A "space" of distributions between the flat and spike (general probabilistic models)

[Ross A. Finlayson]

Hi,

The idea here is that there are prototypical
probability distributions like the uniform for
the flat and the spike for a constant. In the
middle are distributions with centers, and,
say, wavy distributions. Then, there are convergence
results in this space like the Central Limit Theorem,
but only in the regions there, to begin to
explain the alpha-stable convergence and alpha-
astable convergence in terms of this overall
space and with then application of usual models
of dynamical systems.

Anyways I'm wondering who has already made, if
not "spaces" of distributions, beginnings and
ends of paths through distributions (of their
various types usually as over their types of
parameters as of their shapes) for that various
probability distributions may have reasonable
critical points as of chaos theory and general
differential equations where either of the
Central Limit Theorem (as a convergence theorem)
or spike, or some uniform limit theorem or moving
waves, would hold in a quantifiable general model
of a "space" of "all probability distributions",
then for usual general purposes.

[Simon Roberts]

If I understand you correctly, which I probably do not, there is the step function and the impulse function. These are used often in EE.

[Ross A. Finlayson]

This is more about how not every distribution
with an expected value adds up to the Gaussian
bell curve, instead some of them add up to a
uniform flat distribution.

Say there are about fifteen or so distributions,
where 1 pdf = 1 CDF = 1 MGF = 1 distribution, and
that given their various parameters then these
continuous (in R) and discrete (in N) distributions
have various shapes, then where they variously have
centralizing tendencies (gathering toward the mean)
which is a usual workup to the Central Limit Theorem,
or they may have dispersive tendencies (as for
example systematic approaches to remove bias).


Then the idea here is to make more systematic these
probability distributions, each, and then to have
the various centralizing or dispersive tendencies
of the composition of probability distributions or
of their random variables, here for models of
stochastic processes (random things).

https://en.wikipedia.org/wiki/Convergence_of_random_variables

(Here also looking for "dispersion"
as convergence to the flat)

https://en.wikipedia.org/wiki/Category:Probability_distributions

(Wiki's category on probability distributions)

[Ross A. Finlayson]

These days there's much more about
centralizing and dispersive tendencies
where there is to be some analog of the
central limit theorem for some uniformization
limit theorem.

(
Cf https://en.wikipedia.org/wiki/Large_deviations_of_Gaussian_random_functions
)

This reflects not only on the infinitary
vis-a-vis the very large (as about some
"laws of infinite numbers" for a reasonable
complement to "laws of large numbers"),
there is also about various alternatives
to Bayes, and then much for extensions
beyond standard probability theory to
explain as for parastatistics et cetera.