Frequent hypercyclicity of random entire functions for the differentiation operator

In this note we study the random entire functions defined as power series $f(z)={\sum }_{n=0}^{\mathrm{\infty }}\frac{X_n}{n!}{z}^n$ with independent and identically distributed coefficients $(X_n)$ and show that, under very weak assumptions, they are frequently hypercyclic for the differentiation operator $D: H(\C) \to H(\C)$,$f \mapsto Df = f’$. This gives a very simple probabilistic construction of $D$-frequently hypercyclic functions in $H(\text{C})$. Moreover we show that, under more restrictive assumptions on the distribution of the $(X_n)$, these random entire functions have a growth rate that differs from the slowest growth rate possible for $D$-frequently hypercyclic entire functions at most by a factor of a power of a logarithm.