CLT for bounded and dependent sequence

Question

Let $\displaystyle X_1,X_2,...X_n$ be identically distributed such that $\displaystyle Pr\{a \leq X_i\leq b\}=1$ for bounded constants $\displaystyle a,b$. Further Let $\displaystyle cov(X_i,X_j)=\alpha$, $\displaystyle E(X_i^2)=\mu_2<\infty$. What do we then know about the distribution of $$\frac{1}{\delta n}\sum_{i=1}^n(X_i-EX_i)$$ Is there a proper $\delta$ that we could choose to have some "known" (or "anything close to known") asymptotic distribution?

Answer 1

Every square integrable centered distribution can be a limit.

To see this, assume that, for every $i$, $$X_i=B_iZ+(1-B_i)Y_i,$$ where $Z$ is centered, $(Y_i)$ is i.i.d. and centered, $(B_i)$ is i.i.d. Bernoulli with $P(B_i=1)=p$ and $P(B_i=0)=1-p$ for some $p$ in $(0,1)$, and $Z$, $(Y_i)$ and $(B_i)$ are independent.

Then, for every $i\ne j$, $$E(X_i)=0,\qquad\mathrm{var}(X_i^2)=pE(Z^2)+(1-p)E(Y^2),\qquad\mathrm{Cov}(X_i,X_j)=p^2E(Z^2).$$ Furthermore, $$S_n=\frac1n\sum_{i=1}^n(X_i-E(X_i))=R_nZ+T_n,\qquad R_n=\frac1n\sum_{i=1}^nB_i,\quad T_n=\frac1n\sum_{i=1}^n(1-B_i)Y_i.$$ The usual strong law of large numbers applies to $(R_n)$ and $(T_n)$ and shows that $R_n\to p$ and $T_n\to0$ almost surely, hence $$S_n\to pZ\quad\text{ almost surely}.$$ Another example is to pick some $Z$ and $(Y_i)$ with $(Y_i)$ i.i.d. and independent of $Z$ such that $E(Z)=0$ and $E(Y_i)=1$, and to consider $$X_i=Y_iZ.$$ Then $E(X_k)=0$, $E(X_k^2)=E(Y_1^2)E(Z^2)$, $\mathrm{Cov}(X_i,X_j)=E(Z^2)$, and $$S_n\to Z\quad\text{ almost surely}.$$ Edit: More generally (and this might answer a comment), assume that $X_i=u(Y_i,Z)$ for some $Z$ and $(Y_i)$ with $(Y_i)$ i.i.d. and independent of $Z$, then $$\frac1n\sum_{i=1}^nX_i\to v(Z)\quad\text{almost surely},$$ where the function $v$ is defined as $$v(z)=E(u(Y_1,z)).$$ Furthermore, for every $i\ne j$, $\mathrm{Cov}(X_i,X_j)=E(v(Z)^2)$.