A version of the martingale convergence theorem states that

If $\{X_n\ |\ n\in\mathbb{Z}_{\geq0}\}$ is a martingale and $\sup_n\mathbb{E}|X_n|\leq c<\infty$ for some constant $c$, then $X_\infty=\lim_{n\to\infty}X_n$ exists almost surely.

Is this the equivalent to saying that $X_n\xrightarrow{\text{a.s.}}X_\infty$ as $n\to\infty$? If not, what's the difference?

  • 2
    $\begingroup$ No difference. $ $ $\endgroup$ – Did Oct 3 '17 at 5:48

The two are equivalent, and the only difference is in the accent. The statement $X_n \xrightarrow{\text{a.s.}} X_{\infty}$ presupposes the existence of $X_{\infty}$, while "then $X_{\infty} = \lim_{n\to\infty} X_n$ exists almost surely" defines $X_{\infty}$ from the $X_n$.

If we painfully avoid all short-cuts, "$X_n \xrightarrow{\text{a.s.}} X_{\infty}$" is an abbreviation of "There is a random variable $X_{\infty}$ defined on the same space $\Omega$ as the $X_n$, and for almost all $\omega\in \Omega$ we have $X_n(\omega) \to X_{\infty}(\Omega)$". The other one is an abbreviation of "for almost all $\omega\in \Omega$ the limit of the sequence $\bigl(X_n(\omega)\bigr)_{n\in\mathbb{N}}$ exists, and we define the random variable $X_{\infty}$ (almost everywhere) as this limit".

Indeed equivalent. Like saying something is round and green, or that it's green and round.


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