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For a non-Markovian random walk, each step can go up or down. And for the $i_{th}$ step, its step size $S_i$ may depend on the path of walk, and the probability for going up or down may also depend on the path of walk.

Let $T_n = \sum_{i=1}^nS_i$ be the displacement after a fixed number of steps $n$. Assuming the probability distribution: $P(T_n=t)$ will approach a given distribution.

Question: What kind of random walk (maybe with different step sizes, different step probability) can generate the given distribution in the limit (with some scale factors) ?

How can we use neural network to find such a random walk which can generate such expected distribution ?

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