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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

$$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$

for all

$x,y\in {\rm\Omega}$, 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

$$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$

for all

$(z,{\it\zeta})\in {\rm\Omega}$, 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.

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ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

A functional equation and its application to the characterization of the Weibull and stable distributions

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2 results

ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

A functional equation and its application to the characterization of the Weibull and stable distributions