Let $\{X_n\}$ be a sequence of independent random variables converging almost surely to a random variable $X$. Then how to show that $X$ is almost surely a constant ?

I think I somehow have to apply the Borel-Cantelli lemma for independent events, but I don't know how.

Please help.


Suppose that $X_n \to X$ almost surely. Thus if $\Omega_0 = \{X_n \to X\}$ then $\mathbb P(\Omega_0) = 1$. Then for any fixed $c \in \mathbb R$ $$ \Omega_0 \cap \{X < c\} = \Omega_0 \cap\{X_n < c\ \text{infinitely often}\}$$ Then apply Borel-Cantelli to justify that $\{X_n < c\ \text{infinitely often}\}$ happens with probability either 0 or 1, thus $$\mathbb P(X < c) = \mathbb P(\Omega_0 \cap \{X < c\}) = \mathbb P(\Omega _0\cap \{X_n < c\ \text{i.o.}\}) = \mathbb P(\text{$X_n < c$ i.o.})$$ and will equal 0 or 1. Use this to deduce that $X$ is almost surely constant.

  • $\begingroup$ the exercise is to a chapter where only Borel-Cantelli lemmas, Fatou's lemma, MCT, DCT are given ... no Kolmogorov 0-1 law ... $\endgroup$ – user521337 Nov 6 '18 at 0:49
  • $\begingroup$ You could try proving the 0-1 law yourself, but the proof is unrelated to the Borel-Cantelli lemmas, Fatou's lemma or any of the convergence theorems. $\endgroup$ – bitesizebo Nov 6 '18 at 0:53
  • $\begingroup$ I am using the book; Knowing the Odds: An Introduction to Probability , by John B. Walsh; 2012 edition; page 129, Problem 4.23 $\endgroup$ – user521337 Nov 6 '18 at 0:57
  • $\begingroup$ Nevermind, I've figured it out $\endgroup$ – bitesizebo Nov 6 '18 at 1:00
  • $\begingroup$ I don't think your equality is correct ... could you elaborate a bit more please ? $\endgroup$ – user521337 Nov 6 '18 at 1:08

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