Suppose $\sum_i^d{x_i}^2 \leq \sum_i^d{y_i}^2$. What can we say about $\sum_i^da_i^2x_i^2$, compared with $\sum_i^da_i^2y_i^2$. Clearly, as the number of dimensions, d, goes larger, then we can be more confident that $\sum_i^da_i^2x_i^2 \leq \sum_i^da_i^2y_i^2$. How can this be stated in a mathematical/probabilistic way? Also, suppose $x_i,y_i\in \mathbb{R}$, and $a_i \neq 0$, or make other assumptions if it is needed.
Suppose x=(1,1) and y=(4, 0), then the choice of (0.1, 1), would make $\sum_i^da_i^2x_i^2 > \sum_i^da_i^2y_i^2$. But if the number of dimensions was 100, then an a vector like (0.1, 0.1,...,1), with very small numbers for most dimensions, and a large number for one dimension would be very unlikely, considering a is from an assumed distribution.