Consider the following discrete time random walk on $ mathbb Z $: where in the location $ n in mathbb Z $, the walker has a probability $ q_n $ to take a step to the left, probability $ p_n $ to take a step to the right, and the probability $ r_n $ to stay in the same place. Of course for all $ n $, $ p_n + q_n + r_n = 1 $. I'm referring to this triple $ (p_n, q_n, r_n) $ like a *piece of money* Random walking to the location $ n $

Let $ X_t $ to be the expected position of the random walker at the time $ t $ ($ t $ is a positive integer). I say that a random walk is *recurrent* if $ liminf_ {t to infty} X_t = + infty $ with probability $ 1 $ and $ limsup_ {t to infty} X_t = – infty $ with probability $ 1 $. I say that a random walk is *transient* if $ lim_ {t to infty} X_t = infty $ with probability $ 1 $ or $ lim_ {t to infty} X_t = – infty $ with probability $ 1 $.

I would like the following statement to be true:

Consider a random walk $ mathbb Z $ with room at the location $ n $ given

as $ (p_n, q_n, r_n) $ and a random walk $ mathbb Z $ with coin

location $ n $ given as $ ( frac {p_n} {p_n + q_n}, frac {q_n} {p_n + q_n}, 0) $.

Suppose the sequence $ r_n $ is uniformly distant from $ 1 $, and that

the $ p_n, q_n $ are uniformly delimited from $ 0 and $ 1 $.

Then the first random walk is recurrent if and only if the second is,

and the first random walk is transient if and only if the second is.

My hunch is that when we calculate the recurrence / fugacity, we can more or less ignore the possibility that the walker stays in place. When the walker is on the spot $ n $we are only interested if the walker is going left or right, and if this transition occurs after a time step or after 1000 no time does not matter when considering recurrence / transientness. Is this intuition correct, and if so, is there a way to make it rigorous?