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Write $K(t)=\log \mathbb{E}[e^{tY}]$ for the cumulant-generating function in the question's $(\varphi ,\lambda )$ parametrization. Multiplying the density by $e^{ty}$ replaces $\varphi$ in the decaying part of its exponent by $\varphi -2t$. For $t<\varphi /2$, the remaining integral is the normalizing integral for an Inverse Gaussian distribution with parameters $(\varphi -2t,\lambda )$. Comparing the two normalizing constants givesIn particular $K^{\prime }(0)=\sqrt{\lambda /\varphi }$, so the mean parameter used in the more usual Inverse Gaussian distribution notation is $\mu =\sqrt{\lambda /\varphi }$; it must not be confused with the present parameter $\varphi$.

First compute the cumulant-generating function of the sample average, as requested. Independence givesThis is exactly the cumulant-generating function of an Inverse Gaussian distribution with parameters $(n\varphi ,n\lambda )$. The uniqueness theorem for moment-generating functions therefore identifies that law. Scaling $S_n=n{\overline{Y}}_n$ and including the density Jacobian $1/n$ givesEquivalently, the inverse Gaussian sum closure is $S_n\sim IG(\varphi ,n^2\lambda )$ in the parametrization of this question. Its mean is $n\sqrt{\lambda /\varphi }$ and its variance is $n\sqrt{\lambda }/{\varphi }^{3/2}$, in agreement with addition of independent means and variances.

For a general continuous iid sample with cumulant-generating function $K$, finite near zero, set $x=s/n$ and choose a real saddle $\hat{t}$ in the interior of the finite moment domain satisfying $K^{\prime }(\hat{t})=x$. With $K^{\prime \prime }(\hat{t})>0$, the saddlepoint density approximation for the sum isUnder the usual smooth nonlattice conditions, at a fixed interior $x$ the true density has this leading expression times $1+O(n^{-1})$. This is a relative density approximation; the leading expression is not automatically an exactly normalized probability density in every model.

One way to understand the statistical saddlepoint approximation is through exponential tilting. The tilted one-observation density is $f_t(y)=e^{ty-K(t)}f(y)$, with mean $K^{\prime }(t)$ and variance $K^{\prime \prime }(t)$. Undoing this tilt for the sum multiplies its tilted density at $s$ by $e^{nK(t)-ts}$. At $t=\hat{t}$, $s$ is exactly the tilted mean of the sum. A local version of the central limit theorem gives tilted density there approximately $[2\pi n{K}^{\prime \prime }(\hat{t}){]}^{-1/2}$. Multiplying these two factors gives the displayed saddlepoint density approximation. Higher local expansion terms produce the stated relative error order.

For the Inverse Gaussian distribution, differentiation givesThus for every $x>0$ the saddle exists, withAt this saddle,Substitution into the saddlepoint density approximation gives precisely the exact density already found for $S_n$. ThereforeThis exact saddlepoint density of an inverse Gaussian sum is stronger than a relative $O(n^{-1})$ claim: the leading formula is exact and needs no renormalization.

For the final distribution, both tails decay polynomially. Any positive exponential tilt makes the positive tail nonintegrable, and any negative tilt makes the negative tail nonintegrable. Its moment-generating function is therefore infinite away from zero, leaving no open interval on which to form the cumulant-generating function and solve a saddle equation. Here polynomial tails prevent exponential tilting, so the conventional saddlepoint density approximation is unavailable. A finite mean and variance, and even an ordinary central limit theorem, do not supply the missing exponential moments.