24 lines
838 B
TeX
24 lines
838 B
TeX
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\begin{theorem}[Chebyshev's inequality] % TODO Proof
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Let $X$ be a r.v.~with $\Var(x) < \infty$.
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Then $\forall \epsilon > 0 : \bP \left[ \left| X - \bE[X] \right| > \epsilon\right] \le \frac{\Var(x)}{\epsilon^2}$.
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\end{theorem}
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We used Chebyshev's inequality. Linearity of $\bE$, $\Var(cX) = c^2\Var(X)$ and $\Var(X_1 +\ldots + X_n) = \Var(X_1) + \ldots + \Var(X_n)$ for independent $X_i$.
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How do we prove that something happens almost surely?
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\begin{lemma}[Borel-Cantelli]
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If we have a sequence of events $(A_n)_{n \ge 1}$
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such that $\sum_{n \ge 1} \bP(A_n) < \infty$,
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then $\bP[ A_n \text{for infinitely many $n$}] = 0$
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(more precisely: $\bP[\limsup_{n \to \infty} A_n] = 0$).
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The converse also holds for independent events $A_n$.
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\end{lemma}
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Modes of covergence: $L^p$, in probability, a.s.
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