Talk:Lindeberg's condition

Feller's theorem
As stated in Page 348 in 'Measure Theory and Probability Theory' by K. B. Athreya, S. N. Lahiri (2006), Feller's theorem is an alternative method to show that Lindeberg's condition holds and it can also be used to disprove convergence to a normal distribution using proof by contradiction. I think it is a good idea to state Feller's theorem in the article. The theorem states,

If $$\forall \epsilon > 0 $$, $$\lim_{n \rightarrow \infty} \max_{1 \leq j \leq n} P(|X_j| > \epsilon s_n) = 0$$ and $$\frac{S_n}{s_n}$$ weakly converges to a standard normal distribution as $$n \rightarrow \infty$$ then $$\{X_j\}_{j=1}^n$$ satisfies the Lindeberg's condition. Where $$S_n = \sum_{j=1}^n X_j$$ and $$s_n^2 = Var(S_n)$$.

Note that here, I am avoiding triangular array notation to avoid over generalizing and therefore confusing the reader, i.e with respect to the source, I have set $$r_n = n$$

--Beat of the tapan (talk) 10:06, 13 August 2019 (UTC)

Edited by Beat of the tapan (talk) 03:20, 17 August 2019 (UTC)