This Wikipedia page says that the sequence of random variables $X_n$ that assume $1$ with probability $1/n$ and assume $0$ with probability $1 – 1/n$ converges to $0$ in probability but not almost surely. I can't understand why it does not converge to $0$ almost surely.
$$ P(\lim_{n\to\infty}X_n = 0) = \lim_{n\to\infty}(1-1/n)=1.$$
What is wrong with the above equations? To be more concrete, with uniform probability on $[0,1]$, define $X_n(x)=1$ when $0\leqslant x \leqslant 1/n$ and $X_n(x)=0$ when $1/n < x \leqslant 1$. Then $(\lim_{n\to\infty}X_n)(0)=1$ and $(\lim_{n\to\infty}X_n)(x)=0$ for $0<x\leqslant 1$. So, $P\{x:(\lim_{n\to\infty}X_n)(x)=0\} = 1$ and $X_n$ converges to 0 almost surely.
[Math] An example of a sequence of random variables in Wikipedia, that converges in probability but not almost surely
convergence-divergenceprobability
Best Answer
You left out the very important word "independent". Your random variables are not independent.