Var(X1 + X2 +...) = Var(X1) + Var(X2) +...?

**kingwinner** · 07-23-2011 07:08 AM

Suppose the random variables {X_i} are independent,
then is it always true that Var(X_1 + X_2 + X_3 +...) = Var(X_1) + Var(X_2) + Var(X_3)+...?

Note that I'm talking about the case of
∞
Σ X_i
i=1

I'm sure it's true for finite series, but how about infinite series?
I tried searching the internet, but can't find anything...

Any help is appreciated!

**BGM** · 07-23-2011 08:06 AM

I think you need to require the sum of random variables converge to a random variable
with finite variance. But I am not sure the necessary/sufficient condition for it to hold.
It is just my guess.

**Dragan** · 07-23-2011 01:30 PM

I believe this theorem will do (it's a basis for the strong law of large numbers).

If we let $\left \{ x_{n} \right \}_{n=1}^{\infty }$ be a sequence of independent random variables where $E\left ( x_{n} \right )=0$ for all $n\in\mathbb{N}$ and where

$\sum_{n=1}^{\infty }E\left ( x_{n}^{2} \right )< \infty$ .

Then (the sum) $S_{n}$ converges almost surely to a limit $S$ .

**akbar** · 07-25-2011 02:12 AM

First, type of convergence of { Un } should be clarified , if we denote Un = X1+X2+…..+Xn+ …

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