On Estimates of the Rate of Convergence in the Central Limit Theorem in Multidimensional Spaces

Abstract

v. SENATOV (Represented by V. M. Zolotarev) (Translated by V. V. Sazonov) Consider independent and identically distributed random variables X1, X2, with values in the Hilbert space 12. Assume that E X1 0 and E IX 2 < , where I" is the norm in 12. Denote by P the distribution of X1 and by Pn the distribution of Zn (X /... / Xn)n-/2. Let a 2, a22,.., be the eigenvalues of the covariance operator B of P, which we may assume, without loss of generality, to be ordered: a 2 _> a _> and let Af be the normal law with mean zero and covariance operator B. Furthermore, let SR(a) (x: Ixal <_ R} be the ball with radius R and center at a in/2; Cs Cs(P, Af) where Cs(', ") are the ideal metrics introduced by Zolotarev; / tt(P, Af), where tt(., .) is the uniform distance with respect to 2 be the trace of B. the class of convex Borel sets in /2; s f IXl P(dx) < cx; a 2 -]i=1 tri The distribution P may be degenerate in 12. The changes needed in this case in the estimates presented below are quite obvious. We are interested in bounds for ", o) Besides An we shall consider also the distances and pR(Pn, Jf) sup {An(P, r, a): a e/2, r <_ R} Put pd(pn, Af)= sup {An(P, r, a): lal _< d, _> 0}, d>O.Ai Ai(n) a2nl/2 -nl/-----, i-e e(R, n) (R33n-1/2) 1/6 i--1, 2..., 5(R, n) (Rln-1/2)1/2, and fix some (large enough) number -> 0; c, ci (ci _> 1) will denote constants and c(a, b,...) quantities depending only on a, b, THEOREM 1. For any R

Sums of Independent Random Vectors: Proximity Estimating

“…Applying the techniques presented in Refs. [9,10] and Lemma 3.1 in Section 3, it is possible to show the existence constants c * , c * * that depend only on the distribution of the random vector X such that the condition…”

Central limit theorems and bootstrap in high dimensions

“…since m ≥ n/2, and so 1/m 3/2 ≤ (2/n) 3/2 . To bound III 1 , recall the definition of h(w, t) in (33). Note that m jkl (Z (i) + τ Z i ) = 0 for all τ ∈ [0, 1] if both h(Z (i) , 2) = 0 and χ i = 1 hold (indeed, if χ i = 1, then max 1≤j≤p |Z ij | ≤ (3/4)(n/m) 1/2 /β ≤ 2/β, and so when both h(Z (i) , 2) = 0 and χ i = 1 hold, we have that h(Z (i) + τ Z i , 0) = 0, which in turn means that either F β ( (29); in both cases, g (F β (Z (i) + τ Z i )) = 0, so that the claim follows from the definition of m).…”

“…where A is a class of (Borel) sets in R p . Bounding ρ n (A) for various classes A of sets in R p , with a special emphasis on explicit dependence on the dimension p in bounds, has been studied by a number of authors; see, for example, [5,6,7,20,26,31,32,33,34] (see [15] for an exhaustive literature review). Typically, we are interested in how fast p = p n → ∞ is allowed to grow while guaranteeing ρ n (A) → 0.…”

Note on Qualitative Robustness of Multivariate Sample Mean and Median

“…Note that the proof presented in [20] uses the so-called convolution approach (see, e.g., [26]) and induction arguments. This method has been widely used to estimate rates of convergence in multidimensional Central Limit Theorems (see, e.g., [13,14,21,26]). …”