I'm trying to do the following problem from Robert Fernholz's textbook Stochastic Portfolio Theory:

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The assumptions mentioned are:

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and my attempt is as follows:

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I have no idea how to deduce that the limsup of the last expression is strictly positive. The middle term is obviously non-negative and in fact tends to zero in the case where the market is also coherent, in which case the result follows iff the limsup of the first term is strictly positive. We know that the liminf of the first term is non-negative from proposition 2.2.8 of the text as well, but I have no idea how to get this to be strictly positive with the diversity assumption.


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