From Osher Doctorow
From Section 23.3:
1) [exp(-x) + P(AB)]^2 = exp(t) - 1
when the cumulative distribution function (cdf) is chosen rather than
the probability density function (pdf) for the random variable X that
generates time. As the value x of X approaches infinity, exp(t) - 1
approaches P(AB)^2, which I incorrectly stated indicates that t
increases as x increases. In fact, disregarding P(AB) momentarily, t
decreases as x increases.
To try to solve the problem, let's use Gumbel's bivariate exponential
cdf from my sci.physics posting of Jan. 3, 2006, "Gumbel's bivariate
exponential distribution in P(X-->Y)(x,y)," namely:
2) P(AB) = F(x,y) = 1 - exp(-x) - exp(-y) + exp(-x - y - kyx)
Substituting for P(AB) from (2) into (1), we obtain:
3) [1 - exp(-y) + exp(-x - y - kxy)]^2 = exp(t) - 1, k constant in [0,
1]
Since x is the value of the random variable X that generates time,
there is now no question but that as x --> infinity, t decreases from
(3) at least up to y. But what is y? Since y is the "caused" value
of a (random) variable that is caused by X, Y must be time and its
value y is time t. So now:
4) [1 - exp(-t) + exp(-x - t - kxt)]^2 = exp(t) - 1
If t --> infinity, the right hand side of (4) --> infinity but the left
hand side --> 1-, so if the model is correct t does not approach
infinity, suggesting a cyclic Universe of which the model is one part.
Does x --> infinity with fixed t? If it did, then (4) would yield:
5) [1 - exp(-t)]^2 = exp(t) - 1
This has the form u^2 - u + 1 = 0 with u = exp(t) - 1 which only has
complex non-real solutions except if t = 0 in which case we get 0 = 0.
So t cannot be fixed as x --> infinity, unless complex non-real time
has physical reality (not usually assumed) or unless t--> 0 as x -->
infinity.
If t --> 0 as x --> infinity, then (4) also yields 0 = 0 in the limit.
We conclude that in this model, as the random variable X generating
time --> infinity, time t approaches 0, so that for example the Big
Bang generated time (t = 0) at "x = infinity" for the value x of the
random variable X generating time.
Osher Doctorow
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