## Behavior of recurrence relation

Consider the following recurrence:

$$y_i = y_{i-1} + \frac{\log(p^{y_{i-1} - y_{i-2}} - \frac{\lambda}{y_{i-1}^2})}{\log p}$$ where $0 < p < 1$. We are only interested in values for the constant $\lambda$ such that $y_i \geq 0$ for all $i$. We start with $y_0 = 0, y_1 = 1$.

Some experiments with different parameters show that the sequence $y_j$ approaches $y_j \approx a j + b$ for some constant $a \geq 0, b \leq 0$. It seems like you should take $\lambda = O(p)$ but I cannot prove this.

Are there any relatively simple bounds (upper and lower) for the asymptotic behavior of $y_j$?

-
Even though the log term will look ugly, is it not more revealing to write the right hand side as 2y_i-1 - y_i-2 + log(1-stuff)/logp? Gerhard "Ask Me About System Design" Paseman, 2012.02.17 – Gerhard Paseman Feb 17 2012 at 18:37