In the condition of the market being a perfect random walk, the price change (in %) from an arbitrary time t to t+n should be a normal distribution as n->0. If the market is said to be perfectly efficient, the above situation would be true.

The following code attempt to verify this by collecting individual trades and plotting them against price in a volume-weighted histogram during a given time window. We then fit a normal distribution to this data to check the validity of the claim above.
(Technically we would have to fit a log-normal distribution in a price-volume histogram, but as long as n is small, the difference would be minute)

I have not found a rigorous proof for the statement regarding market random walk being normally distributed, but it does intuitively (and empirically) seem to be that way