If the white noise is based on a random number generator, the pattern would restart after a full cycle.
If the random number generator's period is 2^32 and you use one integer per sample, then at 44 kHz, you would have about a million seconds, or twelve days, before the RNG has gone through a full period.
Most RNGs have periods much higher than this. xorshift128 has a period of 2^128-1 and the Mersenne Twister's period is 2^19937-1.
So you could push it out until practically forever.
