The dynamic provisioning of virtualized resources offered by cloud computing infrastructures allows applications deployed in a cloud environment to automatically increase and decrease the amount of used resources. This capability is called auto-scaling and its main purpose is to automatically adjust the scale of the system that is running the application to satisfy the varying workload with minimum resource utilization. The need for auto-scaling is particularly important during workload peaks, in which applications may need to scale up to extremely large-scale systems. Both the research community and the main cloud providers have already developed auto-scaling solutions. However, most research solutions are centralized and not suitable for managing large-scale systems, moreover cloud providers' solutions are bound to the limitations of a specific provider in terms of resource prices, availability, reliability, and connectivity. In this paper we propose DEPAS, a decentralized probabilistic auto-scaling algorithm integrated into a P2P architecture that is cloud provider independent, thus allowing the auto-scaling of services over multiple cloud infrastructures at the same time. Our simulations, which are based on real service traces, show that our approach is capable of: (i) keeping the overall utilization of all the instantiated cloud resources in a target range, (ii) maintaining service response times close to the ones obtained using optimal centralized auto-scaling approaches.