# Sum of independent Lévy processes is a Lévy process

I'm currently reading Cont & Tankov's "Financial Modelling With Jump Processes" and they state the sum of independent Lévy processes is itself a Lévy process but only provide a working example to prove it in the case of Lévy processes. Can anyone point me in the direction of literature containing the proof for the general case? Many thanks

• Not sure about literature, but I think it is a good exercise to go through oneself. Here are hints to get you started in the case of a sum of just two Lévy Processes (by induction, you may extend to finite sums). First, you need to prove that the sum process has independent increments. For this, note that if $X,Y,Z,W$ are all independent random variables, then $X+Y$ and $Z+W$ are also independent. Next you should prove stationarity of increments. For this, note that if $X\stackrel{d}{=}Y$ and $X'\stackrel{d}{=}Y'$, and if $(X,Y)$ is independent of $(X',Y')$, then $X+Y\stackrel{d}{=}X'+Y'$. – Shalop Aug 21 '17 at 11:05