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How to write a latex macro to reference a particular time in a set of youtube videos?

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I have a set of educational youtube videos that I want to reference in a document, with links to a particular time in the video. In Python, the code I want would do something like this:

links = {
    "first": "",
    "second": ""

def youTubeRef(text:str, ref:str h:int, min:int, sec:int):
    link = f"{links[ref]}&t=h{h}m{min}s{sec}"
    return '<a href="{link}">{text}</a>'  # though in latex it would just be a href

Then, in a latex document, I would like to use this like so:

Here is youTubeRef{my text}{first}{0}{12}{13}

And the final rendered document would show:

Here is my text


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This may be a duplicate, but I’ve done some searching and I can’t find exactly this problem setting, probably due to not knowing the right terminology for how to refer to the transition matrix.

I’m considering absorbing random walks ${X_t}$ which have symmetric transition probabilities inside a window of size $k$. For example, on ${0, 1, 2, dots, 100}$, with $k=3$ if $X_t = 5$
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The transition matrix for a walk on the integers in $[0, N]$ has the following properties:

  1. $T$ is (row) stochastic
  2. $T$ is a band matrix with upper and lower band size $k$
  3. $T_{i, j} = 0$ if $|i – j| > max (i, N – i)$
  4. $T_{i,i + r} = T_{i, i – r}$ (this is the main property I don’t know the name of)
  5. $T_{0, 0} = T_{1, 1} = 1$

I would like a proof or counterexample of the following statement:
limlimits_{n to infty} (T^n)_{i, N} = i/N

If the statement above is true, is it still true if the transition distributions are no longer left-right symmetric, but it’s still the case that $E[X_{t + 1}] = X_t$?