Martingale convergence theorems state that under certain conditions, a martingale, submartingale, or supermartingale converges to a limiting random variable. In this chapter, we shall present several martingale convergence theorems.

**Definition 25.1** (Crossing of Real Numbers). Let {*x*_{n} : *n* ≥ 0} be a sequence of real numbers. Let *a* < *b* be real numbers. The sequence {τ_{n}(*a, b*) : *n* ≥ 0} of crossings with respect to the sequence {*x*_{n} : *n* ≥ 0} is defined by

(25.1a)

and for every *n* ≥ 1, we have

(25.1b)

(25.1c)

where *A*_{2n−1} = {*k* ≥ τ_{2n−2} : *x*_{k} ≤ *a*} and *A*_{2n} = {*k* ≥ τ_{2n−1} : *x*_{k} ≥ *b*}.

**Definition 25.2** (Crossing of Random Variables). Let{*X*_{n} : *n* ≥ 0} be a sequence of real random variables. Let *a* < *b* be real numbers. The sequence {τ_{n}(*a*, *b*) : *n* ≥ 0} of crossings with respect to the sequence {*X*_{n} : *n* ≥ 0} is defined as follows. For every ω Ω, {τ_{n}(*a*, *b*)(ω) : *n* ≥ 0} is the sequence of crossings with respect to the sequence {*X*_{n}(ω) : *n* ≥ 0}.

**Definition 25.3** (_{∞}). Let {_{n} : *n* ≥ 0} be a filtration. Then _{∞} is the σ-algebra generated by :

**Theorem ...**

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