Levy Khinchine formula for Gamma distribution
1
1
I am searching for the Levy-Khinchin formula for the Gamma distribution $Gamma(lambda, beta)$ . My attempt is following: For the characteristic function I get $$varphi_{Gamma(lambda,beta)}(t)=Big(frac{beta}{beta -it}Big)^lambda$$ Therefore $$[varphi_{Gamma(frac{lambda}{n},beta)}(t)]^n=Big[Big(frac{beta}{beta -it}Big)^frac{lambda}{n}Big]^n=varphi_{Gamma(lambda,beta)}(t)$$ and the gamma distribution in infinitely divisible. Now the idea is to find a triplet $(b,sigma,nu)$ for the Levy-Khinchine formula. I have a hint to use $nu$ which has a density regarding to the lebesgue measure as follows $$frac{lambda}{x}e^{-beta x}1_{[0,infty)}(x)$$ So the Levy Kinchin formula is the following: $$psi(t)=itb-frac{t^2sigma^2}{2}+int_{mathbb{R}}Big(e^{itx}-1-itxcdot1_{{|x|<1}}(x)Big)frac{lambda}{x}e^{-beta x}1_{[
