Define this 1-D discrete random walk start from 0: roll a die (the die may or may not be fair, the fixed probability of each face is unknown to the observer/guessor), if $d\in\{1,2,3\}$, go left for 1 step, 2 steps or 3 steps respectively; if $d\in\{4,5,6\}$, go right for 1, 2 or 3 step(s). (This is also all the known information to the guessor)
For a finite sequence from this random walk, is there any way we can guess a sequence to certainly/always achieve positive Pearson correlation? If so, what's the maximum Pearson correlation can such a guessing strategy achieve? (Of course the guessor knows the length of the sequence)
Moreover, if the guessor knows the die is fair in advance, is there any simulation based approach guarnteed to achieve positive correlation? What about almost surely positive?
I've read this post, but don't find it very helpful here.
