If $\mu_n$ are probability measures converging weakly to a probability measure $\mu$ and we also have convergence of even moments,$$\int x^{2k} \ d\mu _n\rightarrow c_k+O(1/n)\ ,\ \forall k\in \mathbb{N}$$ Can one conclude this rate of convergence, $\ O(1/n)$, holds for $\int |x|^\alpha\ d\mu_n$ with $\alpha<1$? Are there any known conditions to get such rate of convergence?
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3$\begingroup$ It seems taking dirac measures $\mu_n = \delta_{1/\sqrt{n}}$ is a counterexample: The limit is $\delta_0$ but higher moments converge with a faster rate. $\endgroup$– SteveCommented Jun 10, 2021 at 9:06
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$\begingroup$ Oh God, I am an idiot. Thanks $\endgroup$– jnyanCommented Jun 10, 2021 at 9:48
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