Reference:

Tomás Brázdil, Stefan Kiefer, Antonín Kučera, and Ivana Hutarová Vareková. Runtime analysis of probabilistic programs with unbounded recursion. Journal of Computer and System Sciences, 81(1):288–310, February 2015.

Abstract:

We study the runtime in probabilistic programs with unbounded recursion. As underlying formal model for such programs we use probabilistic pushdown automata (pPDAs) which exactly correspond to recursive Markov chains. We show that every pPDA can be transformed into a stateless pPDA (called pBPA) whose runtime and further properties are closely related to those of the original pPDA. This result substantially simplifies the analysis of runtime and other pPDA properties. We prove that for every pPDA the probability of performing a long run decreases exponentially in the length of the run, if and only if the expected runtime in the pPDA is finite. If the expectation is infinite, then the probability decreases ``polynomially''. We show that these bounds are asymptotically tight. Our tail bounds on the runtime are generic, i.e., applicable to any probabilistic program with unbounded recursion.

Suggested BibTeX entry:

@article{15JCSS-BKKV,
    author = {Tom{\'{a}}s Br{\'{a}}zdil and Stefan Kiefer and Anton{\'{\i}}n Ku\v{c}era and Ivana Hutarov{\'{a}} Varekov{\'{a}}},
    journal = {Journal of Computer and System Sciences},
    month = {February},
    number = {1},
    pages = {288--310},
    title = {Runtime analysis of probabilistic programs with unbounded recursion},
    volume = {81},
    year = {2015}
}