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

AbstractLet $$X_i :(\Omega ,P) \to \mathbb{R}^n $$ be an i.i.d. sequence of rotationally invariant random vectors in $$\mathbb{R}^n $$ . If ∥X1∥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 Sn-1 or the ball Bn. 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.