# Eigenfunction expansion with Markov operators

I'm having trouble comprehending the following.

http://www.stat.berkeley.edu/users/pitman/s205s03/lecture28.pdf in the proof of theorem 28.3.

It says that $H_t(x,y)=H_t\delta_y(x)$, where $\delta_y(x)=1$ if $x=y$ and $0$ if $x\neq y$.

Also why is $E(H_t f)=Ef$ ? I particularly don't know why $K(c)=c$ if $c$ is a constant and $K$ is a Markov operator, because I thought $K$ could only operate on functions. Could we consider $c$ as a function too although it really is just a scalar?

EDIT I know this is another late question but about the same paper, there's an equality I don't understand. Near the end, it says

$$|h_t(x,y)-1|=\left|\sum_{z}(h_{t/2}(x,z)-1)(h_{t/2}(z,y)-1)\pi(z)\right|$$

On the RHS it looks like I should get $|\sum_{z} (h_{t/2}(x,z)h_{t/2}(z,y)-h_{t/2}(x,z)-h_{t/2}(z,y)+1) \pi(z)|.$

However I can't simplify that to be the LHS.

And lastly, the inequality $$\left\|H_t-E\right\|_{2\to\infty}\leq \left\|H_{t_1}\right\|_{2\to\infty} \left\|H_{t_2}-E\right\|_{2\to 2}$$ for $t=t_1+t_2.$

These have been two questions I have been unable to solve and they would help me greatly in understanding Markov bounds. Thanks!

pr.probability

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$c$ is a constant function whose value is $c$ everywhere. – Yuval Filmus Feb 21 '11 at 19:08
Why can we say that if $c$ is really a scalar. What's being mapped to $c$ for this to be considered a function. – Clark Kent Feb 21 '11 at 20:51
Yannik, it's just a convention. Think of $c$ as a vector of the correct dimension, all of whose entries are $c$. – Yuval Filmus Feb 21 '11 at 23:07
To me, it seems near the end they say something different: $|h_t(x,y)-1|=|\sum_{z} (h_{t/2}(x,z)-1) (h_{t/2}(z,y)-1) \pi(z) |$. – Raskolnikov Feb 22 '11 at 18:10
Yeah, that. My bad. – Clark Kent Feb 22 '11 at 20:09

The first is an application of the definition of $H_t$ as an operator which works on functions.

$$H_t f(x) = e^{-t} \sum_{i=0}^{\infty} \frac{t^i K^i f(x)}{i!} \; .$$

Apply it on the function $\delta_y$ to give

$$H_t \delta_y(x) = e^{-t} \sum_{i=0}^{\infty} \frac{t^i K^i \delta_y(x)}{i!} \; .$$

So, the crucial thing to work out is what is the action of $K$ on $\delta_y$. Again using the definition, this becomes

$$K \delta_y (x) = \sum_{z \in \mathcal{X}} K(x,z)\delta_y(z)$$

Since $\delta_y(z)=1$ when $z=y$ and $0$ elsewhere this simplifies to

$$K \delta_y (x) = K(x,y) \; .$$

Applying K $i$ times to $\delta_y(x)$ should give you the $i$^th power of the stochastic matrix $K(x,y)$ which is denoted in the text by $K^i(x,y)$. (This confused me at first, since I though they meant an ordinary power of the $(x,y)$-component.) So for instance, for power two, this means

$$K^2 \delta_y(x) \equiv K^2(x,y) = \sum_{z \in \mathcal{X}} K(x,z)K(z,y) \; .$$

Putting this all together, we get

$$H_t \delta_y(x) = e^{-t} \sum_{i=0}^{\infty} \frac{t^i K^i(x,y)}{i!} \; .$$

But this is exactly how $H_t(x,y)$ is defined. So up till there, much ado about nothing, just some definitions.

EDIT 1 Purely using the definition

$$h_t(x,y)=\frac{H_t(x,y)}{\pi(y)}$$

one can show that

$$\begin{eqnarray} h_t(x,y) & = &\frac{H_t(x,y)}{\pi(y)}=\frac{\sum_z H_{t/2}(x,z)H_{t/2}(z,y)}{\pi(y)} \\ & = &\sum_z \frac{H_{t/2}(x,z)}{\pi(z)}\frac{H_{t/2}(z,y)}{\pi(y)}\pi(z) = \sum_z h_{t/2}(x,z)h_{t/2}(z,y)\pi(z)\end{eqnarray}$$

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