$X_1$ ($(X_n)_{n\geq 1}$ is a skip free random walk, i.e. $(X_n)_{n\geq 1}$ i.i.d takes value in {-1,0,1,2,...} with $\mathbb{E}(X_n)=0$ for all $n\geq 1$ and $\mathbb{P}(X_1+1=k)\sim Ck^{-\alpha-1}$ as $k\rightarrow\infty$ for some $C>0$) has the power law tail behavior $\mathbb{P}(X_1 \geq x) \sim Cx^{-\alpha}$ (where $\alpha\in(1,2)$) as $x\rightarrow\infty$ and so by Tauberian theorems its Laplace transform witnesses the following asymptotic: \begin{equation*} \mathbb{E}(\exp(-\lambda X_1))=1+C'\lambda^{\alpha}+o (\lambda^{\alpha})\qquad \text{as $\lambda\rightarrow 0$} \end{equation*}
Question: I am not sure if defining Laplace transform when it can take value $-1$ is well-defined and I found Lemma 8.1.7 in book written by N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation. But it only works for $\alpha\in(0,1)$. So I don't know how to apply Tauberian theorem again when the probability mass doesn't concentrate on $[0,\infty)$.