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I have a conditional recurrence relational, that is conditioned on a function value (defined on sequence index).

$ X_{i+1} = \begin{cases} X_{i} + a_1. b^i ,& \text{if } X_i \geq c. b^i\\ X_{i} + a_2. b^i, & \text{otherwise} \end{cases} $

where $a_1, a_2, c, b$ are constants.

I am at large as to how to try to solve the problem. Further, I know that $a_1, a_2$ control the rate of change of $X$ such that it remains close to $c. b^i$.

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    What exactly is the problem? Determining $X_i$? Or rather showing that choosing the right constants yields some form of control? – 2017-02-04
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    @mvw I know that $a_1 < c$ and $a_2 > c$, therefore $X_i$ would finally converge to $c. b^i$. However I am unable to show that the convergence time, eg. I am unable to show something like $|X_i - c. b^i| < \lambda$ after $i$ steps with high probability. The\ difference is oscillating a lot initially and subsidizes later. I want to have an approximate idea (order) of the convergence time. I don't need any exact answer. I was hoping to model it as recurrence relation and solving for characteristic equation would help me get convergence time ($i$) later. – 2017-02-04

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