About problem 4.4a page 57 of
https://staff.fnwi.uva.nl/b.j.k.kleijn/AsympStat-LecNotes2010.pdf
For this problem I could check the conditions of lemma 4.9 except the latter with epsilons. First I wrote \begin{align} -\frac{1}{n}\sum_{i=1}^n \psi(X_i-\theta) \xrightarrow{P} -\mathbb{E} \psi(X_1-\theta) \end{align} My thought was $\psi$ is odd, so $\theta_0$ has s.t. to do with that, because otherwise the oddness would be too coincidentally. Anyway we must have $\theta_0$ such that \begin{align} \int_{-\infty}^{\infty} \psi(x-\theta_0)f_{X_1}(x)dx = 0 \end{align} I can imagine that there is such $\theta_0$, but expressing it in the density is my main problem.
Is there anyone with such an expression? Thanks in advance :)