If a distribution contains a non-trivial unbiased statistic of zero, then this distribution does not have a complete statistic?

Question

If a distribution contains a non-trivial unbiased statistic of zero, then this distribution does not have a complete statistic?

Here is what it means

Suppose we have a family of distribution $\mathcal{P}=\{\mathcal{P}_\theta:\theta\in\Theta\}$, and $X\sim\mathcal{P}$. If we there is a statistic $\delta(X)$ so that $\mathbb{E}_\theta(\delta(X))=0,\text{ }\forall\theta\in\Theta$, then $\mathcal{P}$ does not have a complete statistic.

Is this correct? If not, is there any counterexample?

Any reference is also appreciated.

Answer 1

False.
Take the multivariate normal distribution where all the means and correlations are 0 and all the variances are $\theta$.
There is a complete and sufficient statistic for $\theta$- see here.
Now, take $\delta (X)$ as the sum of the components of the random vector $X$.