# When discretizing a stochastic signal , are there random variables independent of those that follow in time and dependent on previous ones in time?

$$X(t)$$ 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).

• Welcome. Sorry, this is very hard to read. I suggest formatting your equations properly using Mathjax. See math.meta.stackexchange.com/questions/5020/… Jan 26 at 21:43
• 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. Jan 27 at 1:47
• Even though some probability texts use caps for the actual random variables, I would discourage that for time-domain random functions. It might confuse. Jan 27 at 2:35

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 $$x$$.
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 $$X(t_{i + m})$$.