Let the random variables $Y_1,\ldots,Y_n$ be independent and identically distributed (i.i.d.) (standard) Inverse Gaussian random variables with parameters $\mu$ and $\lambda$.
Then, let the random variables $\tilde{A}$ and $A$ be given, which are defined as follows: $A = \sqrt n \cdot \tilde{A} = \sqrt n \left(\frac{1}{n} \cdot \sum_{i=1}^n \left[e^{b_i Y_i}\right] - \theta\right)$ ($b_i \in \mathbb{R}$ ($i=1,\ldots,n$) are constants (with $b_i \neq b_j$ for any $i \neq j$) and $\theta$ is a parameter).
Goal: Determine what ultimately follows if one uses the (Lindeberg-Lévy) Central Limit Theorem on $e^{b_i Y_i}$.
The CLT informs us that $\sqrt n (\overline{Y} - E(Y_i)) \xrightarrow[]{D} \mathcal{N}(0,Var(Y_i))$, i.e. $\sqrt n \left(\frac{1}{n} \cdot \sum_{i=1}^n [Y_i] - \mu \right) \xrightarrow[]{D} \mathcal{N}(0,\frac{\mu^3}{\lambda})$.
It seems to me that it follows that $\sqrt n \left(\frac{1}{n} \cdot \sum_{i=1}^n \left[e^{b_i Y_i}\right]- e^{b_i \mu}\right) \xrightarrow[]{D} \cdots$.
Question: How to proceed from here (i.e. how to determine the limit distribution of $\sqrt n \cdot \tilde{A}$)?
You may consider the random variable \begin{align} \frac{1}{s_n} \sum_{i=1}^n (e^{b_i Y_i} - e^{b_i \mu}) \end{align} where $s_n^2 = \sum_{i=1}^n \sigma_i^2$ and $\sigma_i$ is the variance of $e^{b_iY_i}$. Then, you can check whether the Lindeberg's condition is satisfied. If it is satisfied, using the Lindeberg CLT \begin{align} \frac{1}{s_n} \sum_{i=1}^n (e^{b_i Y_i} - e^{b_i \mu}) \xrightarrow[]{D} \mathcal{N}(0,1) \end{align} as $n\to\infty$.