$L_p$ norms

The main result of this section establishes the relationship between the $L_p$ norm of sums of random variables and their tail distributions.

Theorem 6.1   Given $p_0>0$, if $p \ge p_0$, and $(X_n)$ is a sequence of Banach valued independent random variables, then

$${\mathopen\Vert U\mathclose\Vert}_p \approx U^*(e^{-p}/4) + {\mat... ...U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/4) + {\mathopen\Vert\ell\mathclose\Vert}_p ,$$

where the constants of approximation depend only upon $p_0$.

We should note that we are not able to get universal control over the constants as $p_0 \to0$, as is shown by simple examples once one understands that ${\mathopen\Vert Y\mathclose\Vert}_p$ converges to the geometric mean of ${\mathopen\vert Y\mathclose\vert}$ as $p \to 0$.

Combining this with Corollary 3.2, we immediately obtain the following result that compares ${\mathopen\Vert S\mathclose\Vert}_q$ to ${\mathopen\Vert S\mathclose\Vert}_p$. This result extends results of Talagrand, (see Ledoux and Talagrand (1991, Theorem 6.20), Kwapien and Woyczynski (1992, Proposition 1.4.2 and comments following it; see also Hitczenko (1994, Proposition 4.1)) and Johnson, Schechtman and Zinn (1983). If this result is specialized to symmetric or positive real valued random variables, then by considering the cases $p=2$ or $p=1$, it implies the inequality of Rosenthal (1970), including the result of Johnson, Schechtman and Zinn (1983) that gives correct order of the constants as $p\to \infty$. Note that ${\left\Vert\ell\right\Vert}_q \le 2^{1/q} {\left\Vert M\right\Vert}_q$ by Proposition 2.1.

Theorem 6.2   Let $(X_n)$ be a sequence of Banach valued independent random variables and let $p_0>0$. Then there exist positive constants $c_1$, $c_2$ and $c_3$, depending only upon $p_0$, such that for $q \ge p \ge p_0$ we have

$${\mathopen\Vert U\mathclose\Vert}_q \le c_1 {\frac{q }{ \max\{p,\log(e+q)\}}} \bigl({\mathopen\Vert U\mathclose\Vert}_p + M^*(c_2^{-1} e^{-q}) \bigr) + c_1 {\left\Vert M\right\Vert}_q$$

$$\le c_3 {\frac{q }{ \max\{p,\log(e+q)\}}} \bigl({\mathopen\Vert U\mathclose\Vert}_p + {\mathopen\Vert M\mathclose\Vert}_q \bigr).$$

Let us proceed with the proofs. First we need a lemma that allows us to deal with the ``large'' parts of $U$, so that they might be effectively considered as a sum of disjoint random variables.

Lemma 6.3   Let $(X_n)$ be a sequence of Banach valued independent random variables, and let $0<r<1$. Then we may express $U^{(>\ell(r))} = \sum_{k=1}^\infty V_k$, where the random variables $V_k$ are disjoint, and $V_k^*(t) \le k \ell\bigl(t(k-1)!/r^{k-1}\bigr)$.

Proof: In proving this result, we may suppose without loss of generality that $X_n = X_n^{(>\ell(r))}$, that is, we may suppose that $\sum_n\Pr(X_n \ne 0) \le r$.

If $A$ is a finite subset of ${\mathbb{N}}$, define the event

$$E_A = \{ X_n \ne 0$$ if and only if $$n \in A \} .$$

For each positive integer $k$, let $E_k = \bigcup_{\substack{A \subseteq {\mathbb{N}}\\ {\mathopen\vert A\mathclose\vert} = k}} E_A$. Set $V_k = U I_{E_k}$. Notice that if ${\mathopen\vert A\mathclose\vert} = k$, then

$$\Pr(U I_{E_A} > x) \le \sum_{n \in A} \Pr({\mathopen\vert X_n\mathclose\vert} > x/k E_A)$$

$$= \sum_{n\in A} \Pr({\mathopen\vert X_n\mathclose\vert} > x/k) \prod_{m \in A \setminus\{n\}} \Pr(X_m \ne 0) .$$

Hence,

$$\Pr(V_k > x) = \sum_{\substack{A \subseteq {\mathbb{N}}\\ {\mathopen\vert A\mathclose\vert} = k}}\Pr(U I_{E_A} > x)$$

$$\le \sum_{i_1 < \dots < i_k} \sum_{j=1}^k \Pr({\mathopen\vert X_{i_j}\mathclose\vert} > x/k) \prod_{\substack{l=1 \\ l\ne j}}^k \Pr(X_{i_l} \ne 0)$$

$$\le {\frac{1}{ k!}} \sum_{i_1} \dots \sum_{i_k}\sum_{j=1}^k \Pr({\mat... ...j}\mathclose\vert} > x/k) \prod_{\substack{l=1 \\ l\ne j}}^k \Pr(X_{i_l} \ne 0)$$

$$= {\frac{k }{ k!}} \sum_{i_1} \dots \sum_{i_k}\Pr({\mathopen\vert X_{i_1}\mathclose\vert} > x/k) \prod_{l=2}^k \Pr(X_{i_l} \ne 0)$$

$$= {\frac{k }{ k!}} \left(\sum_n \Pr({\mathopen\vert X_n\mathclose\vert} > x/k) \right)\left(\sum_n \Pr(X_n \ne 0)\right)^{k-1}$$

$$\le {\frac{r^{k-1} }{ (k-1)!}} \Pr(\ell > x/k) .$$

Corollary 6.4   Let $(X_n)$ be a sequence of Banach valued independent random variables, let $0<r<1$, and let $0<p<\infty$. Then

$${\mathopen\Vert U^{(>\ell(r))}\mathclose\Vert}_p \le 2 e^{2^p r/p} {\mathopen\Vert\ell\mathclose\Vert}_p .$$

Proof: Apply Lemma 6.3 to obtain the $V_k$. Using the fact that $k \le 2^k$, we obtain that

$${\mathopen\Vert V_k\mathclose\Vert}_p^p \le k^p {\frac{r^{k-1}}{ ... ...e 2^p {\frac{(2^p r)^{k-1}}{ (k-1)!}} {\mathopen\Vert\ell\mathclose\Vert}_p^p .$$

Thus

$${\mathopen\Vert U^{(>\ell(r))}\mathclose\Vert}_p^p = \sum_{k=1}^... ...p r)^{k-1}}{ (k-1)!}} = 2^p e^{2^p r} {\mathopen\Vert\ell\mathclose\Vert}_p^p .$$

Proof of Theorem 6.1: Applying Proposition 2.1, we see that

$${\mathopen\Vert U\mathclose\Vert}_p \ge {\frac{1}{ 2}} {\mathopen\Vert M\mathclose\Vert}_p \ge 2^{-1-1/p} {\mathopen\Vert\ell\mathclose\Vert}_p .$$

Also, we have that

$${\mathopen\Vert U\mathclose\Vert}_p^p = \int_0^1 (U^*(t))^p \, dt \ge 8^{-1} e^{-p} (U^*(e^{-p}/8))^p ,$$

that is, ${\left\Vert U\right\Vert}_p \ge 8^{-1/p} e^{-1} U^*(e^{-p}/8) \ge 8^{-1/p} e^{-1} U^*(e^{-p}/4)$. Hence we have shown that there exists a constant $c_1$, depending only upon $p_0$, such that

$${\mathopen\Vert U\mathclose\Vert}_p \ge c_1^{-1}(U^*(e^{-p}/4) + {\mathopen\Vert\ell\mathclose\Vert}_p) .$$

Furthermore, by Proposition 2.1,

$$\Pr(U \ne U^{(\le \ell(e^{-p}/8))}) \le \Pr(M > \ell(e^{-p}/8)) \le \Pr(M > M^*(e^{-p}/8)) \le e^{-p}/8 .$$

Hence $U^*(e^{-p}/8) \ge (U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/4)$, and so we have shown that there is a constant $c_2>0$, depending only upon $p_0$, such that

$${\mathopen\Vert U\mathclose\Vert}_p \ge c_2^{-1}((U^{(\le \ell(e^{-p}/8))})^* (e^{-p}/4) + {\mathopen\Vert\ell\mathclose\Vert}_p) .$$

Now let us derive the converse inequalities. Corollary 3.2 tells us that for $t \le e^{-p}/2$ that

$$(U^{(\le \ell(e^{-p}/8))})^*(t) \le c_2 {\frac{\log(1/t) }{ p + \log(2)}} \bigl((U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/2) + \ell(e^{-p}/8) \bigr).$$

Thus

$${\mathopen\Vert U^{(\le \ell(e^{-p}/8))}\mathclose\Vert}_p^p \le \int_0^{e^{-p}/2} ((U^{(\le \ell(e^{-p}/8))})^*(t))^p \, dt +(1-e^{-p}/2)(U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/2)$$

$$\le {\frac{1}{ (p + \log(2))^p}} \int_0^1 (\log(1/t))^p \, dt \,\bigl((U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/2) + \ell(e^{-p}/8) \bigr)^p$$

$$\qquad + (U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/2)$$

$$\le c_3^p\bigl((U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/2) + \ell(e^{-p}/8) \bigr)^p ,$$

where $c_3>0$ depends only upon $p_0$. Furthermore,

$$\ell(e^{-p}/8)^p \le 8 e^p \int_0^1 \ell(t)^p \, dt = 8 e^p {\mathopen\Vert\ell\mathclose\Vert}_p^p.$$

Hence, applying Corollary 6.4, and the (quasi-)triangle inequality for $L_p$, we deduce that there exists a constant $c_4$, depending only upon $p_0$, such that

$${\mathopen\Vert U\mathclose\Vert}_p \le c_4((U^{(\le \ell(e^{-p}/8))})^* (e^{-p}/2) + {\mathopen\Vert\ell\mathclose\Vert}_p) .$$

Finally the result follows by noticing that

$$(U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/2) \le (U^{(\le \ell(e^{-p}/8))})^* (e^{-p}/4) ,$$

and also, by an argument similar to one presented above, that

$$(U^{(\le \ell(e^{-p}/8))})^*(e^{-p}/2) \le U^*(3e^{-p}/8) \le U^*(e^{-p}/4) .$$

Finally we remark that from the results mentioned at the end of Section 3 we can obtain one sided versions of Theorem 6.1 with ${\left\vert S\right\vert}$ in place of $U$, for example, given $p \ge p_0$,

$${\mathopen\Vert S\mathclose\Vert}_p \le c S^*(e^{-p}/c) + c {\mathopen\Vert\ell\mathclose\Vert}_p .$$

where the constants depend only upon $p_0$.

Obviously if the sequence of random variables satisfy the Lévy property, then we can obtain the two sided inequality, but otherwise the other side of the inequality need not hold, as is shown by the example $X_1 = 1$, $X_2 = -1$, $X_n = 0$ ($n \ge 2$).