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$X(t)$ is a stochastic process defined on the time interval $(0, T)$. Discretizing the time interval one specify the time instants $t_0=0 < t_1 < t_2,\cdots,< t_{n-1} < t_n = T$.

A random variable $X(t_i)$ may be considered as being dependent on the previous random variables $X(t_1), X(t_2), \cdots , X(t_{i-1})$ and $X(t_i)$ may be considered as independent on the random variables that follows in time $X(t_{i+1}), \cdots, X(t_n)$ ?

The conditional probabilities are written P(X(ti)<x/ X(ti-1))= P(X(ti-1),X(ti)<x) / P(X(ti-1) and P(X(ti)<x/ X(ti+1))= P(X(ti)<x,X(ti+1)<x) / P(X(ti+1)<x)= P(X(ti) but they end up contradicting each other since one has taken P(X(ti)<x/ X(ti+1))= P(X(ti)<x).

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    $\begingroup$ Welcome. Sorry, this is very hard to read. I suggest formatting your equations properly using Mathjax. See math.meta.stackexchange.com/questions/5020/… $\endgroup$
    – Hilmar
    Jan 26 at 21:43
  • $\begingroup$ I have undertaken to edit your first two paragraphs -- consider using that as a template for your last one. Note that $\frac{P(x|y)}{P(y)}$ will render as "$\frac{P(x|y)}{P(y)}$" -- if that helps. $\endgroup$
    – TimWescott
    Jan 27 at 1:47
  • $\begingroup$ Even though some probability texts use caps for the actual random variables, I would discourage that for time-domain random functions. It might confuse. $\endgroup$ Jan 27 at 2:35

1 Answer 1

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If $x$ and $y$ are random variables then they are either mutually independent or they are mutually dependent -- if $x$ depends on $y$ then $y$ depends on $x$.

Normally stochastic processes are modeled as some white process that is the input to some system, resulting in an output process that has time dependency.

If you believe in a world without time travel, then you would model that system as being causal and you would say that any values $X(t_i)$ depend strictly on values of the underlying white process before $t = t_i$. However, for a system with memory, the output process samples at times $t > t_i$ still contain information about the input process at times $t \le t_0$; as a consequence there is correlation -- and lack of statistical independence -- between samples $X(t_{i - m})$ and $X(t_{i + m})$.

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