Which one is more likely to be random walk? Consider the two series in the chart below: $walkA$ and $walkB$.
They are based on the same steps, although the steps come in a different order.
Indeed, $stepsA$ and $stepsB$ have identical sample mean $\hat \mu=0.5$ and sample std dev $\hat \sigma=3$.
However, from a visual inspection, I would intuitively say that $walkA$ is more likely to be a random walk than $walkB$.
Is it there a mathematical method to make such claims?

 A: A random walk has the property that its derivative should be pure uncorrelated noise. In other words, the autocorrelation of the steps for any lag should be 0.
We can permute the order of your steps a couple of times to obtain a distribution of autocorrelations for lag 1:

The lag 1 autocorrelation of your steps A is -0.5606 and for your steps B 0.7000.
I presume the high negative autocorrelation for steps A is because you deliberately tried to make your example appear as 'random' as possible and it therefore zig-zags around more than a truly random walk would? That is, you seem to have constructed something more akin to violet noise?
In any case, this distribution allows us to calculate the probability of finding a lag 1 autocorrelation as extreme as yours. One way to do this is to perhaps take the absolute value of the autocorrelations (as we had no clear hypothesis about the sign) and then to count how many simulated samples are as extreme as the observed ones. In my simulation 6.06% of samples were more extreme than steps A but only 1.16% of samples were more extreme than steps B.

A: The increments in a random walk are exchangeable, so it would be reasonable to use a permutation test to test the underlying hypothesis of exchangeability here.  The blue "walk" will certainly fail that test, which will alert you to the fact that it is not a random walk.
