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you might be interested in the following :

http://library.utia.cas.cz/separaty/2015/E/kristoufek-0452316.pdf

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Thanks for the article Fabrice Clerot.

My question is slightly different though. Owing to the randomness of two time series, the Hurst exponents of the two series will be random variables. I wish to analyze the correlation between the two random variables (Hurst exponents) given the correlation between the two random processes (time series).

I don't know if there are analytic solutions to this question--maybe there are--but a simulation exercise using synthetic time series might be helpful and constructive way of developing some intuitions on the relationships of interest. 

I do agree with your suggestion Matthew Salomom. We have in fact performed a simulation exercise, and have an intuitive explanation of the results. I  need a framework to (possibly) prove it analytically. The challenge is handling the log operation in obtaining the Hurst exponent. Maybe I will start with arguments and see if they can be built into a proof.

Thanks, and I do appreciate and welcome discussions.

Thanks again!!