Chess engines such as StockFish provide centi-pawn evaluation of positions. Clearly the longer the engine runs the deeper it can search and thus provide more accurate centi-pawn evaluation.
Basically, given a random position (say of equal material), I seek to know how the centipawn evaluation changes as a function of the depth parameter. When the depth is very low (say 2-5 moves) the evaluations expected to be erratic and change dramatically, but how does it look like on higher depth (say 20-30 or even higher) - do we tend to see convergence to a value? In other words, is it the case that we learn less and less about the position the deeper we go?
Interestingly, in both cases having the engine run long enough is somewhat futile (unless we are looking for a forcing mate), for (1) if depth 30 is good enough and we learn less and less the deeper we go, why bother to reach depth 40? Or (2) if the other case is true: namely, that there could be some significant difference between depth 30 and depth 40 evaluation, why stop at 40? who promise us that at depth 50 the evaluation would be closer to the 40 value than of the 30?