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world:marcinkiewicz [2022/01/15 10:43] rdouc |
world:marcinkiewicz [2022/03/16 07:40] (current) |
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| - | Using the [[rayan:11_marcinkiewicz_zygmund#g_lemma|$g$-lemma]] with $g \colon x \mapsto x^p$ we deduce | + | Using the [[world:marcinkiewicz#g_lemma|$g$-lemma]] with $g \colon x \mapsto x^p$ we deduce |
| \begin{align*} | \begin{align*} | ||
| \PE \lrb{\lrav{S_n}^p} = p \int_{\rset_+} x^{p-1} \PP\lr{\lrav{S_n} \geq x} \rmd x &\leq \sum_{i=1}^n p \int_{\rset_+} x^{p-1} \PP\lr{\lrav{X_i} \geq x} \rmd x + 2p e^r \int_{\rset_+} \frac {x^{p-1}} {\lr{1 + \frac {x^2} {r B_n}}^r} \rmd x \\ | \PE \lrb{\lrav{S_n}^p} = p \int_{\rset_+} x^{p-1} \PP\lr{\lrav{S_n} \geq x} \rmd x &\leq \sum_{i=1}^n p \int_{\rset_+} x^{p-1} \PP\lr{\lrav{X_i} \geq x} \rmd x + 2p e^r \int_{\rset_+} \frac {x^{p-1}} {\lr{1 + \frac {x^2} {r B_n}}^r} \rmd x \\ | ||
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| \leq 2^p \mathbb{E}\lrb{\norm{X}^p} . | \leq 2^p \mathbb{E}\lrb{\norm{X}^p} . | ||
| \end{equation*} | \end{equation*} | ||
| - | Moreover, by equivalence of norms in finite dimension, the result only needs to be proved for the norm $\norm{\cdot}_p$ on $\rset^d$. Using the [[rayan:11_marcinkiewicz_zygmund#mz|Marcinkiewicz–Zygmund inequality]] in dimension 1 provides | + | Moreover, by equivalence of norms in finite dimension, the result only needs to be proved for the norm $\norm{\cdot}_p$ on $\rset^d$. Using the [[world:marcinkiewicz#mz|Marcinkiewicz–Zygmund inequality]] in dimension 1 provides |
| \begin{align*} | \begin{align*} | ||
| \mathbb{E}\lrb{\norm{\sum_{i=1}^n X_i }_p^p} &= \mathbb{E}\lrb{\sum_{j=1}^d \lrav{ \sum_{i=1}^n X_i(j) }^p} \\ | \mathbb{E}\lrb{\norm{\sum_{i=1}^n X_i }_p^p} &= \mathbb{E}\lrb{\sum_{j=1}^d \lrav{ \sum_{i=1}^n X_i(j) }^p} \\ | ||
world/marcinkiewicz.1642239817.txt.gz · Last modified: 2022/03/16 01:36 (external edit)
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