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Milds # 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 $\{X_{t},{\mathcal {F}}_{t}\}_{-\infty }^{\infty }$ on a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$ . $X_{t}$ is an MDS if it satisfies the following two conditions:
$\mathbb {E} \left|X_{t}\right|<\infty$ , and
$\mathbb {E} \left[X_{t}|{\mathcal {F}}_{t-1}\right]=0,a.s.$ ,
for all $t$ . By construction, this implies that if $Y_{t}$ is a martingale, then $X_{t}=Y_{t}-Y_{t-1}$ will be an MDS—hence the name.