Abstract: If $\varphi$ is a non-negative integrable function on $\mathbb{R}$, satisfying some rather general conditions, then the behavior of the $n$-fold convolutions $\varphi^{(n)}=\underbrace{\varphi \ast \varphi \ast \varphi \ast \cdots \ast \varphi}_{n \; \mathrm{times}}$ as $n\to \infty$ is described by the Central Limit Theorem. However, the problem of describing the behavior of $\varphi^{(n)}$ rises in several contexts when $\varphi$ is not a positive function. Central Limit Theorem was extended to such non-positive probabilities (signed or complex-valued) by a number of authors: A. Zhukov in numerical analysis, V. Krylov in partial differential equations, R. Hersh and K. Hochberg in distribution theory. In these extensions it was typically assumed that $\widehat{\varphi}$ (the Fourier transform of $\varphi$) satisfies the condition $\ln \widehat{\varphi}(t)=(A+o(1))t^q, \; t\to 0$, where $A\neq 0, \; q$ an integer $\geq 2$, and it was shown that $n^{\frac{1}{q}} \varphi^{(n)}(n^{\frac{1}{q}} x)$ converges in some weak sense to the inverse Fourier transform of $\exp(At^q))$. Recently B. Baishanski has argued that in case $Re A=0$, the scaling factor $n^{\frac{1}{q}}$ is not the ``natural'' one. Namely, it is known that in case of probability densities the scaling factor is essentially unique, so the ``natural'' scaling for $\varphi^{(n)}$ in complex-valued case should be the same as the essentially unique scaling of $|\varphi^{(n)}|/\|\varphi^{(n)}\|_{L^1}$. He has considered two examples and shown that under the natural scaling one obtains analogues of the Central Limit Theorem of a new kind when $Re A=0$. We have obtained more general results of the same nature. Our main theorem is \bf Theorem. \it Suppose that $\varphi$ is a complex-valued function such that \begin{eqnarray*} & & \varphi(\nu) \in \mathrm{L}^1(\mathbb{R})\cap \mathrm{L}^s(\mathbb{R})\; \mbox{ for some } \; s>1\\ & & |\widehat{\varphi}(t)|<|\widehat{\varphi}(0)|=1 \; \mbox{ for every } \; t \neq 0\\ & & \nu^2 \varphi(\nu) \in \mathrm{L}^1(\mathbb{R})\\ & & \ln \widehat{\varphi}(t) =i\sum^q_{r=p} a_rt^r-\beta t^q+G(t), \; t\in U-\mbox{a neighborhood of zero }, \\ & & \mbox{where } p, q \in \mathbb{N}, \; 2\leq p 0, \beta>0,\\ & & G\in C^2(U), \; G''(t)=O(t^{q-1}), \;G(t)=O(t^{q+1}), \;t\to 0. \end{eqnarray*} Let the \it scaled $n$-fold convolutions be given by \begin{eqnarray*} \sqrt{\lambda}\psi_{\lambda}(c), \; \mbox{ where }\; \psi_{\lambda}(c)=\sigma_n \varphi^{(n)}(\sigma_n \lambda c), \; \sigma_n=n^{\frac{1}{q}}, \; \lambda=n^{1-\frac{p}{q}} \end{eqnarray*} Then $\sqrt{\lambda}\psi_{\lambda}$ will exhibit one of two types of very regular divergence: \begin{enumerate} \item If $p$ is even, then \begin{eqnarray*} \sqrt{\lambda}\psi_{\lambda}(c)=F(c)u_{\lambda}(c)+r_{\lambda}(c), \end{eqnarray*} where $F(c)$ is an integrable function given by \begin{eqnarray} F(c)=B |c|^{-\gamma}\exp\left\{-E |c|^{\rho}\right\}, \label{opred funkcii F k kot stremyatsya moduli n kr svertok,abstr} \end{eqnarray} $|u_{\lambda}(c)|=1$ and \begin{eqnarray} |r_{\lambda}(c)|\leq \frac{C}{\lambda^{\epsilon}|c|} \; \mbox{ for } \; |c|<\ln^2\lambda, \; \lambda >\Lambda\; , \; where \label{ocenka glavnogo ostatka, ab} \end{eqnarray} $B, \; E, \; C, \;\epsilon, \gamma, \rho$ and $\Lambda$ are positive constants that depend on $p$ and $q$. Moreover, the family $(u_{\lambda})$ satisfies: for every interval $[a,b]$ not containing zero and every arc $l$ of the unit circle \begin{eqnarray*} \lim_{\lambda \rightarrow \infty}\frac{1}{b-a}m\{c: c \in [a,b], u_{\lambda}(c) \in l\}= \frac{1}{2 \pi} (arclength \; \mathrm{ of } \; l) \end{eqnarray*} \item If $p$ is odd, then \begin{eqnarray*} \sqrt{\lambda}|\psi_{\lambda}(c)|=\Phi(c)t_{\lambda}(c)+r_{\lambda}(c), \end{eqnarray*} where \[ \Phi(c)=\left\{ \begin{array}{cc} 0, \mbox{ if } c<0\\ 2 F(c), \mbox{ if } c>0, \end{array} \right. \] $F(c)$ being defined by (\ref{opred funkcii F k kot stremyatsya moduli n kr svertok,abstr}), $0\leq t_{\lambda} (c) \leq 1$ and $r_{\lambda}(c)$ satisfies (\ref{ocenka glavnogo ostatka, ab}). Moreover, the family $(t_{\lambda})$ satisfies: for every interval $[a,b]$ not containing zero and for every subinterval $I$ of $[0,1]$ \begin{eqnarray*} \lim_{\lambda \rightarrow \infty}\frac{1}{b-a}m\{c: c \in [a,b], t_{\lambda}(c) \in I\}= \frac{2}{ \pi} \int_{I}\frac{dx}{\sqrt{1-x^2}} \end{eqnarray*} \end{enumerate} \rm As an application of our main Theorem we obtain a continuous analogue of Girard's asymptotic formulae for norms of powers of absolutely convergent Fourier series.