Problem set 2: Convergence of random variables
1 Problems for seminar
Let \(\Omega\) be countable. Prove that almost sure convergence is equivalent to convergence in probability.
Let \(\xi_1,\xi_2,\ldots\) be independent random variables such that
\[ \xi_n \sim \operatorname{Bern}(p_n). \]
Find all necessary and sufficient conditions on the sequence \(p_1,p_2,\ldots\) such that:
- \(\xi_n \xrightarrow{\mathsf{P}} 0\);
- \(\xi_n \xrightarrow{L^p} 0\).
Let
\[ \xi_n \xrightarrow{d} C, \]
where \(C\) is a constant. Prove that
\[ \xi_n \xrightarrow{\mathsf{P}} C. \]
Let \((\xi_n)_{n\geq 1}\) be a sequence of random variables and define
\[ S_n=\xi_1+\cdots+\xi_n. \]
Prove that if
\[ \xi_n \xrightarrow{\mathrm{a.s.}} \xi \qquad \text{as } n\to\infty, \]
then
\[ \frac{S_n}{n}\xrightarrow{\mathrm{a.s.}}\xi. \]
Suppose that
\[ (\xi_n-\xi)^2\xrightarrow{\mathsf{P}}0. \]
Prove that
\[ \xi_n^2\xrightarrow{\mathsf{P}}\xi^2. \]
Suppose that \(X_n\) and \(X\) take integer values. Prove that convergence in distribution
\[ X_n\xrightarrow{d}X \]
is equivalent to
\[ \mathsf{P}(X_n=k)\longrightarrow\mathsf{P}(X=k) \]
for every \(k\in\mathbb{Z}_+\).
Let \(\xi_1,\xi_2,\ldots\) be independent and identically distributed random variables with
\[ \xi_i\sim\operatorname{Exp}(1). \]
Prove that
\[ \mathsf{P}\left( \limsup_{n\to\infty}\frac{\xi_n}{\ln n}=1 \right)=1. \]
Let \(\xi\) and \(\eta\) be random variables. Define
\[ \rho(\xi,\eta) = \mathsf{E}\left[ \frac{|\xi-\eta|}{1+|\xi-\eta|} \right]. \]
Prove that
\[ \xi_n\xrightarrow{\mathsf{P}}\xi \]
if and only if
\[ \rho(\xi_n,\xi)\longrightarrow0 \qquad\text{as }n\to\infty. \]
2 Extra problems
Let \(\xi_1,\xi_2,\ldots\) be independent random variables such that
\[ \xi_n \sim \operatorname{Bern}(p_n). \]
Find all necessary and sufficient conditions on the sequence \(p_1,p_2,\ldots\) such that:
- \(\xi_n \xrightarrow{\mathrm{a.s.}} 0\);
- \(\xi_n \xrightarrow{d} 0\).
Let \((\xi_n)_{n\geq 1}\) be a sequence of random variables and define
\[ S_n=\xi_1+\cdots+\xi_n. \]
Prove that
\[ \xi_n\xrightarrow{\mathsf{P}}\xi \]
does not necessarily imply
\[ \frac{S_n}{n}\xrightarrow{\mathsf{P}}\xi. \]
Let \(\xi_1,\xi_2,\ldots\) be independent and identically distributed random variables with
\[ \xi_i\sim\mathcal{N}(0,1). \]
Prove that
\[ \mathsf{P}\left( \limsup_{n\to\infty} \frac{\xi_n}{\sqrt{2\ln n}} =1 \right)=1. \]