Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables following a \(\operatorname{ChiSquared}(\theta)\) distribution. A saddlepoint expansion that approximates the density of \(n \bar{X}_{n}\) at a point \(x\) with relative error \(O\left(n^{-1}ight)\) as \(n ightarrow \infty\) was shown in Example 7.12 to have the form \(f_{n}(x)=\left[4 \pi n^{-1} x^{2} \thetaight]^{-1 / 2} \exp \left[-\frac{1}{2} n \theta \log \left(n \theta x^{-1}ight)-\frac{1}{2}\left(1-n x^{-1} \thetaight)ight]\left[1+O\left(n^{-1}ight)ight]\).

Plot the correct GAMmA density of \(n \bar{X}_{n}\) and the saddlepoint approximation for \(\theta=1\) and \(n=2,5,10,25\) and 50. Discuss how well the saddlepoint approximation appears to be doing in each case.

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Example 1.2. Consider the alternating sequence defined by x=(-1)" for all n = N. It is intuitively clear that the alternating sequence does not "settle down" at all, or that it does not converge to any real number. To prove this, let R be any real number. Take 0 < < max{-11, |+1|}, then for any n' N there will exist at least one n">n' such that n-1|> E. Hence, this sequence does not have a limit.