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\lhead{CS 592--ATK, SPRING 2022}               
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\title{Homework 3}

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\item {\bfseries Property of $\sigma$-Fields.} 
  Let $\Omega$ be a sample space. 
  Let $\cF$ be a $\sigma$-field on $\Omega$. 
  Our objective is to prove that the sets $\cF(x)$ partitions the set $\Omega$. 
  
  For any two elements $x,y\in\Omega$, if $\cF(x)\neq\cF(y)$ then prove that $\cF(x)\cap \cF(y) =\emptyset$. 

\noindent{\bfseries Solution.}\newline 
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\item {\bfseries Independent Bounded Difference Inequality.} (20 points)
Let $\Omega=\Omega_1\times\Omega_2\times\dotsi\times\Omega_n$. 
Let $\X_1,\dotsc,\X_n$ be independent random variables over the sample spaces $\Omega_1,\dotsc,\Omega_n$. 
Let $f\colon\Omega\to\bbR$ be a function. 
Suppose there exists $c_1,\dotsc,c_n$ such that the following holds true. 
For every $1\leq i\leq n$, and $x,y\in\Omega$ such that $x$ and $y$ are identical everywhere except at the $i$-th coordinate, then we have $f(x)-f(y)\leq c_i$. 
Let $\mu=\EX{f(\X_1,\dotsc,\X_n)}$. 

Using Azuma's inequality, prove the following concentration inequality. 
  $$ \probX{f(\X_1,\dotsc,\X_n)-\mu \geq E}\leq \exp\left(-2E^2/\sum_{i=1}^nc_i^2\right)$$

\noindent{\bfseries Solution.}\newline 
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\item {\bfseries P\'olya's Urn.} (20 points)
Consider the following experiment. 
There is an urn with $R$ red balls and $B$ blue balls at time $t=0$. 
At every time step $t=1,\dotsc,n$ you sample a ball $\X_t$ from the urn.
Next, you replace the ball that you sample and introduce another ball that has the same color as $\X_t$ into the urn. 
After $n$ times steps, let $\S_n$ represent the total number of red balls that you had sampled. 
Formally, $\S_n = \sum_{i=1}^n \1{\X_i=R}$. 

\begin{enumerate}
\item (5 points) What is $\EX{\S_n}$? 
\item (15 points) State and prove a concentration bound for $\S_n$ around its expected value using Azuma's inequality. 
\end{enumerate}

 


\noindent{\bfseries Solution.}\newline 
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\item {\bfseries Never too Far from Home!} (20 points)
Suppose you start at your home at time $t=0$.
At every time step, you either take one step north or one step south uniformly and independently at random. 
Chernoff bound states that the probability that you ``are far from your home at $t=n$ is small.'' 

In this problem, we want to claim that ``we \ul{never} go far from home at any time $t\leq n$.''

Let me elaborate the difference. 
Suppose $\X_i=+1$ represents a north-step at time $t=i$, and  $\X_i=-1$ represents a south-step at time $t=i$. 
Let $\S_i = \X_1+\dotsi+\X_i$ represent our position at time $t=i$. 
Chernoff bound states that the following probability is small. 
  $$ \probX{\S_n \geq E}$$
We want to show that the following probability is small. 
  $$ \probX{\max_{1\leq i\leq n} \S_i \geq E}$$
The second event is much harsher than the first event. 
For example, if you stray away from home \ul{even once} but, later, return back close to home at time $t=n$ then the first event forgives you but the second event does not!  
  
Prove the following concentration bound using Azuma's inequality. 
  $$ \probX{\max_{1\leq i\leq n} \S_i \geq E} \leq \exp(-E^2/2n)$$

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\item {\bfseries Largest Convex Subset.} (20 points)
Suppose we pick $n$ points $\X_1,\dotsc,\X_n$ uniformly at random from the unit square $[0,1]^2$. 
A set of points is said to be in {\em convex position} if no point in this set can be written as the convex linear combination of other points in the set. 
Let $\S$ represent the size of the largest subset of $\{\X_1,\dotsc,\X_n\}$ that lie in convex position. 
Use the Talagrand inequality to prove a concentration of the random variable $\S$ around its median. 

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%%%%%%%%%%%% PLEASE LIST COLLABORATORS BELOW  %%%%%
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{\bfseries Collaborators :} \newline 
% ENTER THEIR NAMES HERE  

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