Given the time series for a particular stock market, what are the statistical weapons one can bring on to prove, or disprove that random walk hypothesis?
Test your historical time series for both randomness and independence. Understand that a time series may be random and independent; non-random and independent; random with dependencies; and non-random with dependencies. A mistake would be to limit dependency tests to autocorrelation. The most general test I know of is called the differential spectrum by Sherry, which works like this:
- histogram the price changes in your time series
- if the price changes are independent, they should be symmetric about 0
- use Pearson's $\chi^2$ test with one
sign as "observed" and the other as
"expected" for a quantitative measure of symmetry.
However, when you find something with this test, you've still got to hunt for the dependency. But at least the test can tell you if you've got a dependencies or not.
The most important point is that whatever tests you work with, that market can change in the future. So I wouldn't say a market "is" or "isn't" a random walk. Rather, it may phase in and out of random-walkiness for indeterminate amounts of time.
The first chapter of the book Econometrics of Financial Markets by Campbell, Lo and MacKinlay discusses this very well.