APL (Dyalog Unicode), 12 9 bytes
Anonymous tacit prefix function. Assumes ⎕IO (Index Origin) to be 0, which is default on many systems. Returns the final value (0) in addition to printing while run.
{⌊?⎕←⍵}⍣= {…}⍣= apply the following function until stable:
⎕←⍵ output the argument
? return a uniformly distributed random number in the range 0 through that–1
⌊ round down (because ?0 gives a (0,01) float)