2
$\begingroup$

Suppose I have a discrete probability distribution $P$ on $n$ points ($n$ fixed), and I approximate it by sampling it $m$ times to get an estimate $P_m$ which just estimates the probability of getting point $i$ with the number of times we saw $i$ divided by $m$.

How fast can I guarantee with high probability $P_m \rightarrow P$ in $L^1$ and/or $L^\infty$? That is, if I want with high probability $\|P_m - P\| < \epsilon$ what $m$ do I need as a function of $\epsilon$?

  • 2
    One approach: Chebyshev inequality leading to the weak law of large numbers (WLLN). – 2017-02-10
  • 0
    Thanks! This is enough for my purposes, I'd forgotten that weak law has quantitative bounds like that. – 2017-02-14

0 Answers 0