Consider a circle $S$ of length $\theta$. Now suppose, we delete an interval $I$ of length $|I|$ (I'll drop the $|\cdot|$ notation for length and directly write $I$) from it.
Now on $S-I$, I choose a point $U$ uniformly at random and construct around $U$, another interval $J_{U}$ having it's length distributed as a $\min(J^{+}+J^{-}, \theta)$ where $J^{+}$ and $J^{-}$ are iid exponential$(1)$ variates.
What is the expected length of this interval constructed? i.e. what is $E(J_{U})?$
What I did was I considered one end point of $I$ to be the origin(or the pole/reference point of the circle with coordinates measured in anticlockwise direction) and then it is equivalent to constructing $J$ around $0$ and the whole interval $I$ shifted anticlocklwise by $U$. Then $J^{+}$ can be considered to be measured in anticlockwise direction and $J^{-}$ to be measured in clockwise direction about $0$.
Using the above, we get $E\left(\min(U,J^{+})+\min(\theta-I-U,J^{-})\right)$.
Now by conditioning on $U=u$, \begin{align*} &U\int_{U}^{\infty}e^{-z}\,dz+\int_{0}^{U}ze^{-z}\,dz +(\theta-I-U)\int_{\theta-I-U}^{\infty}e^{-z}dz+\int_{0}^{\theta-I-U}ze^{-z}\\ &=(1-e^{-U})+(1-e^{-(x-U)})\end{align*}
Now taking the expectation in $U$, we get $\frac{2x-2(1-e^{-x})}{x}$ where $x=\theta - I$ .
Is the above correct or am I missing some more terms?
Or even going a step backwards, is the conditional expectation of the length of $J_{u}$ given $U=u$ correct?
i.e. if $u$ is a fixed point on $S-I$, then what is the expected length of an interval (distributed as $\min(J^{+}+J^{-},\theta)$ for iid expo$(1)$) that can be constructed about $U$ on the arc $S-I$?
EDIT: I am now sure that I am missing more details. For example, what if $J^{+}>I+u$ and $\theta-J^{-}\leq J^{+}$, then I am certainly ending up ignoring some parts.
Can someone please help me with this.
Edit: To be more explicit, in words, what I want to do is :- We are looking to find the expected length of the interval $I'$ that is built outside of $I$ when we imagine $I$ to be part of the original circle.
This is the mental image I had while doing the problem
EDIT Additionally in the paper, the description given (this was the initial question which attracted close votes for needing more details) that if $I'$ is an independent interval(arc) having length law $\min(J^{+}+J^{-},\theta)$ built around $0$ and if $I$ is a fixed interval not containing $0$ then note the following definition $I\cap_{0} I'=\{x\in I': x\nleftrightarrow 0 \,\text{within}\, I'\setminus I\}$. Then $H_{0}(I):=E(I\cap_{0}I')$ and $UH_{0}=E(H_{0}(I-U))$ for an uniform $U$ variate on the circle.
It is this $UH_{0}$ and $(UH_{0})$ that I want to calculate. It is also claimed later in the paper(page 23) that $E(H_0(I-u)|U=u)$ equals the expected length of part of the interval $I'$ that can be constructed around $u$ outside $I^c$.
This is all the details I have.