Let $X_1, X_2, \dots$ be a sequence of scalar random variables converging in probability to another random variable $X$. \begin{enumerate} \item Suppose that there is a random variable $Y$ which is independent of $X_i$ for each individual $i$. Show that $Y$ is also independent of $X$. \item Suppose that the $X_1,X_2,\dots$ are jointly independent. Show that $X$ is almost surely constant (i.e. there is a deterministic scalar $c$ such that $X=c$ almost surely). \end{enumerate}

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Probability Theory

Exercise 3.3 (Convergence in probability of independent random variables)

Let $X_{1}, X_{2}, \dots$ be a sequence of scalar random variables converging in probability to another random variable $X$ .

1.: Suppose that there is a random variable $Y$ which is independent of $X_{i}$ for each individual $i$ . Show that $Y$ is also independent of $X$ .
2.: Suppose that the $X_{1}, X_{2}, \dots$ are jointly independent. Show that $X$ is almost surely constant (i.e. there is a deterministic scalar $c$ such that $X = c$ almost surely).