Uniform Central Limit Theorem for self normalized sums in high dimensions
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In this article, we are interested in the normal approximation of the self-normalized random vector (∑_i=1^nX_i1/√(∑_i=1^nX_i1^2),…,∑_i=1^nX_ip/√(∑_i=1^nX_ip^2)) in ℛ^p uniformly over the class of hyper-rectangles 𝒜^re={∏_j=1^p[a_j,b_j]∩ℛ:-∞≤ a_j≤ b_j≤∞, j=1,…,p}, where X_1,…,X_n are non-degenerate independent p-dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of log p in the uniform central limit theorem (UCLT) under variety of moment conditions. When X_ij's have (2+δ)th absolute moment for some 0< δ≤ 1, the optimal rate of log p is o(n^δ/(2+δ)). When X_ij's are independent and identically distributed (iid) across (i,j), even (2+δ)th absolute moment of X_11 is not needed. Only under the condition that X_11 is in the domain of attraction of the normal distribution, the growth rate of log p can be made to be o(η_n) for some η_n→ 0 as n→∞. We also establish that the rate of log p can be pushed to log p =o(n^1/2) if we assume the existence of fourth moment of X_ij's. By an example, it is shown however that the rate of growth of log p can not further be improved from n^1/2 as a power of n. As an application, we found respective versions of the high dimensional UCLT for component-wise Student's t-statistic. An important aspect of the these UCLTs is that it does not require the existence of some exponential moments even when dimension p grows exponentially with some power of n, as opposed to the UCLT of normalized sums. Only the existence of some absolute moment of order ∈ [2,4] is sufficient.
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