Best Khintchine Type Inequalities for Sums of Independent, Rotationally Invariant Random Vectors

Best Khintchine Type Inequalities for Sums of Independent, Rotationally Invariant Random Vectors Let $$X_i :(\Omega ,P) \to \mathbb{R}^n $$ be an i.i.d. sequence of rotationally invariant random vectors in $$\mathbb{R}^n $$ . If ∥X 1∥2 is dominated (in the sense defined below) by ∥Z∥2 for a rotationally invariant normal random vector Z in $$\mathbb{R}^n $$ , then for each k∈ ℕ and $$(\alpha ) \subseteq \mathbb{R}$$ $$\left( {\mathbb{E}\left\| {\sum\limits_{i = 1}^k {\alpha _i X_i } } \right\|^p } \right)^{1/p} \leqslant {\text{ (resp}}{\text{.}} \geqslant {\text{)(}}\mathbb{E}\left\| {\text{Z}} \right\|^p )^{1/p} \left( {\sum\limits_{i = 1}^k {\left| {\alpha _i } \right|^2 } } \right)^{1/2}$$ for p≥3 or p,n≥2 (resp. for 1≤p≤2, n≥3). The constant ( $$\mathbb{E}$$ ∥Z∥p)1/p is the best possible. The result applies, in particular, for variables uniformly distributed on the sphere S n-1 or the ball B n. In the case of the sphere, the best constant is $$(\mathbb{E}\left\| {\left. \mathbb{Z} \right\|} \right.^p )^{1/p} = \sqrt {\frac{2}{n}} \left( {\Gamma \left( {\frac{{p + n}}{2}} \right)/\Gamma \left( {\frac{n}{2}} \right)} \right)^{1/p} .$$ With this constant, the Khintchine type inequality in this case also holds for 1≤p≤2,n=2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Best Khintchine Type Inequalities for Sums of Independent, Rotationally Invariant Random Vectors

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2001 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1011434208929
Publisher site
See Article on Publisher Site

Abstract

Let $$X_i :(\Omega ,P) \to \mathbb{R}^n $$ be an i.i.d. sequence of rotationally invariant random vectors in $$\mathbb{R}^n $$ . If ∥X 1∥2 is dominated (in the sense defined below) by ∥Z∥2 for a rotationally invariant normal random vector Z in $$\mathbb{R}^n $$ , then for each k∈ ℕ and $$(\alpha ) \subseteq \mathbb{R}$$ $$\left( {\mathbb{E}\left\| {\sum\limits_{i = 1}^k {\alpha _i X_i } } \right\|^p } \right)^{1/p} \leqslant {\text{ (resp}}{\text{.}} \geqslant {\text{)(}}\mathbb{E}\left\| {\text{Z}} \right\|^p )^{1/p} \left( {\sum\limits_{i = 1}^k {\left| {\alpha _i } \right|^2 } } \right)^{1/2}$$ for p≥3 or p,n≥2 (resp. for 1≤p≤2, n≥3). The constant ( $$\mathbb{E}$$ ∥Z∥p)1/p is the best possible. The result applies, in particular, for variables uniformly distributed on the sphere S n-1 or the ball B n. In the case of the sphere, the best constant is $$(\mathbb{E}\left\| {\left. \mathbb{Z} \right\|} \right.^p )^{1/p} = \sqrt {\frac{2}{n}} \left( {\Gamma \left( {\frac{{p + n}}{2}} \right)/\Gamma \left( {\frac{n}{2}} \right)} \right)^{1/p} .$$ With this constant, the Khintchine type inequality in this case also holds for 1≤p≤2,n=2.

Journal

PositivitySpringer Journals

Published: Oct 3, 2004

References

  • Projection constants of symmetric spaces and variants of Khintchine's inequality
    König, H.; Schütt, C.; Tomczak-Jaegermann, N.

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