Background: Given a sequence of $n$ items, I am working on a sampling method where I slide a window of size $\ll n$ from start to end, and pick the first item within each window with a certain probability. Item $i$ is labeled with a weight number $x_i$ that indicates its relative importance, i.e., it will be picked with probability:
$$\frac{x_i}{\sum_{j=i}^{i+w} x_j}$$
I am trying to analyze the total expected size of my sample using this technique.
Formulation: Let $x_1, x_2, \ldots x_n$ be a sequence of numbers ($x_i \in \mathbb{N}$). What would be tight upper bound of the following summation:
$$\sum_{i=1}^{n-w} \frac{x_i}{\sum_{j=i}^{i+w} x_j}$$
Assume window length ($w+1$) is $\ll n$. I am particularly interested in the case when $x_i \leq c, \forall i \in [1,n]$, where $c \ll n$ is a given constant.