Exercise 3.2 (Modes of convergence on probability spaces)

Let ${(X_{n})}_{n \in N}$ be a sequence of scalar random variables, and let $X$ be another scalar random variable.

1.: If $X_{n} \to X$ almost surely, show that $X_{n} \to X$ in probability. Give a counterexample to show that the converse does not necessarily hold.
2.: Suppose that $\sum_{n} P (| X_{n} - X | > 𝜀) < \infty$ for all $𝜀 > 0$ . Show that $X_{n} \to X$ almost surely. Give a counterexample to show that the converse does not necessarily hold.
3.: If $X_{n} \to X$ in probability, show that there is a subsequence $X_{n_{j}}$ of the $X_{n}$ such that $X_{n_{j}} \to X$ almost surely.
4.: If $X_{n}, X$ are absolutely integrable and $E | X_{n} - X | \to 0$ as $n \to \infty$ , show that $X_{n} \to X$ in probability. Give a counterexample to show that the converse does not necessarily hold.
5.: (Urysohn subsequence principle) Suppose that every subsequence $X_{n_{j}}$ of $X_{n}$ has a further subsequence $X_{n_{j_{k}}}$ that converges to $X$ in probability. Show that $X_{n}$ also converges to $X$ in probability.
6.: Does the Urysohn subsequence principle still hold if "in probability" is replaced with "almost surely" throughout?
7.: If $X_{n}$ converges in probability to $X$ , and $F : R \to R$ or $F : C \to C$ is continuous, show that $F (X_{n})$ converges in probability to $F (X)$ . More generally, if for each $i = 1, \dots, k$ , $X_{n}^{(i)}$ is a sequence of scalar random variables that converge in probability to $X^{(i)}$ , and $F : R^{k} \to R$ or $F : C^{k} \to C$ is continuous, show that $F (X_{n}^{(1)}, \dots, X_{n}^{(k)})$ converges in probability to $F (X^{(1)}, \dots, X^{(k)})$ . (Thus, for instance, if $X_{n}$ and $Y_{n}$ converge in probability to $X$ and $Y$ respectively, then $X_{n} + Y_{n}$ and $X_{n} Y_{n}$ converge in probability to $X + Y$ and $XY$ respectively.
8.: (Fatou’s lemma for convergence in probability) If $X_{n}$ are non-negative and converge in probability to $X$ , show that $EX \leq {liminf}_{n \to \infty} E X_{n}$ .
9.: (Dominated convergence in probability) If $X_{n}$ converge in probability to $X$ , and one almost surely has $| X_{n} | \leq Y$ for all $n$ and some absolutely integrable $Y$ , show that $E X_{n}$ converges to $EX$ .

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Let $(X_n)_{n\in\mathbf{N}}$ be a sequence of scalar random variables, and let $X$ be another scalar random variable.

\begin{enumerate}
\item If $X_n \rightarrow X$ almost surely, show that $X_n \rightarrow X$ in probability. Give a counterexample to show that the converse does not necessarily hold.
\item Suppose that $\sum_n {\bf P}( |X_n-X| > \varepsilon ) < \infty$ for all $\varepsilon > 0$. Show that $X_n \rightarrow X$ almost surely. Give a counterexample to show that the converse does not necessarily hold.
\item If $X_n \rightarrow X$ in probability, show that there is a subsequence $X_{n_j}$ of the $X_n$ such that $X_{n_j} \rightarrow X$ almost surely.
\item If $X_n,X$ are absolutely integrable and ${\bf E} |X_n-X| \rightarrow 0$ as $n \rightarrow \infty$, show that $X_n \rightarrow X$ in probability. Give a counterexample to show that the converse does not necessarily hold.
\item (Urysohn subsequence principle) Suppose that every subsequence $X_{n_j}$ of $X_n$ has a further subsequence $X_{n_{j_k}}$ that converges to $X$ in probability. Show that $X_n$ also converges to $X$ in probability.
\item Does the Urysohn subsequence principle still hold if "in probability" is replaced with "almost surely" throughout?
\item If $X_n$ converges in probability to $X$, and $F: {\bf R} \rightarrow {\bf R}$ or $F: {\bf C} \rightarrow {\bf C}$ is continuous, show that $F(X_n)$ converges in probability to $F(X)$. More generally, if for each $i=1,\dots,k$, $X^{(i)}_n$ is a sequence of scalar random variables that converge in probability to $X^{(i)}$, and $F: {\bf R}^k \rightarrow {\bf R}$ or $F: {\bf C}^k \rightarrow {\bf C}$ is continuous, show that $F(X^{(1)}_n,\dots,X^{(k)}_n)$ converges in probability to $F(X^{(1)},\dots,X^{(k)})$. (Thus, for instance, if $X_n$ and $Y_n$ converge in probability to $X$ and $Y$ respectively, then $X_n + Y_n$ and $X_n Y_n$ converge in probability to $X+Y$ and $XY$ respectively.
\item (Fatou’s lemma for convergence in probability) If $X_n$ are non-negative and converge in probability to $X$, show that ${\bf E} X \leq \liminf_{n \rightarrow \infty} {\bf E} X_n$.
\item (Dominated convergence in probability) If $X_n$ converge in probability to $X$, and one almost surely has $|X_n| \leq Y$ for all $n$ and some absolutely integrable $Y$, show that ${\bf E} X_n$ converges to ${\bf E} X$.
\end{enumerate}

Exercise 3.2 (Modes of convergence on probability spaces)

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