Conjecture on fluctuation regimes for the proportion of heads

Let $p_n\in [0,1]$ for $n\ge 1$, and let $E(N,2)$ denote the quantity governing the covariance contribution in the variance formula

$$\operatorname{Var}(S_N)=N+2N^2E(N,2).$$

The proportion of heads is centered at $1/2$.

Fluctuation-regime conjecture. (i) If $E(N,2)=\mathcal{O}(1/N)$ as $N\to\infty$, then the proportion of heads obeys a central limit theorem. (ii) If $E(N,2)=\mathcal{O}(1/N)$ fails but $E(N,2)=o(1)$ holds, then there is a non-standard central limit theorem for the proportion: its fluctuation about $1/2$ is larger than order $\sqrt{N}$. (iii) If $E(N,2)=o(1)$ fails, then the weak law of large numbers is no longer valid for the proportion, which is not concentrated about $1/2$.

The conjecture proposes a trichotomy linking the decay of the dependence term $E(N,2)$ to central-limit and weak-law behavior. The source provides heuristic motivation and examples, but no resolution of the three assertions; the status is therefore open.