Threshold outcomes from continuous causes: exact statistics for a fast clock and for competing quantities

Many processes decide a discrete outcome by comparing continuous quantities that never repeat. We collect three exact statements about such threshold outcomes and check each by simulation. (i) If the outcome is "the fractional part of fT lies below p", where T is a continuous random time and f a clock frequency, the deviation of its probability from p is an explicit Fourier series in the characteristic function of T. For a Gaussian T with standard deviation sigma and s = f*sigma, the deviation is bounded by (2/pi) exp(-2 pi^2 s^2) / (1 - exp(-4 pi^2 s^2)), and this is sharp up to a factor close to 1: it falls below 1% once s >~ 0.46 and below 0.5% at s = 1/2, while for a Laplace spread it decays only like s^-2. (ii) The winner among several quantities with Gumbel noise has softmax probabilities. (iii) For independent trials of continuous causes, no cause value repeats almost surely, although the outcomes repeat and their frequency converges. The results are elementary consequences of the method of arbitrary functions and of the Gumbel-max property; the contribution is a set of explicit, checked statements with constants. The note makes no claim about any physical system.