I am given $n$ objects and for $n$ times, I pick one of them with uniform probability and put it back after picking it.


For $k\in\{1,\ldots,n\}$ let $f\_k$ denote the number of times that I have picked object number $k$. So we have $f\_k\in \{0,\ldots,n\}$ for all $k$.


We consider $M:= \max\big\{f\_k: k\in\{1,\ldots,n\}\big\}$, so we have $M\in\{1,\ldots,n\}$. We are interested in the expected value $E\_n:= E(M)$.


Does $\lim\_{n\to\infty} E\_n$ exist? If yes, what is its value?



Choosing $n$ times from $n$ objects
Fix some $k$. The probability that the element $1$ is chosen exactly $k$ times is ${n\choose k}\frac1{n^k}\left(1-\frac1n\right)^{n-k}\to \frac1{ek!}$ as $n\to\infty$. So $1$ is chosen at least $k$ times with probability at least, say, $\frac1{2ek!}$


Under the condition that $1$ has been chosen less than $k$ times, the same probability for $2$ is at least ${n-k\choose k}\frac1{n^k}\left(1-\frac1{n-1}\right)^{n-k}\to \frac1{ek!}$, so both $1$ and $2$ are not chosen with the probability at most $\left(1-\frac1{2ek!}\right)^2$ if $n$ is large.


Iterating this, we get that for large enough $n$ the probabitily of $M<k$ is at most $\left(1-\frac1{2ek!}\right)^d$, where $d$ is an arbitrary fixed number. Thus this probability tends to $0$ as $n\to\infty$, and $E\_n\to\infty$.


