Suppose in a probability space $(X,\Omega,\mu)$, let $X_1,X_2,...X_n$ be independent random variables. How do I prove that $X_1+X_2+...+X_{i-1}$ is independent of $X_i+...+X_n$ for some $1\leq i \leq n$ ?

  • $\begingroup$ Write down explicitly the definition of "independence of those two disjoint sums" and see if you can verify it. It may also help to write down explicitly the definition of "$X_1, \ldots, X_n$ are independent." $\endgroup$ – angryavian Nov 1 '18 at 3:28
  • $\begingroup$ math.stackexchange.com/questions/8742/… $\endgroup$ – d.k.o. Nov 1 '18 at 3:55

Demonstrate that $X_a+X_b, X_c$ are independent if $X_a,X_b,X_c$ are mutually independent.

Use mathematical induction.

  • $\begingroup$ I got your point. But how can I show that $X_1+X_2$ independent of $X_3$ here ? $\endgroup$ – Ganesh Gani Nov 1 '18 at 9:29

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