- I have to study a peculiar Rando walk, whose law $ P^\beta (\cdot)$ is depending on a factor $\beta$.
- In parallel, I have another measure $Q^\beta(\cdot) $ depending on the same $\beta$.
- The $\beta$ is actually random, and behaves according to a certain measure $\rho$.
I'm interested in understading the resultin mixed measure: $$ \mathbb{P}(\cdot \in A):= \int P^\beta (\cdot\in A)Q^\beta(\cdot\in A) \ d\rho(\beta)$$
I would like to find back two factor I already know, i.e. the 2 measure separatly mixed w.r. to $\rho$: $$ {P}(\cdot \in A):= \int P^\beta (\cdot\in A) d\rho(\beta)$$ $$ Q(\cdot \in A):= \int Q^\beta(\cdot\in A) \ d\rho(\beta)$$
Is there a way to express the integral of the product into something depending on $P(\cdot)$ and $Q(\cdot)$?
Since $P^\beta,\ Q^\beta$ are less than one and $\rho$ is a probability measure, $$\forall A,\ \ P^\beta(A),Q^\beta(A)\in L^p(\beta)\ \forall p=1,2,...\infty$$ this allows you to apply Holder's inequality in various ways: $$\int P^\beta(A)Q^\beta(A)d\rho(\beta)\le \sup_\beta Q^\beta(A)\int P^\beta(A)d\rho(\beta)$$ or $$\int P^\beta(A)Q^\beta(A)d\rho(\beta)\le \sqrt{\int (Q^\beta(A))^2 d\rho(\beta)} \sqrt{\int (P^\beta(A))^2 d\rho(\beta)}$$ in order to find an equality you need some hypothesis on $\rho$ and on its density function, if exists