I am interested in techniques for comparing two recurrence relations.
The first recurrence is given as:
$$y_n = y_{n-1} (1-1/D) + y_{n-D} /D, ~D \geq 2, ~y_0 = 1, ~y_{m} =0~\forall m<0$$
The second recurrence is given as:
$$x_n = x_{n-1} (1-z_{n-1}) + y_{n-D} z_{n-D}, x_0 = 1, x_{m} =0,~ \forall m<0$$ Here, $D \geq 2$ is the same as the previous recurrence, and $0\leq z_n \leq 1/D$ for all $n\geq 0$ are arbitrary non negative real numbers.
I am trying to compare the two recurrences. Are there known tools to compare two linear recurrence relations?
P.S: I believe $x_n \geq y_n$.
It is easy to note that $x_{n} \geq y_{n}$ for $n\leq (D-1)$ because $0\leq z_n \leq 1/D$ for all $n\leq D-1$.
Experimentally, by choosing $z_n$ uniformly randomly from $[0,1/D]$ for all $n$, I find strong evidence for that statement (over 25 runs of sequences of length 1000).