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Part 1. The equality towards the end of the proof requires computing a conditional expectation $E_\theta (X_1 | X_1^2=y^2)$ where, earlier in the text, we are told that $X_1$ is normally distributed w …

For a given fixed parameter $\theta$, the coverage of a Bayesian credible interval can be greater or lower than the nominal value. For example, coverage may be greater than nominal if the prior is inf …

Spherical symmetry is a special case of exchangeability. The invariance property for spherically symmetric sequences that you describe is indeed the standard definition of spherical symmetry e.g. see …

It looks like you have a good idea of the kind of tools you could use to prove this statement, but I was a bit unclear about your attempted proof, in particular what you meant by the measure/distribut …

Starting from the underlined expression, the next expression is obtained by writing $\frac{1}{i(n-i)}$ as a partial fraction decomposition $\frac{A}{i}+\frac{B}{n-i}$ for some $A$, $B$. The next expre …

For the second equality, think about the integral $$\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} [I(u,X_1)-I(u,X_2)][I(v,Y_1)-I(v,Y_2)]du dv $$ which is a random variable. The structure of the in …

No, the stated Bonferroni style inequality does not hold for characteristic functions. Certainly not in the way you've formulated it, and I can't think of any other way of interpreting the Bonferroni …

There is something not quite right about the equation at the end of your question. On the left you have $n\to\infty$, but on the right $n$ still appears as a finite number. Fortunately, your question …

(1) You are correct, the $T_i$ you defined are not independent. (2) I don't think it's possible to write $U$ as a sum of iid random variables. As $U$ is not a sum of iid random variables, showing that …