Consider the simple symmetric random walk on the integers starting from 
the origin of length $n$. More precisely, I will denote an $n$ step random walk $w$ as 
$$ w:= \omega\_0 \omega\_1 \ldots \omega\_n, $$  

where $\omega\_0 =0$ and $\omega\_i := \omega\_{i-1}+1$ or $\omega\_{i-1}-1$. 
Let me denote the set of all such $w$ as $\Omega\_n$ and endow it with the uniform probability measure. Note that the cardinality of the set $\Omega\_n$ is $2^n$. Define the following random variable
$$ Rt\_n: \Omega\_n \longrightarrow \mathbb{Z} $$  

as follows: $Rt\_n(w)$ is the number of times the walk $w$ returns to the origin. 


$\textbf{Question 1:}$ Does there exist some real number $p>0$ and $C \neq 0$ such that the random variable 
$$w \longrightarrow \frac{Rt\_n(w)}{n^p} $$ 
converges in probability to the constant $C$? More precisely, for every 
$\epsilon >0$ 
$$ \lim\_{n \longrightarrow \infty} Prob\Big( \left|n^{-p}Rt\_n-C\right| > \epsilon \Big) = 0.$$ 
If yes, what is this $p$? I think $p = \frac{1}{2}$, but I do not know if this is true (or for that matter if $p$ exists). 


$\textbf{Question 2}:$ This is weaker than Question $1$; let $\sigma\_n(k)$ be the number of $n$ step random walks starting from the origin, that return to the origin exactly $k$ times. If $n$ is fixed (but large), for what value of $k$ does $\sigma\_n(k)$ attain its maximum? Is it approximately of the order of $C n^p$, where $p= \frac{1}{2}$ and $C$ is some non zero constant? 


$\textbf{Question 3:}$ If the answer to the Question $1$ is yes, does the random variable $\frac{Rt\_n}{n^p}$ converge almost surely to $C$?
In this case, I am going to think of $Rt\_n$ as a map from the space of walks of infinite length; to compute the value of $Rt\_n$ we truncate the walk after $n$-steps. This way the random variables are defined on a common sample space and almost sure convergence makes sense. 



How many times does a simple symmetric random walk of length n return to the origin?
All these questions are answered in paragraph 6 of Chapter III of Volume 1 of "An Introduction to Probability Theory and its Applications" by Feller.


In particular:


(1) $p=1/2$ is indeed the "right" scale, but the (normalized) number of returns only converges in distribution; it does not converge in probability to a constant;


(2) surprisingly, regardless of $n$ the most probable return counts are 0 and 1;


(3) the answer is no, because of (1). 


