world:marcinkiewicz
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world:marcinkiewicz [2022/01/15 10:41] rdouc créée |
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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} \\ | ||
| Line 130: | Line 130: | ||
| \end{align*} | \end{align*} | ||
| </hidden> | </hidden> | ||
| - | |||
| - | ====== Some notation ====== | ||
| - | Reminder of the notation introduced in [[https://projecteuclid.org/download/pdf_1/euclid.aos/1059655912|Fort, Moulines (2003)]] p.12. | ||
| - | |||
| - | Let $p > 0$, $\lr{X_n}_{n \in \nset}$ a sequence of random variables and $\lr{\alpha_n}_{n \in \nset}$ a sequence of nonzero real numbers. We write $X_n = O_{L^p}(\alpha_n)$ if $\lr{\alpha_n^{-1} X_n}_{n \in \nset}$ is bounded in $L^p$. | ||
| - | |||
| - | {{anchor:o_lemma:}} | ||
| - | ====== $O$-lemma ====== | ||
| - | |||
| - | Let $p > 0$ and $\lr{X_n}_{n \in \nset}$ a sequence of random variables such that $X_n = O_{L^p}(\alpha_n)$ with $\sum_{n=0}^{\infty} \alpha_n^p < \infty$. Then, | ||
| - | \begin{equation*} | ||
| - | \lr{X_n}_{n \in \nset} \overset{a.s}{\rightarrow} 0. | ||
| - | \end{equation*} | ||
| - | |||
| - | <hidden> | ||
| - | By assumption, there exists $C \in \rset_+^*$ such that for all $n \in \nset$, $\alpha_n^{-1} \norm{X_n}_{L^p} \leq C$. | ||
| - | |||
| - | Let $\epsilon > 0$. By Markov inequality, for all $n \in \nset$, | ||
| - | \begin{equation*} | ||
| - | \mathbb{P}\lr{\norm{X_n }_p \geq \epsilon} \leq \frac {\PE\lrb{\norm{X_n}_p^p}} {\epsilon^p} | ||
| - | = \frac {\norm{X_n}_{L_p}^p} {\epsilon^p} | ||
| - | \leq \frac {C^p} {\epsilon^p} \alpha_n^p, | ||
| - | \end{equation*} | ||
| - | hence | ||
| - | \begin{equation*} | ||
| - | \sum_{n=0}^{\infty} \mathbb{P}\lr{\norm{X_n }_p \geq \epsilon} \leq \frac {C^p} {\epsilon^p} \sum_{n=0}^{\infty} \alpha_n^p < \infty . | ||
| - | \end{equation*} | ||
| - | By Borel-Cantelli lemma we deduce that almost surely, $\norm{X_n }_p < \epsilon$ for sufficiently large $n$. | ||
| - | That being true for all $\epsilon > 0$, it is true for all $\epsilon = \frac 1 k$ with $k \in \nset^*$, and from a countable intersection of almost sure events $\lr{X_n}_{n \in \nset} \overset{a.s}{\rightarrow} 0$. | ||
| - | </hidden> | ||
| - | |||
| - | {{anchor:mz_slln:}} | ||
| - | ====== Triangular strong law of large numbers ====== | ||
| - | |||
| - | <WRAP center round box 80%> | ||
| - | Let $d \in \nset^*$ and $\lr{X_{n,i}}_{1 \leq i \leq m_n, n\in\nset}$ i.i.d. random variables of $\rset^d$ with $\lr{m_n}_{n \in \nset} \in {\nset^*}^{\nset}$. | ||
| - | Assume the existence of $p \geq 2$ such that $X_{1,1} \in L^p$ and $\sum_{i=1}^{\infty} m_n^{-p/2} < \infty$. Then, | ||
| - | \begin{equation*} | ||
| - | \frac 1 {m_n} \sum_{i=1}^{m_n} X_{n,i} \overset{a.s}{\rightarrow} \PE\lr{X_{1,1}}. | ||
| - | \end{equation*} | ||
| - | </WRAP> | ||
| - | |||
| - | <hidden> | ||
| - | The $\lr{X_{n,i}}_{1 \leq i \leq m_n, n\in\nset}$ being i.i.d. and in $L^p(\rset^d)$, the [[rayan:11_marcinkiewicz_zygmund#multi_mz|generalized Marcinkiewicz–Zygmund inequality]] yields | ||
| - | \begin{equation*} | ||
| - | \frac 1 {m_n} \sum_{i=1}^{m_n} X_{n,i} - \PE\lr{X_{1,1}} = O_{L^p}\lr{m_n^{-1/2}}. | ||
| - | \end{equation*} | ||
| - | As $\sum_{i=1}^{\infty} m_n^{-p/2} < \infty$ by assumption, the [[rayan:11_marcinkiewicz_zygmund#o_lemma|$O$-lemma]] concludes the proof. | ||
| - | </hidden> | ||
| - | |||
| - | **Remark:** The assumptions of the theorem hold as soon as $m_n = n$ for all $n \in \nset^*$ and $X_{1,1} \in L^p$ with $p>2$. | ||
| - | |||
| - | **Remark:** For $p=2$, look at Theorem 2.19 of Hall, Heyde {{ :rayan:hallheyde.pdf |Martingal Limit Theory and its application}}. | ||
| - | |||
| - | |||
| - | {{anchor:compact_slln:}} | ||
| - | ====== Compact strong law of large numbers ====== | ||
| - | |||
| - | Let $\Theta$ be a compact subset of $\rset^d$ with $d \in \nset^*$, $Z$ a measurable space, $\zeta$ a random variable taking its values on $Z$, and $L$ a measurable function defined on $\Theta \times Z$. | ||
| - | Define on $\Theta$ the function $\mathcal{L} \colon \theta \mapsto \mathbb{E}\lrb{L(\theta, \zeta)}$, and for all $\theta \in \Theta$ and $n \in \nset$, the Monte-Carlo average | ||
| - | \begin{equation*} | ||
| - | \hat{\mathcal{L}}^n(\theta) \eqdef \frac 1 {m_n} \sum_{i=1}^{m_n} L(\theta, \zeta_{n,i}), | ||
| - | \end{equation*} | ||
| - | where $\lr{m_n} \in {\nset^*}^{\nset}$ and $\lr{\zeta_{n,i}}_{1 \leq i \leq m_n}$ are i.i.d. random variables with $\zeta_{n,1} \sim \zeta$. | ||
| - | |||
| - | Assume that: | ||
| - | - $\mathcal{L}$ is continuous on $\Theta$, | ||
| - | - there exists a measurable function $\Gamma \colon Z \rightarrow \rset_+$ such that a.s., for all $\theta \in \Theta$, $|L(\theta, \zeta)| \leq \Gamma(\zeta) \in L^p$ with $p \geq 2$, | ||
| - | - $\sum_{i=1}^{\infty} m_n^{-p/2} < \infty$. | ||
| - | Then, | ||
| - | \begin{equation*} | ||
| - | \underset{\Theta}{\sup} \lrav{\mathcal{L} - \hat{\mathcal{L}}^n} \overset{a.s.}{\rightarrow} 0. | ||
| - | \end{equation*} | ||
| - | |||
| - | |||
| - | |||
| - | <hidden> | ||
| - | Let $\delta > 0$ and $\theta_0 \in \Theta$. Write | ||
| - | \begin{equation*} | ||
| - | \mathcal{L} - \hat{\mathcal{L}}^n = \mathcal{L} - \mathcal{L} \lr{\theta_0} + \mathcal{L} \lr{\theta_0} - \hat{\mathcal{L}}^n. | ||
| - | \end{equation*} | ||
| - | By continuity of $\mathcal{L}$ there exists $\epsilon_1 > 0$ such that $\underset{\Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_1}}{\sup} \lr{\mathcal{L} - \mathcal{L} \lr{\theta_0}} \leq \frac {\delta} 3$. For all $\epsilon > 0$, | ||
| - | \begin{align*} | ||
| - | \underset{\Theta \cap \mathbf{B}\lr{\theta_0, \epsilon}}{\sup} \lr{\mathcal{L} \lr{\theta_0} - \hat{\mathcal{L}}^n} | ||
| - | &= \PE\lrb{L\lr{\theta_0, \zeta}} - \underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon}}{\inf} \frac 1 {m_n} \sum_{i=1}^{m_n} L\lr{\theta, \zeta_{n,i}} \\ | ||
| - | &\leq \PE\lrb{L\lr{\theta_0, \zeta}} - \frac 1 {m_n} \sum_{i=1}^{m_n} \underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon}}{\inf} L\lr{\theta, \zeta_{n,i}} \\ | ||
| - | &= \PE\lrb{L\lr{\theta_0, \zeta}} - \PE\lrb{\underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon}}{\inf} L\lr{\theta, \zeta}} + \PE\lrb{\underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon}}{\inf} L\lr{\theta, \zeta}} - \frac 1 {m_n} \sum_{i=1}^{m_n} \underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon}}{\inf} L\lr{\theta, \zeta_{n,i}}. | ||
| - | \end{align*} | ||
| - | By the monotone convergence theorem, there exists $\epsilon_2 \in (0; \epsilon_1)$ such that | ||
| - | \begin{equation*} | ||
| - | \PE\lrb{L\lr{\theta_0, \zeta}} - \PE\lrb{\underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_2}}{\inf} L\lr{\theta, \zeta}} \leq \frac {\delta} 3. | ||
| - | \end{equation*} | ||
| - | We easily prove that a.s., $\lrav{\underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_2}}{\inf} L\lr{\theta, \zeta}} \leq \Gamma(\zeta) \in L^p$. Together with assumption 3. this allows us to apply the [[rayan:11_marcinkiewicz_zygmund#mz_slln|triangular strong law of large numbers]] to $\lr{\underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon}}{\inf} L\lr{\theta, \zeta_{n,i}}}_{1\leq i \leq m_n, n\in\nset}$, which provides a.s. the existence of $n_0 \in \nset$ such that for all $n \geq n_0$, | ||
| - | \begin{equation*} | ||
| - | \PE\lrb{\underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_2}}{\inf} L\lr{\theta, \zeta}} - \frac 1 {m_n} \sum_{i=1}^{m_n} \underset{\theta \in \Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_2}}{\inf} L\lr{\theta, \zeta_{n,i}} \leq \frac {\delta} 3. | ||
| - | \end{equation*} | ||
| - | Together with the definitions of $\epsilon_1$ and $\epsilon_2$, this yields the existence a.s. of $n_0 \in \nset$ such that for all $n_0 \geq n$, | ||
| - | \begin{equation*} | ||
| - | \underset{\Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_2}}{\sup} \lr{\mathcal{L} - \hat{\mathcal{L}}^n} \leq \underset{\Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_1}}{\sup} \lr{\mathcal{L} - \mathcal{L}(\theta_0)} + \underset{\Theta \cap \mathbf{B}\lr{\theta_0, \epsilon_2}}{\sup} \lr{\mathcal{L}(\theta_0) - \hat{\mathcal{L}}^n} \leq \delta. | ||
| - | \end{equation*} | ||
| - | By compacity of $\Theta \subset \underset{\theta \in \Theta}{\cup} \mathbf{B}\lr{\theta, \epsilon_2(\theta)}$, we can extract a finite subcover $\underset{1 \leq i \leq I}{\cup} \mathbf{B}\lr{\theta_i, \epsilon_2(\theta_i)}$ of $\Theta$ with $I \in \nset$. | ||
| - | By finite intersection of almost sure events, there exists a.s. $n_0 \eqdef \max\lr{n_0(\theta_1), \dots, n_0(\theta_I)}$ such that for all $n \geq n_0$, | ||
| - | \begin{equation*} | ||
| - | \underset{\Theta}{\sup} \lr{\mathcal{L} - \hat{\mathcal{L}}^n} = \underset{1 \leq i \leq I}{\max} \underset{\Theta \cap \mathbf{B}\lr{\theta_i, \epsilon_2(\theta_i)}}{\sup} \lr{\mathcal{L} - \hat{\mathcal{L}}^n} \leq \delta. | ||
| - | \end{equation*} | ||
| - | That being true for all $\delta = \frac 1 k$ with $k \in \nset^*$, by countable intersection of almost sure events, $\max\lr{0, \eqsp \underset{\Theta}{\sup} \lr{\mathcal{L} - \hat{\mathcal{L}}^n}} \overset{a.s.}{\rightarrow} 0$. The same reasoning with $L = - L$ provides $\underset{\Theta}{\sup} \lrav{\mathcal{L} - \hat{\mathcal{L}}^n} \overset{a.s.}{\rightarrow} 0$. | ||
| - | </hidden> | ||
| - | |||
| - | **Remark.** If $L$ is continuous with respect to the first variable $\theta$, by Lebesgue's dominated convergence theorem under assumption 2. the function $\mathcal{L}$ is continuous (i.e. assumption 1. is verified). | ||
| - | |||
world/marcinkiewicz.1642239717.txt.gz · Last modified: 2022/03/16 01:36 (external edit)
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