Mean, variance and central limit conjecture for $D_{i,K}$

For each $i\in\mathbb N$ and strip size $K$, let $D_{i,K}$ be the random variable defined in the paper, with expectation $E(D_{i,K})$ and standard deviation $\sigma(D_{i,K})$. For $i\geq1$, define the standardized variable

$$\widetilde D_{i,K}=\frac{D_{i,K}-E(D_{i,K})}{\sigma(D_{i,K})}.$$

Let ${\mathcal L}(\widetilde D_{i,K})$ denote its law, and let ${\mathcal N}(0,1)$ denote the standard normal distribution. Mean, variance and central limit conjecture for $D_{i,K}$. (i) For every $i\in\mathbb N$, there exist $K_i\in\mathbb N$ and positive $a_i,b_i\in\mathbb Q$ such that

$$E(D_{i,K})=a_iK\qquad\text{and}\qquad\sigma(D_{i,K})=b_iK$$

for all $K>K_i$. (ii) For $i\geq1$,

$$\lim_{K\to\infty}{\mathcal L}(\widetilde D_{i,K})={\mathcal N}(0,1).$$

The conjecture is motivated by the generating-function calculations and intensive numerical simulations described in the paper. It asserts eventual exact linearity of the mean and standard deviation in $K$, together with a standard central limit theorem; the supplied text gives no proof or resolution.