Consider the discrete time random process $X_n,n\in \mathbb N$, with $$X_{n+1}=(1-K)\cdot X_n+K\cdot\frac{G_n}{c}\cdot X_n$$ where $G_n$ is a random variable with expectation $\mathbb E[G_n\mid X_n]=\lambda\cdot (1-F(X_n))$ for some strictly increasing and continuous distribution function $F$ and $K,c$ are constants with $K\in (0,1)$ (think of $K=0.1$) and $c\in(0,\lambda)$.

If I substitute the expectation of $G_n$ in the formula above, I get that $$X_{n+1}=X_n\left(1+K\cdot\frac{\lambda\cdot (1-F(X_n))-c}{c}\right)$$

So, it is obvious to me that $X^*=F^{-1}(1-c/\lambda)$, is some kind of fixed point of this random process. Moreover, this is a stable fixed point: if $X_n>X^*$, then $X_{n+1}<X_n$ and if $X_n<X^*$, then $X_{n+1}>X_n$, i.e., $X^*$ is attracting. However, this argument is non-rigorous. So, my question is

• Can I prove rigorously that $X_n$ converges to $X^*$.
• In what sense? Is this convergence in expectation? Can I show something stronger, e.g., convergence in probability?

Footnote: for completeness, $G_n$ is determined as follows: there are Poisson arrivals with rate $\lambda$. Each arrival has a value drawn randomly from a distribution $F$. $G_n$ is the random sum of all arrivals that have a value above $X_n$.