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# Asymptotic Expansions for Distributions

DOI link for Asymptotic Expansions for Distributions

Asymptotic Expansions for Distributions book

# Asymptotic Expansions for Distributions

DOI link for Asymptotic Expansions for Distributions

Asymptotic Expansions for Distributions book

## ABSTRACT

Let us consider the case of a sequence of independent and identically distributed random variables {Xn}∞n=1 from a distribution F whose mean is µ and variance is σ2. Assume that E(|Xn|3) < ∞. Then Theorem 4.20 (Lindeberg and Le´vy) implies that Zn = n

1/2σ−1(X¯n − µ) d−→ Z as n→∞ where Z has a N(0, 1) distribution. This implies that

lim n→∞P (Zn ≤ t) = Φ(t),

for all t ∈ R. Define Rn(t) = Φ(t)− P (Zn ≤ t) for all t ∈ R and n ∈ N. This in turn implies that P (Zn ≤ t) = Φ(t) + Rn(t). Theorem 4.24 (Berry and Esseen) implies that |Rn(t)| ≤ n−1/2Bρ where B is a finite constant that does not depend on n or t and ρ is the third absolute moment about the mean of F . Noting that ρ also does not depend on n and t, we have that

lim n→∞n

1/2|Rn(t)| ≤ Bρ,

uniformly in t. Therefore, Definition 1.7 implies that Rn(t) = O(n −1/2) and

we obtain the asymptotic expansion P (Zn ≤ t) = Φ(t) +O(n−1/2) as n→∞. Similar arguments also lead to the alternate expansion P (Zn ≤ t) = Φ(t)+o(1) as n→∞. The purpose of this chapter is to extend this idea to higher order expansions. Our focus will be on distributions that are asymptotically normal via Theorem 4.20 (Lindeberg and Le´vy). In the case of distributions of a sample mean, it is possible to obtain an asymptotic expansion for the density or distribution function that has an error term that is O(n−(p+1)/2) or o(n−p/2) as n → ∞ with the addition of several assumptions.