Let $\pi$ be a point process on $\mathbb R$ with intensity $\lambda$ (Lebesgue measure). We start at $Y_0=0$ and perform the following algorithm : We select the point closest to $0$ and call it $Y_1$, move to $Y_1$ and remove $Y_1$ from $\pi$. Then, we find the closest point to $Y_1$ in $\pi\backslash\{Y_1\}$, call it $Y_2$, move to $Y_2$ and remove $Y_2$ from $\pi\backslash\{Y_1\}$. We then repeat this algorithm indefinitely.
Next, we define $I_n=[\underset{0\le k\le n}{\min}Y_k, \underset{0\le k\le n}{\max}Y_k]$. I wish to show that there exists a sequence of independant exponential($1)$ random variables $(e_n)_{n\ge1}$ such that $\forall n\ge1$, $\lambda(I_n)=e_1+\cdots +e_n$.
Intuitively this makes lots of sense, since each time a new move is made, we add onto $I_n$ an interval whose length in exponential($1$) by the propoerties of Poisson point processes on $\mathbb R$. My idea to try and formalise this is to let $N_n\in\{0,\cdots,n\}$ the amount of moves that were made to the right, and the amount of moves that were made to the left. Then, let $\pi_+=\pi\cap\mathbb R_+$, $\pi_-=\pi\cap\mathbb R_-^*$ which are independant Poisson point processes whose respective intensities are $\lambda_{|\mathbb R_+}$ and $\lambda_{|\mathbb R_-^*}$. Therefore, $\exists (e_n^+)_{n\in\mathbb N^*},(e_n^-)_{n\in\mathbb N^*}\sim$ exponential($1$) such that $\pi_+=\{e_1^++\cdots+e_n^+\mid n\in\mathbb N^*\}$ and $\pi_-=\{-e_1^--\cdots-e_n^-\mid n\in\mathbb N^*\}$. Therefore, we have $\forall n\in\mathbb N^*$ : $$\lambda(I_n)\overset{d}{=}\sum_{i=1}^{N_n}e_i^++\sum_{i=1}^{n-N_n}e_i^-$$ Except this isn't exactly what is expected. Any help would be greatly appreciated.