Linear combination of random variables, convergence for a large number of variables

Jul 2019
18
0
Mumbai
Hi,

I have positive random variables X1, X2, X3, ..., Xn such that their sum=1 (so they are random, subject to constraints that each Xi is positive their sum has to be 1.. so all are fractions).

Now, I have a function f=C1.X1+C2.X2+C3.X3.....+Cn.Xn

where C1, C2, ....Cn are known coefficients (real numbers).

My question is: if n tends to infinity, will f converge to some number? Is there any law in statistics which talks about such a scenario?

Thanks.
 

romsek

Math Team
Sep 2015
2,969
1,676
USA
Suppose for a moment that your random variables have very small deviation, i.e. that they are essentially just positive real numbers. Further suppose that

$X_n \approx \dfrac{6}{\pi^2 n^2}$

Finally let $C_n = n$

I'll let you convince yourself the infinite sum of the $X_n$'s is $1$, and that
the weighted sequence does not converge.

You'll need to come up with some constraint on the $C_n$'s

After you've done that you'll need to address what type of convergence you want. There are 4 types of convergence with regard to sequences of random variables. You can wiki that.
 
Jul 2019
18
0
Mumbai
Thank you Romsek, I am actually a Chemical Engineer and a novice in statistics. I am learning statistics for a particular research area in my field. The X's I talked about are mass fractions of different chemicals, so their sum is always 1. I did the wiki search, is the "Central Limit Theorem" relevant for the case I described (convergence of linear combination of random variables)? Thanks.
 

romsek

Math Team
Sep 2015
2,969
1,676
USA
Thank you Romsek, I am actually a Chemical Engineer and a novice in statistics. I am learning statistics for a particular research area in my field. The X's I talked about are mass fractions of different chemicals, so their sum is always 1. I did the wiki search, is the "Central Limit Theorem" relevant for the case I described (convergence of linear combination of random variables)? Thanks.
Ok, so you want to cast these $X_n$'s as a discrete distribution.

And then ask if we take a zillion copies and average them will this converge to a normal distribution.

Again the problem is that unconstrained weights can cause divergence of the averaging sum as n goes to infinity.

If you can specify weights such that this doesn't occur then yes, the Central Limit Theorem will apply.