For 1. According to the definition of convergence in probability, $X_n \overset{p} \to X$ iff $$P(|X_n-X|>\epsilon) \to 0$$ as $n\to \infty$. But since $X_n$ is equal either to $0$ or to $1$ you have for $X\equiv0$ and for all $\epsilon>0$ that: $$P(|X_n-0|>\epsilon)=P(X_n=1)=p_n$$ which obviousl...
Thanks for the answer! I am trying to understand your proof of 1). I dont understand the sentence "But since $X_n$ is equal to one or zero..." and the steps after this. Can you explain it to me in more detail please?
Probably then you have just copied the exercise from somewhere then, without having read it! It is in the text that you posted, first line top right... $P(X_n=0)$ and $P(X_n=1)$. No other values.
But what about the $p_n$?I thought this notation does mean: $P(X_n=0)=P(X_n^{-1}(0))=p_n$ I thought thats just a property on a certain set and we don't know anything about the other values. I am reading it like: "The measure of the fiber under all $X_n$ of zero is exactly $p_n$"
It does not fit that you are doing probability theory with convergence, Borel-Cantelli etc, and you do not know basic notation. What you are asking is elementary notation. I am sorry.
Discussion between Stefanos and Epsil…
Discussion between Stefanos and Epsil…