# Thread: Var(X1 + X2 +...) = Var(X1) + Var(X2) +...?

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

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!

2. ## Re: Var(X1 + X2 +...) = Var(X1) + Var(X2) +...?

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.

3. ## Re: Var(X1 + X2 +...) = Var(X1) + Var(X2) +...?

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

If we let be a sequence of independent random variables where for all and where

.

Then (the sum) converges almost surely to a limit .

4. ## The Following User Says Thank You to Dragan For This Useful Post:

kingwinner (08-10-2011)

5. ## Re: Var(X1 + X2 +...) = Var(X1) + Var(X2) +...?

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

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts