Suppose we have a birth-death chain with a state space
$$ S = \{0,1,2,\ldots\} $$
and transition probailities:
$$p(x,y)=\begin{cases}q_x, &\text{if } y = x-1, &\text{i.e. death}\\ p_x,&\text{if } y=x+1, &\text{i.e. birth}\\ \end{cases}$$
Here, $p_x+q_x = 1$
Now,
there is a theorem that states that if $q_x \geq p_x$ $\forall x \geq 1$, then the birth and death chain is recurrent.
I am working with a Markov chain that has a more complex relationship between $p_x$ and $q_x$. Namely, I am considering the case where $q_x < p_x$ for infinitely many $x>0$ (but not necessarily for all $x>0$). and $q_x \leq p_x$ at all other $x$. Thus, suppose
$$ 0 <n_1<n_2< \cdots $$
is an infinite set of strictly increasing numbers (i.e. positive time points). I am trying to produce two examples of birth death chains (i.e., of probabilities $q_x, p_x$) such that:
- $0<q_{n_k}<p_{n_k}$ for all $k>0$
- $q_x \geq p_x >0$ when $x \neq n_k$ for any $k$.
and the condition that one chain is recurrent while the other is transient.
I am not sure if such chains exist. Would anyone know any examples of the top of their head or lead me in a direction? Thank you so much!
Choose $p_{2x}=\frac12\left(1-a\right)$ and $p_{2x-1}=\frac12\left(1+b\right)$ for every $x\geqslant1$, with $a$ and $b$ in $(0,1)$. For every $x\geqslant1$, the chain starting from $2x$ hits $2x+2$ before $2x-2$ with probability $$p=\frac12\left(1+\frac{b-a}{1-ab}\right).$$ Thus, the chain is transient if $p\gt\frac12$ and (positive) recurrent if $p\lt\frac12$. This solves the question because one can adjust the parameters $a$ and $b$ to get both cases (with $p_{2x}\lt p_{2x-1}$, as required) since $p-\frac12$ has the sign of $b-a$.