So, I have read over (note: I don't necessarily know very well) some measure-theoretic probability. One problem with the (regular) CLT is that it assumes independent and identically distributed random variables, but by removing the identically distributed assumption and imposing some conditions, we can get some nice extensions of the CLT:
Lyapunov CLT: http://ift.tt/1xFfHgj
Lindeberg CLT: http://ift.tt/1xFfHgl
It also so happens that Lyapunov implies Lindeberg, but the converse is not true. Looked through the two measure-theoretic probability books I have, and couldn't find a counterexample, but someone finally asked on MathSE and a nice answer has come up: http://ift.tt/1xFfHwD . :)
Lyapunov CLT: http://ift.tt/1xFfHgj
Lindeberg CLT: http://ift.tt/1xFfHgl
It also so happens that Lyapunov implies Lindeberg, but the converse is not true. Looked through the two measure-theoretic probability books I have, and couldn't find a counterexample, but someone finally asked on MathSE and a nice answer has come up: http://ift.tt/1xFfHwD . :)
Extending the CLT (for nerds)
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