5

According to Billingsley, let $P$ and $Q$ be two probability measures. Then the Prokhorov metric $\pi (P,Q)$ is the infimum of those positive $ \epsilon $ for which the two inequalities $PA \le QA^{\epsilon}+ \epsilon$ and $QA \le PA^{\epsilon}+ \epsilon$ for all Borel sets $A$.

Then theorem 6.8 says that if the set on which probability measures is separable and complete, weak convergence is equivalent to $ \pi $ convergence.

Then,

Question: what is an intuitive explanation that the Prokhorov metric metrizes weak convergence? How can I explain it using an example?

user729424
  • 5,061
webster
  • 389
  • 2
  • 7
  • https://math.stackexchange.com/questions/133774/what-is-the-motivation-of-levy-prokhorov-metric – Alex R. Apr 14 '20 at 20:03

0 Answers0