Inhomogeneous Mandelbrot cascades

We consider a modification of the classical Mandelbrot cascade in which the ”inverse temperature” parameter may be different on each scale. The main result is a sufficient and almost necessary condition for the inhomogeneous Mandelbrot martingale to converge in $\mathcal{L}^1$ . Near-critical inhomogeneous cascade measures can be used to study fine multifractal properties of critical Mandelbrot cascades.