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I am interested in random walks with momentum in 1D, ie., random walks in which there is a certain probability $p$ to take a step in the same direction as the last step, and a probability $q = 1-p$ to take a step in the other direction (switch direction). In a previous thread, there was an outline proof that the distribution of positions converges to a normal distribution, and that the variance is equal to $2p/(1-p).$ I have had a hard time to find literature on this type of random walk. I'd be interested in the following point and would be grateful if you can point me to papers, lecture notes, or text books on the matter.

  1. Average # steps from starting point should scale with $\sqrt n$ and can be adapted from standard random walks to scale as $2\sqrt{2pn/(PI\times(1-p))}$ in the normal limit. Can anything be set about the evolution for small $n$?
  2. The law of iterated logs should apply and can be adapted to $2pn/(1-p)\log(\log(pn/(1-p))).$ Anything on the evolution of small $n$?
  3. return to origin should apply, with the expected value going to infinity. Is there a proof of this? Can the distribution of # steps for returning to the origin be derived?
User1865345
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Jens
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    The literature is huge. By collecting the current state and previous state in a vector, you describe an ordinary random walk in 2D. Moreover, it decomposes into two 1D walks when you rotate the coordinates by an eighth-circle. But this is merely an application of the technique described in detail in answers to the thread you reference. – whuber Sep 08 '23 at 12:58
  • Your link seems to suggest a variance of $np/(1-p)$ rather than $2p/(1-p)$ – Henry Sep 08 '23 at 14:51

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