Let $(X_i)_{i \geq 1}$ be independent random variables which follow the uniform law on $\{1,...,n\}$. Now we defined the random variable $\forall k \leq n, Y_k = \inf_{m \geq 1} \{ \mid \{X_1, ..., X_m \} \mid = k\}$. Now I need to prove that the random variables : $(Y_k-Y_{k-1})_{ 2 \leq k \leq n}$ are independent.
Some thoughts :
First I think the random variables are following a geometric law of parameter : $\frac{n-k+1}{n}$. Doest it help ? I don't know... It's not really intuitive to me that they are independent since th variables $Y_k$ are not independent.
Thank you !