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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3@marty *Most versio$n$s 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).