I know that if $(X_j , j \geq 1)$ is a sequence of i.i.d. process with $\sigma$ being the probability distribution of $X_j's$ and $N \sim \text{Poisson}(\lambda)$ independent of the $X_j's$, then the compund Poisson random variable $$Z = \sum_j^N X_j \sim CP(\lambda, \sigma)$$ is such that its characteristic function is given by: $$\varphi_X(t)= e^{\lambda(\,\phi_X(t)- 1 \,)}, \varphi_X(t) = \int_{\mathbb{R}}e^{itx}d\sigma(x)$$ Moreover, for the centered compound Poisson variable $\tilde{Z} = Z - \mathbb{E}Z = Z - \lambda \mathbb{E}X$, its ch. f. is given by \begin{equation}\label{eq1} \varphi_{\tilde{Z}}(t)= e^{\lambda\phi(t)},\quad \phi(t)= \int_{\mathbb{R}}\Big(e^itx - 1 - itx\Big)d \sigma(x)\tag{1} \end{equation} But I would like to know if there is any $\bar{X}_j$ so that the characteristic function of $$\bar{Z}= \sum_{j=1}^N \bar{X}_j$$
is given, directly (without centering), by equation (\ref{eq1}). I tried putting $\bar{X}_j = X_j - \mathbb{E}X_j$, but it didn't work.
some help!