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Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let us just take $a_i=1$ and $b_i=i$. Write $X=\sum_{i=1}^{n}a_iX_i$ and $Y=\sum_{i=1}^{n}b_iX_i$. Questions:

1) What is the simplest way to show (maybe a result to quote) that the vector $(X,Y)$, appropriately scaled, is asymptotically jointly Gaussian?

2) Can one show that for "decent" (smooth, bounded) functions $f,g$ we have $E f(X)g(Y)= (1+o(1))E f(Z_1)g(Z_2)$, where $Z_i$ are Gaussian variables such that $E Z_{1}^2=E X^2$, $E Z_{2}^2=E Y^2$ and $E Z_1Z_2=E XY$?

I am sure this is fairly standard, but I would very much appreciate a useful reference.

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    $\begingroup$ 1) Using Carmer-Wold criterion and Lyapunov CLT it is easy to show that $(Xn^{-1/2},Yn^{-3/2})$ converges to a centered Gaussian vector with covariance matrix $\begin{pmatrix} 1 & 1/2\\ 1/2 &1/3 \end{pmatrix}$. 2) No, because $X$ and $Y$ are integer valued. If, however, you scale them, then the convergence would follow from the CLT. $\endgroup$
    – zhoraster
    Commented Dec 14, 2014 at 12:56
  • $\begingroup$ zhoraster: thank you! The answer to #1 is very helpful. I am not sure why it is not true for #2. Say $f,g$ are both bounded and continuous, or, if necessary - Lipschitz, why would discreteness matter? I would see why it would fail for "bump" functions that look like an indicator of a very short interval, but what about polynomials of fixed degree? $\endgroup$
    – TOM
    Commented Dec 15, 2014 at 10:23
  • $\begingroup$ And, if, say, I rescale them, is there a corresponding Berry-Esseen bound available (proved in some paper)? $\endgroup$
    – TOM
    Commented Dec 15, 2014 at 11:29

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