Convergence in probability

A sequence $(𝑋 𝑖) \infty 𝑖 = 1$ of real random variables is said to converge in probability to $𝑋$ iff either of the equivalent conditions are met for all $𝜖 > 0$ : #m/def/prob

l i m 𝑖 \to \infty ℙ (| 𝑋 𝑖 - 𝑋 | > 𝜖) = 0 l i m 𝑖 \to \infty ℙ (| 𝑋 𝑖 - 𝑋 | < 𝜖) = 1

This is precisely convergence under the Ky Fan metric.

#state/tidy | #lang/en | #SemBr