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S13g. How probable are realizations of stochastic processes?
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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.