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Suppose $ X_1,X_2,…$ are Bernoulli random variables with $ P(X_i=1)=p_i$ and $ X_i$ have negative correlation. Is there a CLT in this case, i.e. does $ \frac{Z_n-(\Sigma^n_{i=1}p_i)}{\sqrt{n}}$ converge to a Gaussian in distribution? And if we add the assumption that $ \underset{n \longrightarrow \infty}{lim} \frac{\Sigma^n_{i=1}p_i}{n}=p\in(0,1)$ , is it true then?Read more

Consider the distribution given by taking the top singular vector of a matrix whose entries are equally likely to be +1 or -1. This is not well-defined for matrices where the subspace achieving the top singular value is >1-dimensional, but we can ignore this case since it occurs with negligible probability. Can we say anythingRead more