Suppose you are given $88$ i.i.d random variables (e.g. gamma distribution) and $1$ other random variable with another distribution. Together, it seems that they don't converge in distribution to the normal distribution. But the $88$ together do. Is there a term for this?
Central Limit Theorem Question
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1No idea about the meaning of the assertion that *88 i.i.d random variables converge* (or do not converge). A family of 88 beasts is not an infinite sequence of beasts and one needs the latter for the notion of convergence to even make sense. – 2012-03-28
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0Most versions of the CLT I have seen require all variables to have the same distribution. So the term might be "Not satisfying the hypotheses of the theorem." As to the convergence, there are version of the CLT which give bounds for the difference between the limit distribution and the actual distribution that depend on the number of variables and the type of distribution. – 2012-03-28
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3@marty *Most versions of the CLT I have seen require all variables to have the same distribution*... Really? [Lyapunov](http://en.wikipedia.org/wiki/Central_limit_theorem#Lyapunov_CLT). [Lindeberg](http://en.wikipedia.org/wiki/Central_limit_theorem#Lindeberg_CLT). – 2012-03-28
1 Answers
It really makes little sense to say that "88 variables converge to..." What "converges" is always (probability or whatever) a sequence $S_N$ for $N \to \infty$.
Informally, however, you might say that 88 is a large number (but, again, this requires justification, in your context), and hence you might be confident in saying that your $Z=\sum_{i=1}^{88} X_i$ will be "approximately normal". Now, you are asking what happens if you have one extra different variable: $W=Y+\sum_{i=1}^{88} X_i$ , where $Y$ has different probability function. Howewer, if the variance of $Y$ is "comparable" to that of $X$, then this variance will be very small as compared to the 88-sum, hence it will approximately a constant, and the total sum will still be "approximately normal".
More in general, the CLT does not require iid variables (but it does require some restrictions about the variances).