Which coupling minimises the following cyclic sum?

Question

We recall that a coupling of probability distributions $\mu_1, \dots, \mu_n$ on $\mathbb R$ is a set of random variables $X_1, \dots, X_n$ defined on the same probability space such that $X_i$ is distributed according to $\mu_i$ for each $i$.

Let $\mu_i = \text{Uniform}([0, 1])$ for all $1 \leq i \leq n$. What is the value of the following optimisation problem, and by what coupling is it achieved?

$$\min_{X_i \text{ a coupling of } \mu_i} \mathbb E \left [\sum_{i = 1}^n \frac{X_i}{X_{i+1}} \right].$$

Here by convention we set $X_{n+1} = X_1$.

Answer 1

The sum in the excpected value is bounded below by $n$ (by the arithmetic-geometric mean inequality). The lower bound is achieved when all the variables are equal. Whence the value of you minimum is $n$ and is achieved for the coupling $X_1=\dots=X_n=Y$, $Y\sim \mathrm{U}[0,1]$.