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Let $U_1$, $U_2, \dots$ be i.i.d. random variables taking values uniformly in $\{1, \dots, n\}$. Let

$$T:=\text{min} \{j\in\{1, \dots, n\}: U_j=U_k \text{ for some } k < j\}.$$

So $T$ can be thought of as the smallest positive integer $j$ such that the value $U_j$ has been obtained before and is hence the first to repeat.

I want to show that $\mathbb{P}(T<n^{c})\rightarrow0$ if $c<1/2$ and $\mathbb{P}(T<n^{c})\rightarrow1$ if $c>1/2$ as $n\rightarrow\infty$.

I have a suspicion that for both I should use how $n^{c}$ compares to $\sqrt{n}$ but I'm not entirely sure.

Edit: For future clarity, I initially incorrectly wrote $\mathbb{P}(T<n^{1/c})$ instead of $\mathbb{P}(T<n^{c})$ (as how it is written now) so please interpret the accepted answer accordingly.

elysian-peace
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1 Answers1

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Note that $$ P(T>k)=\frac{n!}{(n-k)!n^k} $$ This is because the event $\{T >k\}$ occurs if and only if $U_1,\dots,U_k$ are distinct numbers.

Using this question, we have $$ P(T>k)= e^{-k^2/2n}+ o(1)\qquad \text{as $k,n\to\infty$} $$ Setting $k=n^{1/c}$ gives the desired answer.

Mike Earnest
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