In cotype-2 Banach spaces, weak convergence of S_n/sqrt(n) is equivalent to convergence along a subsequence m_n, and failure of the full sequence implies no rescaling yields a non-degenerate limit.
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Some Results on the Central Limit Theorem for Subsequences in Banach Spaces
In cotype-2 Banach spaces, weak convergence of S_n/sqrt(n) is equivalent to convergence along a subsequence m_n, and failure of the full sequence implies no rescaling yields a non-degenerate limit.