In probability theory, a martingale difference sequence (MDS) is related to the concept of the martingale. A stochastic series X is an MDS if its expectation with respect to the past is zero. Formally, consider an adapted sequence
for all
The MDS is an extremely useful construct in modern probability theory because it implies much milder restrictions on the memory of the sequence than independence, yet most limit theorems that hold for an independent sequence will also hold for an MDS.