I want to show, that for a $\lambda$-biased random walk $(Y_k)$ on the the integers $−1, 0, 1, 2, 3, ...$ with probability transitions
$P(Y_{k+1}=x+1|Y_k=x)=\frac{1}{1+\lambda}$ and
$P(Y_{k+1}=x-1|Y_k=x)=\frac{\lambda}{1+\lambda}$
the probability that $(Y_k)$, starting from $0$ and never hits $−1$ is more than $1 − λ$.
The tip is: Markov property.
But i don’t know how to use the markov property for this problem.
Thank you in advance.
Kind regards!
I assume that $\lambda\in [0,1]$. First, for $n\ge 0$, $$ \mathsf{P}_1(Y_{2n+1}=0)=\binom{2n+1}{n}\frac{\lambda^{n+1}}{(1+\lambda)^{2n+1}}, $$ where $\mathsf{P}_k$ denotes the law of a random walk starting at $k$ (i.e., $Y_0=k$). Let $T_0:=\inf\{n:Y_n=0\}$. Then $$ \mathsf{P}_1(T_0= 2n+1)=\frac{1}{2n+1}\times \mathsf{P}_1(Y_{2n+1}=0), $$ and $$ \mathsf{P}_1(T_0<\infty)=\sum_{n=0}^\infty\mathsf{P}_1(T_0= 2n+1)=\lambda, $$ or $\mathsf{P}_1(T_0=\infty)=1-\lambda$.