[Math] Does this condition imply the Lindeberg condition

probability theory

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!

Best Answer

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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