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\centerline {\bf Math 563 - Fall 15 - Homework 5} 

\medskip

\no 1. (from Durrett) 
All the weak and strong laws of large numbers we have seen
require that the random variables are independent. 
In this problem we study relaxing this assumption. 
Let $X_n$ be identically distributed but not necessarily independent. 
Assume $E X_n^2 < \infty$. Let $\mu=E X_n$. The covariance  of random variables 
$X$ and $Y$ is $Cov(X,Y)=E[XY]-E[X] E[Y]$. If $X$ and $Y$ are independent
then $Cov(X,Y)=0$, but the converse is not true. 
Suppose $|Cov(X_n,X_m)| \le r(|n-m|)$ where $r(k)$ is a function that 
converges to zero as $k \ra \infty$. Let $S_n=X_1+\cdots X_n$.
Prove that $S_n/n \ra \mu$ in probability. 

\medskip
\no 2. (from Resnick) Define a sequence of random variables recursively 
as follows. Let $X_0$ be uniformly distributed on $[0,1]$. 
Having defined $X_0,X_1, \cdots, X_n$ we let $X_{n+1}$ be uniformly 
distributed on $[0,X_n]$. Prove that $\frac{1}{n} \log X_n$
converges a.s. and find the limit. 

\medskip
\no 3. (from Resnick) Let $A_n$ be independent events. Prove that 
\bea
\frac{1}{n} \sum_{i=1}^n 1_{A_i} - \frac{1}{n} \sum_{i=1}^n P(A_i)
\ra 0 \, in \, probability
\eea

\medskip

\no 4. (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 that the 
family $\{X_n,Y_n: n=1,2,3,\cdots\}$
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 5. 
We look at the set of all random variables and consider two random
variables to be the same if they are equal a.s.
Define
\bea
d(X,Y) = E\left[ \frac{|X-Y|}{|X-Y|+1} \right]
\eea
We proved in class that 
$X_n \rightarrow X$ in probability if and only if 
$d(X_n,X) \rightarrow 0$. 
Prove that with this metric the set of random variables is a complete
metric space. 

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