We consider for each $n\in \mathbb{N}$ a sequence $(X^n_i)_{i \in \mathbb{N}}$ of non-negative real-valued random variables. We define $F^n_i$ to be the natural filtration, that is $F^n_i = \sigma(X^n_1, ... , X^n_i)$. We let $\mu$ be a finite measure on $\mathbb{R}_+$, and assume that we have a sequence of $(a_n)_{n \in \mathbb{N}}$ of linear time-scales, such that: $$\lim_{n \to \infty} \sum_{i=1}^{\left \lceil{a_n t}\right \rceil } P(X^n_i > x | F^n_{i-1}) = t\mu((x,\infty)),$$ $$\lim_{n \to \infty} \sum_{i=1}^{\left \lceil{a_n t}\right \rceil } \left(P(X^n_i > x | F^n_{i-1})\right)^2 = 0,$$ where $t,x > 0$, and the limits are meant as convergence in probability. Now, the Durrett-Resnick criterion states that $$\sum_{i \in \mathbb{N}} \delta_{(i/a_n, X^n_i)} \rightarrow \mathfrak{P},$$ where $\mathfrak{P}$ is a Poisson Point Process with intensity measure $dt \times d\mu$. I would now like to prove this. First off, I computed $$\lim_{n \to \infty} P\left(\max_{i=1,...,\left \lceil{a_n t}\right \rceil} X^n_i \leq x\right) = e^{-\mu((x,\infty))}.$$ I don't really know how this fact can be applied here. I tried to compute the Laplace functional:
$$L_{\xi_n}(f) = \mathbb{E} \left[ \exp\left(-\int_{\mathbb{R}^2_+} f(x,t) \sum_{i \in \mathbb{N}} \delta_{(i/a_n, X^n_i)} \right)\right] = \prod_{i \in \mathbb{N}}\mathbb{E} \left[ \exp\left(- f(X^n_i,i/a_n) \right)\right]$$ Now I have learned a cool trick, which is: $$\prod_{i \in \mathbb{N}}\mathbb{E} \left[ \exp\left(- f(X^n_i,i/a_n) \right)\right] = \prod_{i \in \mathbb{N}} \left( 1 +\mathbb{E} \left[ \exp\left(- f(X^n_i,i/a_n) \right) - 1\right] \right) \sim \exp\left(\frac{1}{n} \sum_{i \in \mathbb{N}} n \mathbb{E} \left[ \exp\left(- f(X^n_i,i/a_n) \right) - 1\right] \right).$$ We assume $f$ to have compact support, so only finitely many summands are non-zero. I would now like to further compute $$n \mathbb{E} \left[ \exp\left(- f(X^n_i,i/a_n) \right) - 1\right] .$$ I could write $$n \mathbb{E} \left[ \exp\left(- f(X^n_i,i/a_n) \right) - 1\right] = -\int_0^\infty \left(\exp\left(- f(\tau,i/a_n) \right) - 1 \right)nP(X^n_i = \tau) d\tau$$ but this seems rather useless. I am unsure how to incorporate the given conditions. Could someone give me a hint?