Using Expander Graphs to test whether samples are i.i.d

The purpose of this note is to point out that the theory of expander graphs leads to an interesting test whether $n$ real numbers $x_1, \dots, x_n$ could be $n$ independent samples of a random variable. To any distinct, real numbers $x_1, \dots, x_n$, we associate a 4-regular graph $G$ as follows: using $π$ to denote the permutation ordering the elements, $x_{π(1)} < x_{π(2)} < \dots < x_{π(n)}$, we build a graph on $\left\{1, \dots, n\right\}$ by connecting $i$ and $i+1$ (cyclically) and $π(i)$ and $π(i+1)$ (cyclically). If the numbers are i.i.d. samples, then a result of Friedman implies that $G$ is close to Ramanujan. This suggests a test for whether these numbers are i.i.d: compute the second largest (in absolute value) eigenvalue of the adjacency matrix. The larger $λ- 2\sqrt{3}$, the less likely it is for the numbers to be i.i.d. We explain why this is a reasonable test and give many examples.

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