Convergence almost surely

Convergence almost surely is convergence almost everywhere on real random variables. prob A sequence $(𝑋 𝑖) \infty 𝑖 = 1$ of real random variables converges almost surely to a real random variable $𝑋$ iff

ℙ (l i m 𝑖 \to \infty 𝑋 𝑖 = 𝑋) = 1

which is equivalent to either of the following conditions holding for all $𝜖 > 0$

ℙ (l i m 𝑖 \to \infty | 𝑋 𝑛 - 𝑋 | < 𝜖) = 1 l i m 𝑖 \to \infty ℙ (s u p 𝑗 \geq 𝑖 | 𝑋 𝑖 - 𝑋 | > 𝜖) = 0 \infty \sum 𝑖 = 1 ℙ (| 𝑋 𝑖 - 𝑋 | > 𝜖) < \infty

Proof

proof

Properties

See Relation between convergence concepts

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Convergence almost surely

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