Ky Fan metric

The Ky Fan metric is a metric $𝑑 : R V \times R V \to ℝ$ on the space of real random variables defined by prob

𝑑 (𝑋, 𝑌) = i n f {𝜖 > 0 : ℙ (| 𝑋 - 𝑌 | > 𝜖) \leq 𝜖} = 𝔼 [m i n (| 𝑋 - 𝑌 |, 1)]

The convergence given by the Ky Fan metric is precisely Convergence in probability.

Proof

proof