From 4c4a8ad7ed25b342e63ab0f0eaca977f49bd3710 Mon Sep 17 00:00:00 2001 From: Zentrik Date: Mon, 15 Jan 2024 16:56:24 +0000 Subject: [PATCH] Fix typo --- ProbAndMeasure/07_ergodic_theory.tex | 2 +- ProbAndMeasure/probmeasure.pdf | Bin 415785 -> 415771 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/ProbAndMeasure/07_ergodic_theory.tex b/ProbAndMeasure/07_ergodic_theory.tex index f43f6f9..b30a04d 100644 --- a/ProbAndMeasure/07_ergodic_theory.tex +++ b/ProbAndMeasure/07_ergodic_theory.tex @@ -33,7 +33,7 @@ \subsection{Laws of Large Numbers} So we can assume $\mu = 0$. For distinct indices $i, j, k, \ell$, by independence and the Cauchy--Schwarz inequality, we have \begin{align*} - 0 = \expect{X_i X_j X_k X_\ell} = \expect{X_i^2 X_j X_j} = \expect{X_i^3 X_j};\quad \expect{X_i^2 X_j^2} \leq \sqrt{\expect{X_i^4}}\sqrt{\expect{X_j^4}} \leq M + 0 = \expect{X_i X_j X_k X_\ell} = \expect{X_i^2 X_j X_k} = \expect{X_i^3 X_j};\quad \expect{X_i^2 X_j^2} \leq \sqrt{\expect{X_i^4}}\sqrt{\expect{X_j^4}} \leq M \end{align*} So we can compute \begin{align*} diff --git a/ProbAndMeasure/probmeasure.pdf b/ProbAndMeasure/probmeasure.pdf index 160759938e51119a24ff35348803de44a10a57df..471f5dac720d4b3d2a3e7394fab6a735e5c91d49 100644 GIT binary patch delta 66390 zcmV)3K+C_Wuo;`M8GwWVgaWh!xW|7sGd?~Fb98cLVQmU{+T~kYj~vAne$TIH!^>J| zDplvyRV@1ui6BKH3c-qq011}O5}VkP{gS_*?^JblSNBxU^w`T{6CrlSHK%Upeyid2 zU%ad>w3?X@QJlh)qeeE;k(!{^VYrH?0Dtt{4GOxA2LS>y9oYiDi_tT9OU zT3f>BEe3qRh))2)C{qT@P%Hij8R!m&>^6nHqmf|mgHFe 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