Harman Patil (Editor)

Martingale central limit theorem

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In probability theory, the central limit theorem says that, under certain conditions, the sum of many independent identically-distributed random variables, when scaled appropriately, converges in distribution to a standard normal distribution. The martingale central limit theorem generalizes this result for random variables to martingales, which are stochastic processes where the change in the value of the process from time t to time t + 1 has expectation zero, even conditioned on previous outcomes.

Statement

Here is a simple version of the martingale central limit theorem: Let

X 1 , X 2 , … -- be a martingale with bounded increments, i.e., suppose E ⁡ [ X t + 1 − X t | X 1 , … , X t ] = 0 ,

and

| X t + 1 − X t | ≤ k

almost surely for some fixed bound k and all t. Also assume that | X 1 | ≤ k almost surely.

Define

σ t 2 = E ⁡ [ ( X t + 1 − X t ) 2 | X 1 , … , X t ] ,

and let

τ ν = min { t : ∑ i = 1 t σ i 2 ≥ ν } .

Then

X τ ν ν

converges in distribution to the normal distribution with mean 0 and variance 1 as ν → + ∞ . More explicitly,

lim ν → + ∞ P ⁡ ( X τ ν ν < x ) = Φ ( x ) = 1 2 π ∫ − ∞ x exp ⁡ ( − u 2 2 ) d u , x ∈ R .

References

Martingale central limit theorem Wikipedia
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