Re: How Chapman-Kolmogorov implies Markov ??
- From: "rayohauno@xxxxxxxxxxxx" <juanpool@xxxxxxxxx>
- Date: 10 Feb 2007 15:06:52 -0800
On 10 feb, 12:37, "Edward Green" <spamspamsp...@xxxxxxxxxxx> wrote:
On Feb 9, 7:56 am, "rayoha...@xxxxxxxxxxxx" <juanp...@xxxxxxxxx>
wrote:
It's easy to see that a stochastic process that follows the Markov
property, then follows the Chapman-Kolmogorov equation. But the visce
versa holds too, and I can find a proof of that. Some one of you can
show me a proof ??
Before you fade back into obscurity, perhaps you would help me
understand the question.
Starting with
http://en.wikipedia.org/wiki/Chapman-Kolmogorov_equation
I run into some conceptual problems. An "indexed set of random
variables" is almost an empty concept, being, I suppose, a set of
random variables each identified with an associated element of an
index set. Perhaps the notation "f_i" is supposed to imply there is
something equivalent about all of these random variables, though what
this is is undefined. Identical marginal distributions?
Now, when we begin labeling a set of the indexes, "i_1, i_2, ...,
i_n", I get lost. We have in mind again, of course, an almost empty
concept, that we index the index set iself, or else write a finite
sequence in it, explicitly using the integers as our indices.
It occurs to me now (for the first time? well... experience is
infinite/life finite), that indexing a set by the integers and writing
a sequence in it are related but distinct concepts: the sequence may
reuse elements, while indexing implies a one-to-one relation -- we
don't say "oh, x_3 and x_7 are the same variable... did I forget to
mention that"?
Which is intended?
Anyway, I get the feeling, not unknown in mathematical arguments, that
the formal exposition has run ahead of the sense -- the author knows
what he is trying to capture, I don't! Rather than trying to reverse
engineer his intentions through the ambiguities, would you possibly be
so kind as to fill in story?
Please assume I am just sophisticated enough to understand the
intention, though not conversant with it.
(P.S. Thanks for posting a real question, anyway)
sorry by my horrible notation .. it´s dificult to do so with only ISO
text ...
i will define my notation :
p_(k|r)(y_1,t_1;...;y_k,t_k|y_(k+1),t_(k+1);...;y_(k+r),t_(k+r))
is the conditional probability that the stochastics variables
Y_1,...,Y_k gives values
y_1,...,y_k at respectives times t_1,...,t_k if the stocastics
variables Y_(k+1),...,Y_(k+r) gives
values y_(k+1),...,y_(k+r) at respectives times t_(k+1),...,t_(k+r)
also
p_n(y_1,t_1;...;y_n,t_n)
is just a joint probability that stochastics variables Y_1,...,Y_n
gives values y_1,...,y_n at times t_1,...,t_n respectively.
.
- References:
- How Chapman-Kolmogorov implies Markov ??
- From: rayohauno@xxxxxxxxxxxx
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