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Convergence of series, sequences and functions and different modes of convergence.

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Central limit theorem with limit of functions

Suppose that $$\sqrt{n}(X_n - \theta)\xrightarrow{d} X,$$ according to the delta method, we have $$\sqrt{n}(g(X_n)-g(X))\xrightarrow{d} g'(\theta)X$$ when $g$ is differentiable. My question is, if $$\ …
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