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Fix typo
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Zentrik committed Jan 15, 2024
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2 changes: 1 addition & 1 deletion ProbAndMeasure/07_ergodic_theory.tex
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Expand Up @@ -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*}
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