I have the following product:
$$\prod_{n=1}^N \frac{A_n}{B_n}$$
, where $A_n$ and $B_n$ are the nth element of the finite sequences {$A_x$} and {$B_x$} respectively. I'd like to know the conditions under which this product is higher than one. Note: the two sequences are just random numbers and do not follow some logic pattern.
It probably has to do with the comparison of some central value of the two series. But I don't know if I have to compare the two arithmetic means, the harmonic means, the geometric means, …
At the risk of belaboring the obvious, let us note that this holds if and only if $$ \left(\prod_{n=1}^NA_n\right)^{1/N}\gt\left(\prod_{n=1}^NB_n\right)^{1/N}. $$ The LHS is the geometric mean of $(A_n)_{1\leqslant n\leqslant N}$ and the RHS is the geometric mean of $(B_n)_{1\leqslant n\leqslant N}$.