Is this probabilistic principle for stochastic processes known?

Question

In the course of a proof, I used the following principle, which seems so intuitive that it should have a name:

Suppose one has a stochastic process $X_t$, for $t \in \omega$, on a (possibly infinite) state space $S$. Suppose one has $n$ Boolean functions $A_i, B_i: S \rightarrow \{ 0, 1 \}$. Suppose that, for all $s \in S$ and all $t \in \omega$, one has the condition that $P(B_i(X_{t+1}) = 1 \mid X_t = s) \leq p_i$.

Now consider the following event $E$:

1. For all $i = 1, ..., n$, there is a unique time $t_i$ such that $A(X_{t_i}) = 1$;

2. For all $i \neq i'$ we have $t_i \neq t_{i'}$

3. For all $i = 1, ..., n$ we have $B(X_{t_i+1}) = 1$

Then the probability of the event $E$ is at most $p_1 p_2 \dots p_n$.

Is the principle already known? Would one consider it "obvious" ?

Answer 1

I don't think this is true.

Consider one dimensional Brownian motion with $X_0 = 1$ and let $B_i$ be the indicator of the event that the $k$th decimal place is a $0$ (so all of our $B_i$'s are the same). Let $k$ be large enough that we may choose $p_i = p< 1$.

Now let $A_i$ be the indicator of the interval $\left(0,\frac {i\cdot\varepsilon}n\right)$. Abuse notation and let $t_0$ be the first hitting time of $0$.

Now let $E'$ be the event that $B_i = 1$ for the entire time interval $(t_n+1,t_0+1)$. Then $P(E')$ is strictly positive and we have $E'\supset E$.

But as $t_n$ is the first hitting time of $\varepsilon$ the event $E'$ does not depend on $n$, hence we may choose $n$ large enough that $P(E') > p^n$.

So I would say that you're missing at least one assumption.