Nonhomogeneous simple random walk in $\mathbb{N}$. Gambler's ruin. Consider a gambler with an initial fortune $a> 0$ (say an integer $a$ in $\mathbb{N}$ ) playing against a gambling hause with initial fortune of $b> 0$ (say an integer $b$ in $\mathbb{N}$). Supose that the stochastic process $\{X_n\}_{n=0,1,2,3,\ldots}$ denote the cumulative fortune of gambler. If the gambler's winnings in each game is given by a jump random variable $J_n:\Omega \to \{-1,0,+1\}$ then $$ X_0=J_0=a,\quad X_1=J_0+J_1,\quad X_2=J_0+J_1+J_2,\ldots\quad\ldots, X_n=J_0+J_1+\ldots+J_n $$ Fix $p,q$ and $r$ such that $0< p,q<1$; $0\leq r<1$ and $p+q+r=1$. If that for all $n\in\mathbb{N}$ and for all $k\in\{1,2,3,\ldots,n\}$ we have $$ \mathbb{P}(J_k=+1)= p, \quad \mathbb{P}(J_k=-1)= q \quad \mbox{ and } \quad \mathbb{P}(J_k=0)=r $$ then it is well known that the probability of ruin of the gambler or ruin of the gambling hause is finite. In more precise terms if we denote by $T$ the following stopping time: $$ \min\{ n\in\mathbb{N}: X_n\leq 0 \mbox{ or } X_n\geq a+b\} $$ we have $\mathbb{P}(T<\infty)=1$.
But now suppose that the gambler's winnings in each game is given by $$ \mathbb{P}(J_k=+1)= e^{-k}, \quad \mathbb{P}(J_k=-1)= e^{-k} \quad \mbox{ and } \quad \mathbb{P}(J_k=0)=1-2e^{-k} $$
Question. With the new odds of gains that the gambler has what is the probability of the game ending in finite time? We have $\mathbb{P}(T<\infty)=1$?