[markov_chains_II] adding the definition of accessibility

[longye-tian]

Dear John @jstac ,

As we discussed in #460 , this issue is to discuss the definition of accessibility.

I propose using the definition in the QuantEcon Book Economic Networks: Theory and Computation on page 31.

I modified this definition slightly from 'vertex' in the book to 'state' in this lecture to match the context as follows:

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State $x$ is called accessible (or reachable) from state $y$ if either $x=y$ or there exists a directed path from $x$ to $y$.

Two states, $x$ and $y$, are said to communicate if $x$ and $y$ are accessible from each other.

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I have one question about the above definition:

Should we also include the definition of a directed path or provide a link to this definition?

Best ❤️

Longye

[jstac]

Many thanks @longye-tian .

Should we also include the definition of a directed path or provide a link to this definition?

Let's change the definition of accessible to $P^t(x,y) > 0$ for some $t \geq 0$.

I like the idea of discussing directed paths but this option is simpler.

[longye-tian]

Many thanks @longye-tian .

Should we also include the definition of a directed path or provide a link to this definition?

Let's change the definition of accessible to Pt(x,y)>0 for some t≥0.

I like the idea of discussing directed paths but this option is simpler.

Thank you for the reply!

I will add the definition to the lecture as follows:

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State $x$ is called accessible (or reachable) from state $y$ if $P^t(x,y)>0$ for some $t\ge 0$.

Two states, $x$ and $y$, are said to communicate if $x$ and $y$ are accessible from each other.

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Best ❤️ Longye