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\begin{document}

\centerline {\bf Math 563 - Fall '14 - Homework 4} 

\smallskip


\no 1. (from Durrett) {\it Monte Carlo integration:}
Let $f(x)$ be a Lebesgue integrable function on $[0,1]$. 
Let $U_n$ be an i.i.d. sequence with each $U_n$ uniformly 
distributed on $[0,1]$. Prove that 
\bea
\frac{1}{n} \sum_{k=1}^n f(U_k) \rightarrow \int_0^1 f(x) \, dx
\eea
with probability one. 
This is a really bad way to compute a one-dimensional integral,
but there are analogs of this in higher dimension that are sometimes
one of the best ways to compute high dimensional integrals.

\medskip

\no 2. (from Durrett) {\it metric for convergence in probability:}
We look at the set of all random variables and consider two random
variables to be the same if there are equal a.s.
Define
\bea
d(X,Y) = E\left[ \frac{|X-Y|}{|X-Y|+1} \right]
\eea

\no (a) It is obvious that $d$ is reflexive and $d(X,Y)=0$ if and only 
if $X=Y$ a.s. Prove that $d$ satisfies the triangle inequality and so 
defines a metric. 

\no (b) Prove that $X_n \rightarrow X$ in probability if and only if 
$d(X_n,X) \rightarrow 0$. 

\medskip

\no 3. (from Durrett) Prove that the metric space in the previous problem is 
complete. 

\medskip

\no 4. (from Durrett) Let $X_n$ be an independent sequence of RV's. 
Prove that $sup_n X_n < \infty$ a.s. if and only if there is a constant 
$C$ such that 
\bea
\sum_{n=1}^\infty P(X_n > C) < \infty. 
\eea

\medskip


\no 5. (from Durrett) Let $X_n$ be an i.i.d. sequence of non-negative
random variables that represent the lifetimes of a sequence of identical
light bulbs. Let $Y_n$ be another i.i.d. sequence of non-negative random 
variables. $Y_n$ is the time we must wait after the $n$th bulb
burns out before it is replaced. (We also assume $\{X_n,Y_n: n=1,2,3,\cdots\}$
is independent.) Assume that $E X_1$ and $E Y_1$ are both finite. 
Let $W_t$ be the amount of time in $[0,t]$ that we have a working light 
bulb. Prove that 
\bea
{W_t \over t} \ra {E[X_1] \over E[X_1] + E[Y_1]} \quad a.s.
\nonumber
\eea

\medskip

\no 6. (from Durrett) Let $X_n$ be an i.i.d. sequence of random variables 
with distribution function $F(x)$. Let $\lambda_n$ be an increasing
sequence of real numbers which converges to $\infty$. Let 
\bea
A_n = \{ \max_{1 \le m \le n} X_m > \lambda_n \}
\eea
Prove that if $\sum_n (1-F(\lambda_n)) <\infty$, 
then $P(A_n \, \, i.o.)=0$, and that if the sum is $\infty$ then the 
probability is $1$.

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