Variance ratio and Hurst exponent tests

I have used the Chow Denning test and the Hurst exponent (Peng, Whittle and R/S methods) to examine if a particular time series follows a random walk. My results are conflicting between the 2 tests. From what I can fathom, the Hurst exponent does account for multiple variances (although I have not seen any journal article that explains this)

1) Does this imply that a time series can be "random" under the mean but "non-random" under the variance (or vice versa)? 2) Is the Hurst exponent seen as a superior test compared to the variance ratio when testing for random walk behaviour?