Suppose that \(X_{1}, \ldots, X_{n}\) are a set of independent and identically distributed random variables from a distribution \(F\) that has mean equal to zero, unit variance, and cumulant generating function \(c(t)\). Find the form of the polynomial \(p_{3}(x)\) from Theorem 7.5 by considering the form of an expansion for the cumulant generating function of \(n^{1 / 2} \bar{X}_{n}\) that has an error term equal to \(o\left(n^{-3 / 2}ight)\) as \(n ightarrow \infty\). What assumptions must be made in order to apply Theorem 7.5 to this problem?

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Theorem 7.5. Let {X} be a sequence of independent and identically distributed random variables. Let F(x) = P[n/20-1(Xn-) x], with density fn(x) for all x = R. Assume that E(|X|P) < , " is integrable for some v1, and that fn(x) erists for n v. Then, P fn(x) (x)-(x)nk/2pk(x) = o(nP/2), k=1 (7.17) uniformly in x as n o, where p(x) is a real polynomial whose coefficients depend only on 1,...,k. In particular, p(x) does not depend on n, p or on other properties of F.