Iirc, bounded functions are finite and not all finite functions are bounded.
From Williams' Probability w/ Martingales:
Why 'finite 4th moment' ? It's technically right I guess, but why not 'bounded 4th moment' ?
2026-03-29 17:26:59.1774805219
First version of Strong Law - finite vs bounded
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1 up voted flag
In my view "bounded" is better as otherwise you might have XnXn taking the values ±n±n with equal probability, so with fourth moment n4n4 which is finite for each nn, but where Sn/nSn/n does not converge. – Henry Nov 3 at 13:26
@Henry Guess so. Thanks. But the book says finite instead of bounded... :| – BCLC Nov 7 at 8:48
It does, but it then uses KK as an upper bound – Henry Nov 7 at 14:59