Martingale Difference Sequence

Last updated Nov 1, 2022

Let $(d_{j})$ be a Discrete-Time Process adapted to Filtration $(\mathcal{B}_{j})$ with State Space $\mathbb{R}$. Then, it is a Martingale Difference Sequence if

1. $d_{j} \in L^{1}$ $\forall j \in \mathbb{N}$
2. $\mathbb{E}(d_{j+1} | \mathcal{B}_{j}) = 0$ $\forall j \in \mathbb{N}$

1. A Discrete-Time Process is a Difference Sequence iff it is the difference of Martingale increments
2. (1) in other words: Accumulation of Martingale Difference Sequence is a Martingale

• Conditional Expectation
• Natural Numbers
• L1 Integrable Functions
• Real Numbers