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Let 
$G$ be a commutative group, 
$Y$ a real Banach space and 
$f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation for all 
$$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$
$x,y\in {\rm\Omega}$, where 
$H$ is a finite cyclic subgroup of 
$\text{Aut}(G)$ and 
${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation for all 
$$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$
$(z,{\it\zeta})\in {\rm\Omega}$, where 
$f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and 
${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure 
$0$.
The Cauchy functional equation Φ(x + y) = Φ(x) + Φ(y) is generalized to the form 
, assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by 
, for all x, y ≧ 0;(2) a characterization of the symmetric α-stable distribution by the equidistribution of linear statistics.
