Given a double array of random variables $X_{nj}, j=1,\dots, k_n, n\in\mathbb{N}$ with $k_n \to \infty$ as $n \to \infty$, suppose

  • for each $n$, $X_{nj}, j=1,\dots, k_n$ are independent,
  • each $X_{nj}$ has finite mean $\mu_{nj}$ and finite variance $\sigma_{nj}$.

Define $S_n := \sum_{j=1}^{k_n} X_{nj}$ and $s_n := \sqrt{\sum_{j=1}^{k_n} \mathrm{var} X_{nj}}$.

Now there are two versions of sufficient conditions for Lindeberg-Feller Central Limit Theorem:

  1. From Kai Lai Chung's A Course in Probability Theory, the Lindeberg condition is defined as $$ \forall \epsilon >0, \quad \lim_{n \to \infty} \frac{\sum_{j=1}^{k_n} \mathrm{E} [(X_{nj} - \mu_{nj})^2 I_{\{|X_{nj} - \mu_{nj}| > \epsilon s_n\}} ] }{s_n^2} = 0.$$
  2. In Theorem D.19 of William Greene's Econometric Analysis (p112 of his appendix D file or Theorem 11 on page 14 of this note ), the Lindeberg condition is replaced with $$ \lim_{n\to\infty} \frac{\max_{j=1,\dots,k_n}\sigma_{nj}^2}{s_n^2} = 0$$ $$\lim_{n \to \infty} \frac{s_n^2}{n} < \infty. $$ (Note: (1) The book deals with a sequence of random variables, but here I take the liberty to generalize it for a double array of random variables. Please correct me if I am wrong. (2) I have also changed the notation a bit.)

    Added: In the book, instead of $\frac{\max_{j=1,\dots,k_n}\sigma_{nj}^2}{s_n^2}$, it writes $ \frac{\max_{j=1,\dots,k_n}\sigma_{nj}}{\sqrt{k_n} s_n}$. This is said to be a typo, and it should be $ \frac{\max_{j=1,\dots,k_n}\sigma_{nj}}{ s_n}$. And $ \frac{\max_{j=1,\dots,k_n}\sigma_{nj}}{ s_n}$ converges to finite, if and only if $ \frac{\max_{j=1,\dots,k_n}\sigma_{nj}^2}{ s_n^2}$ converges to finite. (Note: I have changed the notation a bit.)

I wonder what relations are between these two versions of Lindeberg(-Feller) CLT?

Is the version in Greene's book a special case of that in Chung's? (I can see this is true when the double array is a sequence of identically distributed random variables.)

Thanks and regards!


The Lindeberg condition is weaker than the one given in Greene's book, i.e. Greene's condition implies the Lindeberg condition. The Lindeberg-Feller CLT states that (let's just use one sequence) the Lindeberg condition holds if and only if (1) $\frac{\sum (X_i - \mu_i)}{s_n} \stackrel{d}{\to} N(0, 1)$ and (2) $\frac{\max_{1 \le i \le n} \sigma_i}{s_n} \to 0$. Greene's condition takes (2) as an assumption and derives (1), so the Lindeberg condition must hold. The same thing works for triangular arrays.

EDIT: The above is predicated on the fact that the condition in Greene, as given, is correct, which I was somewhat wary of; the other question posted by OP related shows similar reservations from other answerers.


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