# Sieving random iterative function systems

## TL;DR

This paper introduces a sieving procedure for random iterative function systems, resulting in a scale-invariant process with cadlag paths and finite total variation, and explores its properties and examples.

## Contribution

It develops a novel sieving method to construct and analyze scale-invariant stochastic processes from random Lipschitz functions, extending understanding of their distribution and path properties.

## Key findings

- The sieved process is cadlag and has finite total variation.
- Examples include perpetuities, Bernoulli convolutions, and random continued fractions.
- The process exhibits scale invariance and specific path properties.

## Abstract

It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space components, it is possible to construct a scale invariant stochastic process. We study its distribution and paths properties. In particular, we show that it is c\`adl\`ag and has finite total variation. We also provide examples and analyse various properties of particular sieved iterative function systems including perpetuities and infinite Bernoulli convolutions, iterations of maximum, and random continued fractions.