Consider a functional of the form [T(F)=int_{-infty}^{infty} int_{-infty}^{infty} hleft(x_{1}, x_{2}ight) d Fleft(x_{1}ight) d Fleft(X_{2}ight)] where (F in

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Consider a functional of the form

\[T(F)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h\left(x_{1}, x_{2}ight) d F\left(x_{1}ight) d F\left(X_{2}ight)\]

where \(F \in \mathcal{F}\), a collection of distribution functions. Find an expression for the \(k^{\text {th }}\) Gâteaux differential of \(T(F)\) without using the assumption that \(h\) is a symmetric function of its arguments. Compare this result to that of Theorem 9.1.

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