Lindeberg condition

The verbal interpretation of (5.5) is that the process is continuous — this is the Lindeberg condition. The function a x, t) is the velocity of conditional expectation ( drift vector ), and bjj(x, t) is the matrix of the velocity of conditional covariance ( diffusion matrix ). The latter is positive semidefinite and symmetric as a result of its definition (5.7). 

The terminology is nonstandard, and in physical literature the Kramers-Mpyal expansion is given as a (nonsystematic) procedure to approximate discrete state-space processes by continuous processes. The point that we want to emphasise here is the clear fact that, even in the case of a continuous state-space, the process itself can be noncontinuous, when the Lindeberg condition is not fulfilled. The functions for the higher coefficients do not necessarily have to vanish. 

Equations (4.3-4) and (4.3-5) are the first of several important limit theorems that establish conditions for asymptotic convergence to normal distributions as the sample space grows large. Such results are known as central limit theorems, because the convergence is strongest when the random variable is near its central (expectation) value. The following two theorems of Lindeberg (1922) illustrate why normal distributions are so widely useful. 

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