If $\frac{p_0(x)}{p_1(x)}$ is a non-decreasing function of $x$, how can I prove that $\frac{p_0(x)^{*n}}{p_1(x)^{*n}}$ is a non-decreasing function of $x$, where $f^{{*n}}(x)=\overbrace {f(x)*f(x)*\cdots *f(x)}^{n}$ and $*$ denotes the convolution.

The question comes from statistical hypothesis testing, specifically the likelihood ratio test (LRT):

The random variables $X_i$ are drawn from $p_0(x)$ (under $H_0$) and $Y_i$ are drawn from $p_1(x)$, under $H_1$. If I observe a single random variable and want to decide whether $H_0$ is true or $H_1$, I can use the LRT $\frac{p_0(x)}{p_1(x)}$ that yields the optimal test because I know it is a non-decreasing function of $x$.

Now, if I observe $n$ random variables, how can I show that the LRT based on the sum of them is still optimal?


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