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From Wikipedia, Anderson's theorem is stated as:

Let $K$ be a convex body in n-dimensional Euclidean space $\mathbb{R}^n$ that is symmetric with respect to reflection in the origin, i.e. $K = −K$. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a non-negative, symmetric, globally integrable, unimodal function. Then, for any $0 ≤ c ≤ 1$ and $y ∈ \mathbb{R}^n$, $$ \int_{K} f(x + c y) \, \mathrm{d} x \geq \int_{K} f(x + y) \, \mathrm{d} x. $$

Then Wikipedia also gives its application to probability theory:

Given a probability space $(Ω, Σ, P)$, suppose that $X : Ω → \mathbb{R}^n$ is a random variable with probability density function $f : \mathbb{R}^n → [0, +∞)$ and that $Y : Ω → \mathbb{R}^n$ is an independent random variable. The probability density functions of many well-known probability distributions are p-concave for some p, and hence unimodal. If they are also symmetric, then Anderson's theorem applies, in which case

$$ P ( X \in K ) \geq P ( X + Y \in K ) $$ for any origin-symmetric convex body $K ⊆ \mathbb{R}^n$.

I was wondering

  1. How does Anderson's theorem lead to the inequality for probability, given $Y$ is not a constant but a random variable? The density function of $X+Y$ is the convolution of the densities of $X$ and of $Y$, which has a different form from the density of $X$. Also I am not sure if $Y$ may allow to not have a density.
  2. What are the requirements on the distributions of $X$ and $Y$? The quote is a bit vague to me? Does it say that the distributions of $X$ and of $Y$ should be both unimodal and symmetric?
  3. How is $p$-concavity defined for a distribution and for a general function? I found different definitions on the internet and am not sure which one is assumed here.

I also appreciate if there can be some pointers to references regarding Anderson's theorem and its proof.

Thanks and regards!

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