## Can we derive the density function from the property below?

Hi,

Let p(x) be a density function of a random variable. Suppose p(x) satisfies the following property. There are two small positive numbers $\epsilon$ and $\Delta$, such that for all $-\Delta\leq\theta\leq\Delta$, for all x, exp$(-\epsilon)p(x+\theta)\leq p(x)\leq$exp$(\epsilon)p(x+\theta)$. Then what can we get about this p(x)?

It is obvious that the Laplace distribution satisfies this property. How about the converse? Must this p(x) be some version of Laplace distribution?

Thank you!

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If $p(x)=\exp (\phi(x))$ the stated property just says that $|\phi(x)-\phi(y)|\le\epsilon$ whenever $|x-y|\le\Delta$. A lot of functions satisfy this, e.g. any Lipschitz function with constant $\epsilon/\Delta$. Imposing $\int_\mathbb{R} p(x)dx=1$ you still get quite a huge class. – Pietro Majer Jul 2 at 10:10