Let $X_n$ a sequence such that $P(X_n=1)=P(X_n=n)=\frac{1}{n^3+1}$ and $P(X_n=2)=\frac{n^3-1}{n^3+1}$
a) Check if $X_n \xrightarrow{p} X$, identifying the random variable X.
b) Check if $X_n \xrightarrow{L^2} X$, identifying the random variable X.
c) Check if the Law of larger numbers applies in this case.
How do I find where a random variable is converging to ?
I tried to solve the first item using the definition of convergence of probability and markov...
I can see that $X_n$ does not converge when $X = 0$
$P(|X_n-X|\ge \epsilon) = P(X_n\ge \epsilon) \le E(X_n)/\epsilon$
And $E(X_n)/\epsilon$ goes to 2 when $n \rightarrow \infty$
I found, by guessing, that $X_n \xrightarrow{p} X$ when $X=n$, but is that correct ? And $n$ surely is not the only value, right ? How can i make it formal ?
Thank you