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Let the nonnegative random variables $X$ and $Y$ have distribution functions $F$ and $G$ and density functions $f$ and $g$, respectively. Suppose $X$ is second-order stochastically dominant over $Y$, that is for all $a\in\mathbb{R}$ $$ \int_a^\infty [1-F(x)] dx \geq \int_a^\infty [1-G(x)] dx $$ (This implies $\mathbb{E}[X^k]\geq\mathbb{E}[Y^k]$ for any $k\in\mathbb{N}$, see here.) Further, $$ \lim_{x\rightarrow\bar x} \frac{g(x)}{f(x)} = 0 \hspace{1cm} \bar x = \max X $$

Define the nonnegative random variables $X_k$ and $Y_k$ by the density functions $$ f_{X_k}(x) = \frac{x^k f(x)}{\int_0^\infty y^k f(y) dy} = \frac{x^k f(x)}{\mathbb{E}[X^k]} \hspace{1cm} g_{X_k}(x) = \frac{x^k g(x)}{\int_0^\infty y^k g(y) dy} = \frac{x^k g(x)}{\mathbb{E}[Y^k]} $$ I am stuck on proving that $X_k$ is second-order stochastically dominant over $Y_k$ for all $k\in\mathbb{N}$.

Note: I cannot find a discrete counterexample.

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