the formula $$E(\lambda):=\sum_{n=0}^\infty \frac{\lambda^n}{w_0\cdots w_n}\, e_n$$ defines a continuous $\TT$-eigenvector field $E:\TT\to X_p$ such that $\overline{\rm span}\, E(\TT)=X_p$. Hence $B_{\mathbf w}$ is strongly mixing in the Gaussian sense. This is known since [@BG2] if $p<\infty$, but it appears to be new for weighted shifts on $c_0(\NN)$.

The converse is true if $p\leq 2$ ( (\[shift\]) is satisfied if $B_{\mathbf w}$ is strongly mixing in the Gaussian sense) since in this case $X_p$ has cotype 2, but the case $p>2$ is not covered by Theorem \[WS\]. However, it turns out that the converse does hold true for any $p<\infty$. In fact, (\[shift\]) is satisfied as soon as the weighted shift $B_{\mathbf w}$ is frequently hypercyclic ([@BR]). As shown in [@BG2], this breaks down completely when $p=\infty$: there is a frequently hypercyclic weighted shift $B_{\mathbf w}$ on $c_0(\NN)$ whose weight sequence satisfies $w_1\cdots w_n=1$ for infinitely many $n$. Such a weighted shift does not admit any (nontrivial) invariant Gaussian measure.

Let us also recall that, in contrast with the ergodic properties, the hypercyclicity of $B_{\bf w}$ does not depend on $p$: by a well known result of H. Salas ([@Sal]), $B_{\mathbf w}$ is hypercyclic on $X_p$ for any $p$ if and only 