Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number of $i$'s such that $$ X_{(i)} - X_{(i-1)} > L $$ that is, how many backward-intervals have length at least $L$.
I tried to calculate the characteristic function of the random variable $$ \sum_{i=2}^N \mathbb{1}\lbrace X_{(i)} - X_{(i-1)} > L\rbrace $$
with the aid of the density of the order statistics $$ n! \, \mathbb{1}\lbrace x_1 \leq \ldots \leq x_n \rbrace \, g(x_1) \cdots g(x_n) $$ but to no avail (it becomes very tricky). Any ideas on this? Can the characteristic function be obtained explicitly in terms of $g$ and $L$?
Thanks in advance
PS: I am aware that the distribution of the range of size 1 is well known. However, I am studying an event that depends of all ranges of size 1 simultaneously.