Fixed points of inhomogeneous smoothing transforms

Abstract

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation , where means equality in distribution, is a given sequence of non-negative random variables and is a sequence of i.i.d. copies of the non-negative random variable X independent of . In this situation, X (or, more precisely, the distribution of X) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Furthermore, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C = 0. Using this correspondence and the known theory on the homogeneous equation, we present a full characterization of the set of fixed points under mild assumptions.

Keywords:

• branching random walk
• fixed point
• multiplicative martingales
• smoothing transform
• stochastic fixed-point equation
• weighted branching process

Keywords:

• 60E05
• 60J80
• 39B22

Acknowledgements

The authors are grateful to Takis Konstantopoulos for pointing out a reference, and to Ralph Neininger for pointing out an error in an earlier version of this paper. This research was supported by DFG grant Me 3625/1-1.