\documentclass{article}
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\title{MATH 418/544, Assignment 3}
\date{Due 2023-11-18, 23:59} %replace by your name and student id

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\noindent \thetitle \hfill \thedate

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\begin{problem}
  The Total variation distance $d_{TV}(X,Y)$ is defined as $\inf \P(X'\neq  Y')$ over all couplings of $X,Y$.
  How does convergence in the total variation distance relate to the other notions of convergence discussed in class?
  (Including distribution, probability, a.s., $L^1$.)
\end{problem}

%your solution here

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\begin{problem}
  Let $X_n$ be i.i.d. uniform in $[0,1]$ random variables.
  Let $Y_n = 1$ if $X_n$ is a local maximum: $X_n = \max\{X_n,X_{n-1},X_{n+1}$.
  Let $S_n = \sum_{i=1}^n Y_i$.
  Find constants $a,b$ so that the sum of $Y_n$ has a CLT:
  \[ \frac{S_n-an}{b\sqrt{n}} \to N(0,1), \]
  and prove this convergence.
  (Hint: The $Y_n$ are not independent, but only depndent for nearby $n$'s.
  Approximate $S_n$ by a sum of independent R.V.s.)
\end{problem}

%your solution here

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\begin{problem}
  Prove that $\phi_X$ is periodic if and only if $X$ takes values in $a\Z$  (multiples of $a$) for some $a$.
\end{problem}

%your solution here

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\begin{problem}
  Write a proof of the following statement from class: If $X\geq 0$ is a R.V. then $\E X = \int_0^\infty \P(X>t) dt$.
  Moreover
  $\E X^a = \int_0^\infty t^{a-1} \P(X>t) dt$.
\end{problem}

%your solution here

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\begin{problem}
  Prove that for any random variable with finite mean, $t \P(X>t)\to 0$ as $t\to\infty$. Moreover, if $\E|X|^a<\infty$ then $t^a \P(X>t)\to 0$.
\end{problem}

%your solution here

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\begin{problem}
  If $X_n\xrightarrow[]{prob} X$ are independent, show that $X$ must be a  constant random variable.
\end{problem}

%your solution here

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\begin{problem}
  Show that convergence in probability is equivalent to convergence in the
  metric
  \[
  d(X,Y) = \E (|X-Y|\wedge 1).
  \]
\end{problem}

%your solution here

\end{document}
