sum of stochastically continuous processes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:29:37Z http://mathoverflow.net/feeds/question/115582 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115582/sum-of-stochastically-continuous-processes sum of stochastically continuous processes Paolo 2012-12-06T07:00:27Z 2012-12-06T08:29:42Z <p>Hallo, is the sum of two stochastically continuous processes again a stochastically continuous process? why? Thank you very much, Paolo.</p> http://mathoverflow.net/questions/115582/sum-of-stochastically-continuous-processes/115588#115588 Answer by Alexander Shamov for sum of stochastically continuous processes Alexander Shamov 2012-12-06T08:29:42Z 2012-12-06T08:29:42Z <p>Yes. A stochastically continuous process is a mapping $t \mapsto X_t \in L^0$ into the space of random variables that is continuous with $L^0$ given the topology of convergence in probability. Since $L^0$ is a topological vector space for this topology - in particular, addition is continuous - it is clear that $t \mapsto (X_t, Y_t) \mapsto X_t + Y_t$ is continuous as a composition of continuous mappings.</p>