------------------------------------------------------------ S13g. How probable are realizations of stochastic processes? ------------------------------------------------------------ In a stochastic setting, _every_ realization of a stochastic process typically has probability 0; nevertheless, exactly one of them actually happens. Taking for simplicity the stochastic process defined by independent flips of a fair coin, a realization is an infinite binary sequence, and each of these has probability zero. (Partial realizations of finite length N each have a probability of 2^-N which is extremely tiny for large N.) For discrete stochastic processes having a continuum of allowed values at each time step, even partial realizations have zero probability, except in degenerate situations. The same holds for continuous-time stochastic processes. The case of measuring electron spin, say, is more difficult to analyze because as stated, it is not yet a well-defined stochastic process. If it is taken as a continuous measurement, the flips occur at random times, and so even a single flip at a definite time has probability zero. If it is taken as a discrete process, we need to specify a measuring protocol that applies at definite, equidistant times. Then it is likely that there are some correlations, and probabilities even of finite pieces of a particular realization are hard to get by. Nevertheless, under reasonably random circumstances (for example, when measuring spins of independent electrons), the probability of the most likely sequence of N measurements decreases exponentially with N, and the probability of a complete realization (infinite sequence) is again zero.