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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
$$\ …