Normal approximation to the hypergeometric distribution in ...

S.N. Lahiri et al. / Journal of Statistical Planning and Inference 137 (2007) 3570 –3590 3585
Next consider the case where 0 ∈[b− h/2,b+ h/2). Then, by Taylor’s expansion,
b−h/2
h(b) − (x) dx
h3 | ′′ (0)|/24.
b−h/2
Now using similar arguments for the case ‘ √ 3 ∈ (b − h/2,b + (j0 + 1 2 )h)’ and using the above bounds, one can
complete the proof of the lemma.
Proof of Theorem 1. Let r ∈ N be an integer such that (2.4) holds. Since r will be held fixed all through the proof,
we shall drop r from the notation for simplicity, and write fr = f , r = , pr = p, qr = q, nr − n, etc. First, suppose
that f 1 2 . Consider the case x 0. Let ˜xk = xk/ √ 1 − f = (k − np)/, k = 0, 1,...,n. Define
and
K0 = sup{k ∈ Z+ :˜xk 0},
K1 = inf{k ∈ Z+ :˜xk − 1},
K2 = inf{k ∈ Z+ :˜xk − }
Jx =⌊np + x⌋, x ∈ R,
where ≡ r ∈ (0, 1 2 ] is as in (2.3). Note that by definition,
K1 − 1