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An urn contains $N$ red balls and $N$ black balls. Consider the game in which you sequentially draw balls from the urn:

a) with replacement; b) without replacement, until the $2N$ balls are all drawn; c) replacing the opposite color.

For each red ball, you gain $1\$$ dollar, and get $1\$$ fined for each black ball.

It is known that the evolution of the capital at time $n\in\mathbb{N}$ is a discrete version of the: a) Brownian motion b) Brownian bridge c) Ornstein-Uhlenbeck. I wonder what should be the mechanics of the game for getting the discrete version of an Ornstein-Uhlenbeck bridge. Moreover, can we design dynamics like a)-c) to match any Gaussian Process (with replacement) and Bridges (without replacement) derived from them.

I know that my question is not that specific. But I just wonder how far can we extend this type of "urn game". I think that, at the core, this question gets boiled down to Random Walks approximating continuous tiume Gauss-Markov processes. Some version of the CLT, like the functional Donsker's theorem

Aguazz
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  • Can you provide detailed references for what you described as "It is known that ...". Intuitively, I think these questions/games can be ultimately related to some sort of "mean-field limits". – Fei Cao Jan 05 '22 at 00:44
  • Designing "games" to match pre-determined target (random) objects is usually a very hard type of "inverse problem". – Fei Cao Jan 05 '22 at 00:51
  • Hi @FeiCao. I attached links to parts b) and c) to back up that "It is known that" claim. For part a) is easier to see the similarity with the Brownian motion. I'm not aware of any other reference for c). However, for b) the problem traced back to this paper by L.A. Shepp. – Aguazz Jan 05 '22 at 11:19
  • Thanks, based on the references you shared, I still have no clue how to tackle you problem. I wish someone can help you on this problem – Fei Cao Jan 05 '22 at 17:35
  • (b) is hardly a game: you end up in the end with $$0$. But another version for the Brownian bridge may be to play (a) and then retrospectively spread the final result as equal negative amounts across the $2N$ rounds. Can you do something similar with (c) for the Ornstein-Uhlenbeck bridge? – Henry Jan 06 '22 at 00:41

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