Does one-to-one correspondence of the set of sample paths of $X$ and $Y$ imply that $X$ and $Y$ are indistinguishable?

Question

I have two discrete stochastic processes (urn models) $X(t)$ and $Y(t)$. There is a function $\varphi(.)$ such that whenever $\{x_0,x_1,...\}$ is a sample path of $X$ then $\{\varphi(x_0),\varphi(x_1),...\}$ is a sample path of $Y$. Similarly, whenever $\{y_0,y_1,...\}$ is a sample path of $Y$, $\{\varphi^{-1}(y_0),\varphi^{-1}(y_1),...\}$ is a sample path of $X$. There is, therefore, a one-to-one correspondence between the set of sample paths of $X$ and $Y$.
(1) Can we say that the set of sample paths are of equal size?
(2) There are two functions $f$ and $g$ that map $X$ and $Y$ into a discrete binary process $Z$ such that whenever $f$ maps a sample path $\mathbf{x}:\{x_0,x_1,...\}$ of $X$ into a sample path of binary variables $z=\{f(x_0),f(x_1),...\}$; $g$ maps $\{\varphi(x_0),\varphi(x_1),...\}$ to the same binary sample path $z$. i.e. $\mathbf{z}=\{f(x_0),f(x_1),...\}= \{g(\varphi(x_0)),g(\varphi(x_1)),...\}$. What can we say about $X$ and $Y$? Do we say that $X$ and $Y$ are projection-equivalent? Examples of $f: f(x)=1$ if $x>0.4, 0$ otherwise.
(3) The probability that each sample path $\mathbf{x}$ of $X$ and $\mathbf{y}=\varphi(\mathbf{x})$ of $Y$ occurs depends on the binary projected sample path $\mathbf{z}$. Do we have that $X$ and $Y$ have the same finite n-dimensional distributions?
Thanks.

Answer 1

Presumably $\phi$ is invertible correct? Also we are referring to continuous time discrete state space stochastic processes (some people refer to random sequences as being a sub-type of stochastic process, so I want to make sure I understand the question).

(1) If there is a bijection between the two sets of sample paths, then by the definition of cardinality they have the same cardinality.

(2) This seems to be more a property of the function $\phi$ than of X and Y.

(3) How would this necessarily be the case if f and g are not one-to-one? If the range of f and g has cardinality two (which you seem to be implying by describing z as being a binary stochastic process), then it seems unlikely that either f or g could be bijections.

Then there isn't a unique sample path of X and Y corresponding to any given binary sample path z (since the projection function isn't bijective), so although it might make sense to say that z depends on X or Y, it doesn't make sense to say that X or Y depends on z (since z doesn't determine any sample path X or Y uniquely).

$\phi(x)$ and y have the same finite dimensional distributions, as do x and $\phi^{-1}(y)$.

The fact that $\phi$ is an invertible Borel function means that we can conceivably describe the distributions of X vis a vis Y and vice versa as push-forward/pull-backs of one another, but otherwise that doesn't make them equally distributed or equivalent.