So , after posting a large number of questions on this Math.SE and Stats.SE , I have finally compiled a set of alterantive definitions of almost sure convergence. They are

$$X_n\xrightarrow{a.s.} X\\\Longleftrightarrow \Bbb{P}(\omega \ : \ X_n(\omega)\to X(\omega)\text{ as }n\to \infty)=1\\\Longleftrightarrow \lim_{n\to\infty}\Bbb{P}(\omega \ :\ \sup_{k>n}|X_k(\omega)-X(\omega)|>\epsilon)=0\ \ \forall \epsilon>0\\\Longleftrightarrow \Bbb{P}(\cap_{n=1}^{\infty} \cup_{k=n}^{\infty} \{ |X_k - X| > \epsilon \}) = 0\ \ \forall \epsilon>0$$

Are all these correct?

Specifically what happens when $X_n(\omega)\to X$ is replaced be $X_n(\omega)= X$ in the second definition (OR am I makinga fundamental mistake)?

  • $\begingroup$ All the equivalence seems fine to me. Also, when you replace $X_n \to X$ by $X_n = X$, what you get is a sequence which is eventually constant $\Bbb{P}$-a.s. $\endgroup$ – Sangchul Lee Oct 12 '16 at 16:50
  • 1
    $\begingroup$ In the second line you want to write $X(\omega)$, not just plain $X$. $\endgroup$ – user940 Oct 12 '16 at 16:52
  • $\begingroup$ @ByronSchmuland Thanks for that! $\endgroup$ – Qwerty Oct 12 '16 at 17:13

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