Let
- $(E,d)$ be a metric space
- $\lambda$ be a measure on $\mathcal B(E)$
- $p:E\to[0,\infty)$ with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $$\mu:=\frac{p\lambda}c$$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(X_n)_{n\in\mathbb N}$ be an $E$-valued independent identically $\mu$-distributed process on $(\Omega,\mathcal A,\operatorname P)$
- $(x_0,\varepsilon)\in E\times(0,\infty)$ and $$I:=\left\{n\in\mathbb N:d(x_0,X_n)\ge\varepsilon\right\}$$
How can we determine the distribution of $X_I$? Does the distribution admit a density with respect to $\lambda$? If so, what is this density?
Let $$A:=\left\{x\in E:d(x_0,x)<\varepsilon\right\}.$$ Assume $$\mu(A)<1.$$ We should have $$\operatorname P\left[I=n,X_n\in B\right]=\mu(A)^{n-1}\mu(A^c\cap B)\tag1\;\;\;\text{for all }n\in\mathbb N$$ and hence $$\operatorname P\left[X_I\in B\right]=\frac{\mu(A^c\cap B)}{1-\mu(A)}\tag2$$ for all $B\in\mathcal B(E)$.
But how can we determine the density of the distribution? (I would like to evaluate this density in a computer program and hence need a suitable expression for it.)