see Definition 2.1 and Proposition 2.4 below. Under this weakened assumption, we establish in Theorem 4.3 and Corollary 4.4, the validity of the Edgeworth expansion in the Kolmogorov metric. In order to facilitate the reading of the paper, the proofs of the results stated in Sections 3 and 4 are postponed in the last Section 5
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