7.5 Extensions of Central Limit Theorem. Limit Theorems for Other Types of Statistics

As we will see in the second part of this book concerning stochastic processes, the Central Limit Theorem is a great tool and there exist variants of CLT for renewal processes, martingales, and many other types of processes. Extensions of this theorem have also been proven for m-dependent random variables. An m-dependent sequence is a sequence where the random vector is independent of the vector for any t, s, such that . Further extensions to independent sequences but not identically distributed and many other situations have been considered, and variants of CLT have been proven, primarily due to the great importance of this theorem in practical applications.

All these theorems and extensions are looking at the distribution of the sum of random variables. However, in practical applications, more often than not, the statistics of interest is not a sum. In this situation, when the random variable may not be expressed as a sum of independent random variables, we can try to use Slutsky's theorem 7.34 on page 231. An alternative approach is the method presented next.

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