 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 215

The squared $L^2(\pi)$ distance has the diagonal identity
$$
\left\lVert\frac{P^t(x,\mathord\cdot)}{\pi(\mathord\cdot)}-1\right\rVert_{2,\pi}^2
=\frac{P^{2t}(x,x)-\pi(x)}{\pi(x)}.
$$
The diagonal excess is nonnegative and decreases with time. Therefore
$$
(2t+1)\{P^{2t}(x,x)-\pi(x)\}
\leq\sum_{k=0}^\infty\{P^k(x,x)-\pi(x)\}.
$$
Using the identity supplied in the question gives
$$
\left\lVert\frac{P^t(x,\mathord\cdot)}{\pi(\mathord\cdot)}-1\right\rVert_{2,\pi}^2
\leq\frac{\mathbb E_\pi\tau_x}{2t+1}.
$$
At $t=8\mathbb E_\pi\tau_x$ the right side is at most $1/16$, up to the immaterial integer rounding. Thus
$$
\boxed{t_{\mathrm{mix}}^{(2)}(x,1/4)\leq8\mathbb E_\pi\tau_x}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 1 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 1 / c / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 1 / c / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)