I have two questions related to positive real numbers:
- If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner product can be approximated as:
<a, b> = sum(a) sum(b)/N
a and b are independent but not necessarily identically distributed and N can be as large as needed. checking this in Matlab, it seems to work all the time, except when either or both of a and b contain negative values.
- We have c1 and c2 are positive scalar constants, while x, y and z are vectors of size N by 1 each containing positive real values. If (Matlab notations)
c1 sum(x.*y./z.^2) > c2 sum(y./z.^2),
is it possible to prove that
c1 sum(x.*y./z.^3) > c2 sum(y./z.^3)?
Any form of help will be much appreciated.
For any vector $\mathbf x$ let $r^k(\mathbf x)$ be the vector obtained by rotating the components by $k$ positions. Then $\sum\mathbf a\cdot\sum\mathbf b=\sum_{k=0}^{N-1}\langle \mathbf a,r^k(\mathbf b)\rangle$. Hence at least if the distribution for the various components are the same and are independent (but otherwise unspecified), the approximation $\langle \mathbf a,\mathbf b\rangle=\frac 1N\sum \mathbf a\sum\mathbf b$ is somewhat justified: It is the expected scalar product between random coordinate permutations of the given vectors.