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I have looked around the literature but I've not found a clean answer to the following. Imagine that you have a generalized PĆ³lya urn (GPU) in the sense of Pemantle's survey (Section 2.1 in Pemantle07). What is the expected time until the ratio of white balls to black balls is equal to $p$ (assuming that the initial ratio is $p_0$)? An answer for the much simpler case of the GPU with a diagonal $A$ matrix would also be interesting. A non-trivial upper bound would also suffice.

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There is more than one way to generalise a Pólya urn, so you may need to give more detail. In many cases I would think the answer to your question will be an infinite expectation or worse. –  Henry Sep 11 '11 at 17:00

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