Rates of convergence in the free central limit theorem

Convergence rates for the central limit theorem.

Rates of convergence for minimal distances in the central limit theorem under projective criteria

In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given.

The Edgeworth Expansion and Convergence in the Central Limit Theorem

For regularity of φ, proceed by induction, write the derivative difference quotient, use the mean value theorem on the corresponding integrand and apply LDCT. • Corollary 1 If X has m finite moment and law μ, then φ(ξ) = ∫ R e (ix)dμ(x) for k ≤ m. • Theorem 2 If S = ∑n k=1 Xk where {Xk} are independent with c.f.’s φk the c.f. of S, φ satisfies φ = ∏n k=1 φk. Proof: The law of S is the n-fold co...

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.

Convergence of Moments in a Markov-chain Central Limit Theorem

Let (Xi)1i=0 be a V -uniformly ergodic Markov chain on a general state space, and let be its stationary distribution. For g : X! R, de ne Wk(g) := k 1=2 k 1 X i=0 g(Xi) (g) : It is shown that if jgj V 1=n for a positive integer n, then ExWk(g) n converges to the n-th moment of a normal random variable with expectation 0 and variance 2 g := (g ) (g) + 1 X j=1 Z g(x)Exg(Xj) (g) 2 : This extends t...