This paper presents an investigation of the asymptotic behavior within natural exponential families, with particular focus on the limiting distributions of scaled family. We examine the scaled random variable \(Y_t = \frac{X_t}{t}\) , where \(X_t\) belongs to a natural exponential family with mean parameter tm ( \(m > 0\) ) as \(t \rightarrow 0^+\) . Under the fundamental assumption that the second derivative of the variance function V extends continuously to zero with \(V''(0) \ne 0\) , we establish that \(Y_t\) converges in distribution to a Gamma law with explicitly determined shape and rate parameters \(\alpha = \frac{2}{V''(0)}\) and \(\beta = \frac{2}{m V''(0)}\) , respectively. The proposed proof integrates tools from convex analysis, probability theory, and functional analysis, including the Arzelà-Ascoli theorem. The paper also provides a detailed illustrative example using the uniform distribution on [0, 1], where the limiting distribution simplifies to an exponential law.
