TIL that the sequence of increasingly finegrained random walks that converges to Wiener process has walks that increase their speed as they get finer (speed grows with square root of scale). Post factum it seems kinda obvious: you'd otherwise converge to a constant function. I haven't yet figured out how this changes as one adjusts the exponent.
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Silly me, obviously for any larger exponent the thing diverges and for any smaller it converges to constant 0 (at least pointwise). It's sufficient to look at stddev of position at some fixed time.
