Past exam of the probability course of the University of Oxford / 2015 / 3 / b / i

Put $q=1-p$ and ${\xi }_i=Y_i-Y_{i-1}$. The Markov property and the state-independent transition probabilities show that, conditional on the whole previous history, ${\xi }_i$ is $+1$ with probability $p$ and $-1$ with probability $q$. Multiplying these conditional probabilities for any finite step sequence proves that the ${\xi }_i$ are independent and identically distributed random variables, withBecause $Y_n=k+{\sum }_{i=1}^n{\xi }_i$, the strong law of large numbers givesIn particular, $Y_n\to +\infty$ almost surely. This stronger pathwise consequence will determine both eventual crossings and the asymptotic first-passage times.