How can I, formally, explain why this sequence of random variable below converges in probability but does not converge in distribution?
Let $X_{1},X_{2},X_{3},...$ random variables i.i.d such that: $P(X_{n}=1)=1/n$, $P(X_{n}=0)=1-1/n$.
Show that $X_{n} \xrightarrow{P} 0$, but $P(X_{n}\rightarrow 0)=0$.
I am facing difficulties to express myself formally to explain the first part. I know and I understand that converges in probability to zero. The second part I am also facing difficulties.
Many thanks, guys
My hint for the second part is: For $n \rightarrow \infty$ the sequence $X_{n}$ will takes values 1 and 0. Thats why $X_{n}\rightarrow 0$ does not happen and $P(X_{n}\rightarrow 0)=0$. Am I right?
First of all you cannot have i.i.d random variables with the stated properties since they don't have the same distributiom. Assuming that you drop the 'ídentcally distributed' part you can use Borel Cantelli Lemma. Since $\sum_n P(X_n=0)=\sum_n P(X_n=1)=\infty$ Borel Cantelli Lemma tells you that for almost all $\omega$ the sequence $\{X_n(\omega)\}$ has infinitely many $1$'s and infinitely many $0$'s.