Testing Random Walk in the stock market

-
Well the weak form efficiency in the stock market hinges on the assumption that information flow is random. Information flow is the previous stock price.
So my current stock prices are random since they were previously random since they were previously random....
Ah, the familiar iterative cycle.

Now what if I set up my stock market such that, at the start prices are not random.
Or rather wouldn't the best test for weak form be to check the condtions at the origins rather than just checking the stationarity/non-stationarity of current market returns?

Have got my basics horribly mixed up???