Let us consider a Levy process $X_t$ with the characteristic function of the form: $$ \hat{\mu}(t) = \text{exp}\biggl( i\gamma t - \frac{\sigma^2}{2} t^2 + \int_{\mathbb{R}} (e^{itx} - 1 - itx \mathbb{1}_{|x|\leq 1})\nu(dx) \biggr), $$
where $(\gamma, \sigma, \nu)$ are parameters called the generating triplet.
It is well known-that we have to assume that $$ \nu(\{0\}) = 0 \quad \text{and} \quad \int_{\mathbb{R}}(1 \wedge x^2)\nu(dx)<\infty $$ These conditions guarantee that the integral $\int_{\mathbb{R}} (e^{itx} - 1 - itx \mathbb{1}_{|x|\leq 1})\nu(dx)$ is finite.
The indicator function occurring in the formula on characteristic function can be replaced by some function $c(x)$ and then: $$ \hat{\mu}(t) = \text{exp}\biggl( i\gamma_c t - \frac{\sigma^2}{2} t^2 + \int_{\mathbb{R}} (e^{itx} - 1 - itx c(x))\nu(dx) \biggr), $$ where $$ \gamma_c = \gamma + \int_{\mathbb{R}} x (c(x) - \mathbb{1}_{|x| \leq 1})\nu(dx) $$
My question is: What are necessary and sufficient conditions for $c(x)$?
I found in Sato's book titled 'Levy Processes and Infinitely Divisible Distributions' that sufficient conditions are as follows:
$$ c(x) = 1 + o(|x|) \quad \text{as} \quad |x|\rightarrow 0 $$ $$ c(x) = O\biggl(\frac{1}{|x|}\biggr) \quad \text{as} \quad |x|\rightarrow \infty $$
However, I did not find anything about necessary conditions.
In the formula on characteristic function of $X_1$, the Levy process $X=\{X_t,t\ge0\} $ at $t=1$, \begin{align*} \hat{\mu}_1(u)&=\exp\Big(i\gamma u-\frac{\sigma^2}{2}u^2+\int_{\mathbb{R}}(e^{iux}-1-iux1_{\{|x|\le 1\}}) \nu(dx) \Big)\\ &=\exp\Big(i\gamma_c u-\frac{\sigma^2}{2}u^2+\int_{\mathbb{R}}(e^{iux}-1-iuxc(x)) \nu(dx) \Big), \end{align*} since \begin{equation*} \int_{\mathbb{R}}(1\wedge x^2)\nu(dx)<\infty, \tag{1} \end{equation*} to guarantee the integral $\int_{\mathbb{R}}(e^{iux}-1-iuxc(x)) \nu(dx)$ is finite for each $\nu$ satisfying (1), the necessary (and sufficient) conditions are \begin{gather*} c(x)=O\Big(\frac{1}{|x|}\Big)\quad\text{as}\quad x\to\infty,\tag{2}\\ c(x)=1+O(|x|), \quad\text{as}\quad\text{as}\quad x\to0.\tag{3} \end{gather*} On the other hand, in the discussion of weak convergence of IDDF (infinitively divisible distributions), the (2)(and (3)) are not sufficient(cf. Sato's book pp38 and pp41, Theorem 8.7), (2) and \begin{gather*} c(x)=1+o(|x|), \quad\text{as}\quad\text{as}\quad x\to0.\tag{3} \end{gather*} are necessary.