Metric entropy and the central limit theorem in $C(S)$

Entropy inequalities and the Central Limit Theorem

Motivated by Barron (1986, Ann. Probab. 14, 336–342), Brown (1982, Statistics and Probability: Essays in Honour of C.R. Rao, pp. 141–148) and Carlen and So er (1991, Comm. Math. Phys. 140, 339–371), we prove a version of the Lindeberg–Feller Theorem, showing normal convergence of the normalised sum of independent, not necessarily identically distributed random variables, under standard conditio...

Entropy and the Central Limit Theorem

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

The Martingale Central Limit Theorem

One of the most useful generalizations of the central limit theorem is the martingale central limit theorem of Paul Lévy. Lévy was in part inspired by Lindeberg’s treatment of the central limit theorem for sums of independent – but not necessarily identically distributed – random variables. Lindeberg formulated what, in retrospect, is the right hypothesis, now known as the Lindeberg condition,1...

The Lindeberg central limit theorem

Theorem 1. If μ ∈P(R) has finite kth moment, k ≥ 0, then, writing φ = μ̃: 1. φ ∈ C(R). 2. φ(v) = (i) ∫ R x edμ(x). 3. φ is uniformly continuous. 4. |φ(v)| ≤ ∫ R |x| dμ(x). 1Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, third ed., p. 515, Theorem 15.15; http://individual.utoronto.ca/ jordanbell/notes/narrow.pdf 2Onno van Gaans, Probability measu...