Let us consider a boolean hypercube $C = \{-1, 1\}^n$. Let $S = \{x \in C \mid |\{i \mid x_i = -1\}| = \varepsilon n\}$ be a Hamming sphere in $C$ (here $\varepsilon$ stands for the fixed parameter from $(0, 1/2)$). Let us sample $X \in S$ uniformly at random. And we are interested in estimating $\mathrm{E}[X_1 X_2 \ldots X_{\alpha n}]$, where $\alpha$ is a sufficiently small constant.

We would like to argue that this expectation is somewhat close to $(1 - 2\varepsilon)^{\alpha n}$ (that could have been the case if $X_i$'s were independent).

Is there any way to avoid tedious computations here? It is tempting to say that $X_i$'s are "almost independent" and use some "limit theorem".

Or is it possible to replace $X_i$'s by properly correlated gaussians and argue that expectation doesn't change much?

After all, I feel that this question must be considered somewhere, so it would be nice to find an appropriate reference.