PROBABILITIES Course

Convergence in probability

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In French, it's Convergence en probabilité. The sequence $X_n$ converges in probability if

@ \mathbb{P}(\lim_{n \to+ \infty} |X_n-x| \gt \epsilon) = 0 @

It's that's the case, then $x$ is a constant $c$ and the notation is

@ X_n \xrightarrow{p} c @

It's almost the same as "almost surely", $X^n$ values when $n \to+ \infty$ are around c, like $[c-\epsilon,\ c+\epsilon]$.

  • $\lim_{n \to+ \infty} \mathbb{E}(X_n) = c$
  • $\lim_{n \to+ \infty} \mathbb{V}(X_n) = 0$