Almost sure convergence via pairwise independence

If {A_1,A_2,...} are pairwise independent and {\sum_{n=1}^{\infty}P(A_n)=\infty} then as {n \rightarrow \infty}

\displaystyle \boxed{ \frac{\sum_{m=1}^{n}\mathbb{I}_{A_m}}{\sum_{m=1}^{n}P(A_m)} \xrightarrow{a.s.} 1 }


Let {X_m = \mathbb{I}_{A_m}} and {S_n = X_1+...+Xn}. Since {A_m} are pairwise independent, the {X_m} are uncorrelated and thus

\displaystyle var(S_n) = var(X_1) + ... + var(X_n)

Since {X_m \in \{0,1 \}}

\displaystyle var(X_m) \leq \mathbb{E}[X_m^2] = \mathbb{E}[X_m] \Rightarrow var(S_n) \leq \mathbb{E} [S_n]

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Expectation: Useful properties and inequalities

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If {X \geq 0} is a random variable on {(\Omega, \mathcal{F}, P)}. The expected value of {X } is defined as

\displaystyle \mathbb{E}(X) \equiv \int_{\Omega} X dP = \int_{\Omega} X(\omega) P (d \omega)


  • Jensen’s inequality. If {\varphi} is convex and {E|X|, E|\varphi(X)| < \infty}

\displaystyle \mathbb{E} (\varphi(X)) \geq \varphi(\mathbb{E}X)

  • Holder’s inequality. If {p,q \in [1, \infty]} with {1/p + 1/q =1} then

\displaystyle \mathbb{E}|XY| \leq \|X\|_p \|Y\|_q

  • Cauchy-Schwarz Inequality: For {p=q=2}

\displaystyle \mathbb{E}|XY| \leq \left( \mathbb{E}(X^2) \mathbb{E}(Y^2) \right)^{1/2}

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Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

Definition 1. A {\sigma}-field {\mathcal{F}} is a non-empty collection of subsets of the sample space {\Omega} closed under the formation of complements and countable unions (or equivalently of countable intesections – note {\bigcap_{i} A_i = (\bigcup_i A_i^c)^c}). Hence {\mathcal{F}} is a {\sigma}-field if

1. {A^c \in \mathcal{F}} whenever {A \in \mathcal{F}}
2. {\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}} whenever {A_i \in \mathcal{F}, n \geq 1}

Definition 2. Set functions and measures. Let {S} be a set and {\Sigma_0} be an algebra on {S}, and let {\mu_0} be a non-negative set function

\displaystyle \mu_0: \Sigma_0 \rightarrow [0, \infty]

  • {\mu_0} is additive if {\mu_0 (\varnothing) =0} and, for {F,G \in \Sigma_0},

    \displaystyle F \cap G = \varnothing \qquad \Rightarrow \qquad \mu_0(F \cup G ) = \mu_0(F) + \mu_0(G)

  • The map {\mu_0} is called countably additive (or {\sigma}-additive) if {\mu (\varnothing)=0} and whenever {(F_n: n \in \mathbb{N})} is a sequence of disjoint sets in {\Sigma_0} with union {F = \cup F_n} in {\Sigma_0}, then

    \displaystyle \mu_0 (F) = \sum_{n}\mu_0 (F_n)

  • Let {(S, \Sigma)} be a measurable space, so that {\Sigma} is a {\sigma}-algebra on {S}.
  • A map \displaystyle \mu: \Sigma \rightarrow [0,\infty]. is called a measure on {(S, \Sigma)} if {\mu} is countable additive. The triple {(S, \Sigma, \mu)} is called a measure space.
  • The measure {\mu} is called finite if

    \displaystyle \mu(S) < \infty,

    and {\sigma}finite if

    {\exists \{S_n\} \in \Sigma}, ({n \in \mathbb{N}})\displaystyle \mu(S_n)< \infty, \forall n \in \mathbb{N} \text{ and } \cup S_n = S.

  • Measure {\mu} is called a probability measure if \displaystyle \mu(S) = 1, and {(S, \Sigma, \mu)} is then called a probability triple.
  • An element {F} of {\Sigma} is called {\mu}-null if {\mu(F)=0}.
  • A statement {\mathcal{S}} about points {s} of {\mathcal{S}} is said to hold almost everywhere (a.s.) if

    \displaystyle F \equiv \{ s: \mathcal{S}(s) \text{ is false} \} \in \Sigma \text{ and } \mu(F)=0.

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