Section 16.1

Hitting Probabilities and Times

Biased and symmetric walks on Z: the probability of ever reaching a level, the mean and variance of hitting times.

Toolkit

Core technique: first-step (one-step) analysis conditioning on the first move, exploiting that reaching a level $k>1$ decomposes into $k$ i.i.d. copies of reaching one level up, by the strong Markov property.

For a walk with step $+1$ w.p. $p$ , $-1$ w.p. $q=1-p$ , let $h$ be the probability of ever reaching $+1$ starting from $0$ , and let $T_1$ be the hitting time of $+1$ (possibly infinite).

Hitting probability: $h = p + q h^2$ , so $h=1$ if $p\ge 1/2$ , and $h = p/q$ if $p<1/2$ .
For hitting level $n>0$ from $0$ : probability is $h^n$ .
If $p>1/2$ : $\mathbb{E}[T_1] = \dfrac{1}{p-q} = \dfrac{1}{2p-1}$ , and $\mathbb{E}[T_n] = n\,\mathbb{E}[T_1]$ .
Variance: write $T_1 = 1 + T' + T''$ with $T',T''$ i.i.d. copies of $T_1$ (if the first step is $-1$ ); take $\mathbb{E}[T_1^2]$ via this recursion to get
$\operatorname{Var}(T_1) = \frac{4pq}{(p-q)^3}=\frac{4p(1-p)}{(2p-1)^3},\qquad \operatorname{Var}(T_n)=n\operatorname{Var}(T_1).$
Symmetric case $p=1/2$ : $h=1$ but $\mathbb{E}[T_n]=\infty$ .

Problems