I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid continuous symmetric random variables in $\mathbb{R}$.
How to check if a symmetric random variables is the difference of two iid symmetric random variables
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1$\begingroup$ Can you give some examples and non-examples to help find reasonable conditions? $\endgroup$– Joonas IlmavirtaCommented Aug 21, 2014 at 13:10
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2$\begingroup$ One obvious necessary condition is that the characteristic function (aka Fourier transform) $\int e^{itx} dF(x)$ must be nonnegative. $\endgroup$– Nate EldredgeCommented Aug 21, 2014 at 15:40
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4$\begingroup$ ... and (some branch of) its square root will then be the characteristic function of $Y_i$. By Bochner's theorem you just have to check that this square root is positive definite. $\endgroup$– Robert IsraelCommented Aug 21, 2014 at 16:47
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1 Answer
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No conditions are necessary. Let $Y_2$ be any continuous symmetric random variable independent of $X$. Then $X + Y_2$ is a continuous symmetric random variable.
Perhaps you wanted $Y_1$ and $Y_2$ to be independent?
answered Aug 21, 2014 at 15:20