I tried to review a few posts related to the infinite-countable state space Markov chain and its stationary distribution. However, I could not solve the problem myself.
It relates to my previous post Markov chain for in-flow and out-flow of mass? and another's Find stationary distribution for infinite space Markov chain.
Imagine there is a mass of customers in a department store receiving a good or bad signal at each discrete time. A good signal arrives with probability $p$ and a bad signal with $q$, such that $p+q <1$. Furthermore, at each period, with probability $\phi$, a customer dies. In that case a new customer enters the store having no signal (or, a customer loses all the signals with probability $\phi$).
Customers have their evaluation point in mind. A new customer's, who had no signal, initial point is $0.5$. With a good signal, it increases by $70\%$, whereas, with a bad signal, it decreases by $30\%$. This is just for an example.
My state space is $(g,b)$ where $g$ denotes the number of good signals received and $b$ the number of bad signals. If there is a customer who has $(g=10,b=5)$, his evaluation point $G$ will be $G=0.5 *1.7^{10}*0.7^{5}$. I assume that customers reaching $G<L$ such that $0.5>L>0$ gives up evaluating the store and forever have such $G$ until he dies.
These are what I characterized.
(1) I can classify $(g,b)$ into three categories.
First, those of which evaluation point $G>(L$ / $0.7$). For those states, the following relation holds. $S_{(g,b)}$ denotes the share of customers having signal $(g,b)$.
$$S_{(g,b)}= p \cdot S_{(g,b-1)} + q \cdot S_{(g+1,b)} $$
Second, those $(g,b)$ with $G$ such that $(L$ / $0.7)>G>L$. Now there is no movement from $(g,b-1)$ to $(g,b)$, because at such $(g,b-1)$, customers stop evalutating, having $G<L$. Therefore, there is only movement from $(g+1,b)$
$$S_{(g,b)}= q \cdot S_{(g+1,b)} $$
Last, the mass of customers stopped evaluating with $L>G>0$.
(2) Particulaly for $S_{(0,0)}$, the share of customers at $(g=0,b=0)$, satisfies :
$$S_{(0,0)} = p \cdot S_{(0,-1)} + q \cdot S_{(1,0)} + \phi \sum_{(g',b')} S_{(g,b)} $$
where the last term comes from consumers' death. Here I assumed $L$ is low enough, so that $S_{(0,0)}$ is in the first category.
(3) The summation of shares equal to one.
$$ \sum_{(g',b')} S_{(g,b)} = 1$$
I'm interested in the stationary values of $S_{(g,b)}$, the share of customers at each state $(g,b)$.
My hunch is that, unless it's a special case like the evaluation point increases by $50\%$ (by 3/2 times) and decreases by $66\%$ (by 2/3 times), the threshold $L$ generating the absorbing state complicates the problem massively. Still, I was wondering if I could get a closed-form, or an intuitive analytical solution based on the concept of detailed balance or Kolmogorov criterion, as in the second post. Furthermore, as the stationary distribution is invariant (if it exists), I was wondering if it could be derived through the function-space fixed-point argument.
Any comments or thoughts would be more than helpful and appreciated.