$\xi$ has modification $\eta$ with continuous paths $\iff \exists c \in \mathbb R$ s.t. $P(\xi_{0}=c)=1$

Question

Let $\xi:=(\xi_{n})_{n\geq 0}$ be IID and $\eta:=(\eta_{n})_{n \geq 0}$:

A path is defined as a map for fixed $\omega$ that $[0,\infty[\ni t\mapsto\xi_{t}(\omega)$

Show that:

$\xi$ has modification $\eta$ with continuous paths $\iff \exists c \in \mathbb R$ s.t. $P(\xi_{0}=c)=1$

I assume that the $\Rightarrow$ is easiest to prove and I guess it is easiest via contradiction.

So assume that for any $c \in \mathbb R$ that $P(\xi_{0}=c)<1$ then there must exist some set $A$ where $\{\xi_{0}=c\}\cap A=\varnothing$ and $P(A)>0$, but since $\xi$ has modification $\eta$ with continuous paths, then $P(\eta_{0}=c)<1$ and thus $\{\eta_{0}=c\}\cap A=\varnothing$. Now we need to use the continuity of our map, but I am not sure whether I am even on the right path. Any hints?

Answer 1

No generality is lost by assuming that all of the random variables $\xi_n$, $\eta_n$ are bounded by one fixed constant. If not, replace them by $\arctan\xi_n$ and $\arctan\eta_n$ and proceed.

The continuity of $\eta$ implies that $(\omega,n)\to\eta_n(\omega)$ is jointly measurable, permitting the use of Fubini's theorem. Let $c=\Bbb E[\eta_n]=\Bbb E[\xi_n]$. Use Fubini to show that $$ \Bbb E\left[\left(\int_a^b\eta_t\,dt\right)^2\right]=(b-a)^2c^2, $$ for each choice of $0\le a<b<\infty$. From this you see that the variance of the random variable $\int_a^b\eta_t\,dt$ is zero. Therefore $\int_a^b\eta_t\,dt = (b-a)c$, for all $0\le a<b<\infty$, a.s. From this and the continuity of $\eta$ it follows that $\Bbb P[\eta_t=c$ for all $t\ge 0]=1$. In particular, $\Bbb P[\xi_0=c]=\Bbb P[\eta_0=c]=1$.

Answer 2

If your variables are bounded, you may easily get the implication by showing that covariance converges to variance by dominated convergence, so variance must be zero.

But in the general case where $\xi_0$ may not even have a variance. Take $I$ and $J$ two disjoint closed intervals, and $t_n, s_n$ two sequences both converging to the same $t$, with disjoint supports. By second Borel-Cantelli lemma, the events $E_n=\{\eta_{t_n}\in I \text{ and } \eta_{s_n} \in J\}$ are independent with the same probably, so if this probability is non zero, $$\mathbb P(\limsup E_n)=1$$ which is clearly incompatible with the continuity hypothesis.

Yet unless a random variable is a.s. constant, one may always find such $I$ and $J$. So here $\xi_0$ must be a.s. constant