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The independence of all sub-paths between adjacent returns to origin in random walks

In random walks, a path may return to origin for the $r$-th time in $n$-th step ($r$ is given but $n$ is not given), and under this condition, these $n$ steps can be split into $r$ sections where each …
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The independence of all sub-paths between adjacent returns to origin in random walks

Thanks to @Especially Lime's reminder of the difference between the median and the mean (in probability, the equivalents are quantile and expectation). And now I think this is exactly the key point le …
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