A Cauchy mean is another Cauchy, so the average does not settle

Let $X_1,\ldots,X_n$ be independent Cauchy random variables with location $\theta\in\mathbb R$ and known scale $1$, each having density $$ f_\theta(x)=\frac1{\pi\bigl(1+(x-\theta)^2\bigr)}, \qquad x\in\mathbb R. $$ Using characteristic functions, prove that $\bar X$ has exactly the same Cauchy$(\theta,1)$ law for every $n\ge1$. Conclude that $\bar X$ is not consistent for $\theta$ and that the ordinary CLT-plus-WLLN justification for the sample mean does not apply. Explain why a method-of-moments estimator based on $\mathbb E[X_1]$ is not defined. Write the log-likelihood and the score equation for the MLE of $\theta$. Prove that the one-observation score is bounded and that its expectation is zero, and compute the one-observation Fisher information $$ I(\theta)=\mathbb E_\theta\bigl[(\partial_\theta\log f_\theta(X))^2\bigr]. $$ (The value is the constant $1/2$.) Contrast this regular score behavior with the inconsistent sample mean. Do not construct a Neyman–Pearson test of $\theta=\theta_0$, and do not replace the model by $N(\theta,1)$.

The characteristic function of $\bar X$ is $[\varphi(t/n)]^n$.