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\title{MATH 418/544, Assignment 2}
\date{Due 2023-11-04, 23:59} %replace by your name and student id

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\begin{problem}
  Consider percolation on a graph with vertex set $\N$ with $m_n$ parallel edges between $n$ and $n+1$.
  \begin{enumerate}
    \item Determine a necessary and sufficient condition on the sequence $(m_n)$ for which there is an infinite connected cluster.
    \item Use this to find a sequence so that $p_c=1/2$ and at $1/2$ there is a.s. an infinite cluster.
  \end{enumerate}
\end{problem}

%your solution here

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\begin{problem}
  Let $X_1,X_2,\dots$ be i.i.d.\ random variables with standard normal
  distribution, and $M_n = \max(X_1,\dots,X_n)$. 
  \begin{enumerate}
  \item Find $\limsup \frac{X_n}{\sqrt{\log n}}$.
  \item Show that  $M_n/\sqrt{\log n}$ converges a.s.\ to the same value.
    (Hint: You need some estimates on the tail of the normal distribution.)
  \end{enumerate}
\end{problem}


%your solution here

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\begin{problem}
  Show 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}
  Let $X_i$ be i.i.d. random variables, and $S_n=\sum_{i\leq n} X_i$.
  \begin{enumerate}
  \item Show that if $\E X_i = +\infty$ then $\frac1n S_n \to \infty$ a.s.
  \item Show that if $\E X_i$ is not defined (positive and negative parts both infinite) then a.s. $\limsup \frac1n S_n = \infty$ and $\liminf \frac1n S_n = -\infty$.
\end{problem}

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\begin{defn}
  A random variable has density $f$ if it's distribution function is $F(t) = \int_{-\infty}^t f(x) dx$.
  (In that case, $f(x) = F'(x)$.)
  A random variable with a density is called continuous.
\end{defn}

\begin{problem}
  If $X$ is a continuous random variable with distribution
  function $F$ and density $f=F'$, show (from the definition of
  expectation in class) that $\E X = \int_{-\infty} x f(x) dx$.
\end{problem}

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% your solution here

\begin{defn}
  A pair of random variables $X,Y$ has join density function $f:\R^2\to\R$ if
  $\P((X,Y)\in A) = \iint_A f(x,y)\,dx\,dy$.
\end{defn}

\begin{problem}
  Suppose continuous random variables $X_1,X_2$ have densities $f_1,f_2$ respectively.
  Prove from the definition of independence that they are independent if and only if they have a joint density of the form $f(x,y) = f_1(x)f_2(y)$.
\end{problem}

% your solution here

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\begin{problem}
  Let $X,Y$ are two independent random variables.
  For an angle $\theta$, the rotation by theta is the pair of random variables
  \[
    X' = X \cos \theta + Y \sin\theta,  \qquad 
    Y' = - X \sin \theta + Y \cos\theta.
  \]
  Find a joint distribution of independent variables $X,Y$, with $\P(X=0)<1$ so that for any $\theta$ the rotation $(X',Y')$ has the same (joint) distribution as $(X,Y)$. \\
  \textbf{Bonus}: Find (with proof) all such distributions.
\end{problem}

\begin{starproblem}
  \begin{enumerate}
  \item For given $p\in(0,1)$, construct a tree where $p_c=p$.
  \item Find a tree where $p_c=p$ and $\theta(p_c)>0$.
  \end{enumerate}
  (Hint: if a vertex in level $n$ has $d_n$ children, write recursions for the probability that $0$ is connected to level $n$, and analyse these.
\end{starproblem}



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