Expander walk sampled and moving sets

Let $J$ be the constant projection. On the mean-zero subspace $M-J$ has norm $\alpha$, and $M^g=J+(M-J{)}^g$ for $g\ge 1$. The rank-one operator $P_{T}J{P}_S$ has norm $\sqrt{{\beta }_T{\beta }_S}$, by the norms of its two indicator vectors. The remaining term has norm at most ${\alpha }^g$, since projections are contractions. Thus $\mathrm{\parallel }{P}_T{M}^g{P}_S\mathrm{\parallel }\le \sqrt{{\beta }_T{\beta }_S}+{\alpha }^g$. This is the same supported-operator framework as confinement.

Expand the finite path sum with the successive projections, using stationarity to start at $t_0$ uniformly. It is the normalized inner product of the endpoint indicators with the product of the intermediate restricted operators. Bound each operator by the first step and the endpoint norms by $\sqrt{{\beta }_0}$ and $\sqrt{{\beta }_r}$. If $r=0$ this is simply ${\beta }_0$; an empty target makes the actual probability zero and the inequality remains valid. No independence of successive visits is used.