Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 2 / 12F (Probability)

Write $K=k+1$ and $h_j={\mathbb{E}}_j{D}_j$. The expected duration of symmetric gambler's ruin satisfies $h_0=h_K=0$ and, by conditioning on the first step,These expected values are finite: from any interior state, $K$ consecutive right steps ensure absorption, an event of probability at least $2^{-K}$. Blocks of $K$ steps therefore bound the survival probability by a decreasing geometric sequence. The candidate $h_j=j(K-j)$ has the required boundary values and satisfies $h_{j+1}-2h_j+h_{j-1}=-2$. HenceFor completeness, the difference of two solutions of this linear recurrence relation has zero second difference and is affine in $j$; its two zero boundary values force it to vanish.

The vertices visited by a simple symmetric random walk form an integer interval: nearest-neighbour steps cannot skip a vertex. At the stopping time $T_k$, write that interval as $[\ell ,r]$, with $r-\ell +1=k$. The walk is at one of its endpoints, because the $k$th new vertex must extend the earlier interval. This also covers $k=1$, when both endpoints coincide.

The Strong Markov property restarts the walk from this endpoint. Visiting a new vertex is exactly exiting $[\ell ,r]$, or reaching $\ell -1$ or $r+1$. Translate these absorbing boundaries to $0,k+1$. The starting point becomes either $1$ or $k$, so the expected duration of symmetric gambler's ruin is $k$ in both cases. Taking conditional expectations and then expected values gives the expected time to expand a random-walk range:

Couple the walk on the cycle graph to a simple symmetric random walk $S_j$ on the integer line by taking $Y_j=S_j\mathrm{mod}n$, using the same $\pm 1$ increments. The visited integer interval has distinct residues until its length reaches $n$; an interval of exactly $n$ consecutive integers contains every residue once. Consequently the cover time on the cycle is exactly $T_n$ in this coupling.

Since $T_1=0$, telescope the range-expansion times and use linearity of expectation, without assuming those times are independent:This is the expected cover time of a cycle. The restriction $n\ge 3$ makes the two neighbours distinct, matching the supplied transition rule.