A truncated distribution is simply a limitation of some other probability distribution to lie on a restricted support. The idea is simple, however carrying out efficient probability, moment, and expectation computation with such distributions is sometimes a challenge.
To date I have only this publication related to truncated distributions. However, I am working on further research in this area ad creating software for truncated multi-variate distributions.
A dynamic view to moment matching of truncated distributions, by
Benoit Liquet, Yoni Nazarathy. Published in Statistics & Probability Letters.