Martingale difference sequence  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 formula_1 on a probability space formula_2. formula_3 is an MDS if it satisfies the following two conditions:  for all formula_6. By construction, this implies that if formula_7 is a martingale, then formula_8 will be an MDS—hence the name. 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.