# real analysis – Proof of Levy-Khintchine formula: Question on the existence of an infinitely divisible distribution defined by the formula

I am reading the proof of the Levy-Khintchine formula from Ken Iti Sato’s Levy Processes, however, I cannot understand a line from the proof that given a symmetric nonnegative definite $$d times d$$ matrix $$A$$, and a measure $$nu$$ on $$mathbb{R}^d$$ with $$nu({0})=0$$, $$int (1 wedge |x|^2) nu(dx), and $$gamma in mathbb{R}^d$$, we get an infinitely divisible distribution $$mu$$ whose characteristic function is given by the formula.

In the proof below, it states that $$phi_n$$ is the convolution of a Gaussian and a compound Poisson distribution. A distribution $$mu$$ on $$mathbb{R}^d$$ is compound Poisson if there exists $$c>0$$ and $$sigma$$ on $$mathbb{R}^d$$ with $$sigma ({0})=0$$ and the characteristic function $$hat mu(z)= exp(c(hat sigma(z)-1)).$$ $$D$$ here is the closed unit ball.

But I cannot see from below why we get this form for the inner integral. That is, it seems like we should have $$int_{|x|>1/n} (1-ilangle z,xrangle 1_D(x)) nu(dx)=1$$ from the definition above but I don’t see how we get this. Why does this integral define the characteristic function of a compound Poisson? Posted on