# 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

, Volume 5 (2) – Oct 3, 2004
38 pages

/lp/springer_journal/best-khintchine-type-inequalities-for-sums-of-independent-rotationally-fT3G9H6TYR
Publisher
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

### References

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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### Monthly Plan • Read unlimited articles • Personalized recommendations • No expiration • Print 20 pages per month • 20% off on PDF purchases • Organize your research • Get updates on your journals and topic searches$49/month

14-day Free Trial

Best Deal — 39% off

### Annual Plan

• All the features of the Professional Plan, but for 39% off!
• Billed annually
• No expiration
• For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588$360/year

billed annually

14-day Free Trial