I have a problem understanding why the concept of right continuity of a cdf has to decrease a sequence $x_n$ or $y_n$ to a limiting value $x$ or $y$ respectively.
I do not understand why in this statement $x_n$ or $y_n$ has to decrease to $x$ or $y$:-
$\mathbf{F}_{X,Y}(x,y)$ is right continuous in both co-ordinates, i.e. if $x_n \searrow x$ and $y_n \searrow y$, then $\mathbf{F}_{X_{n},Y_{n}}(x_{n},y_{n}) \searrow \mathbf{F}_{X,Y}(x,y)$
In fact in case of univariate cdfs the sequences $x_n$ or $y_n$ increase to infinity.
Take any sequence, $y_n \nearrow \infty \\ \text{i.e. } \ y_1 \leq y_2 \leq y_3 \leq \cdots \\ \text{hence }\ \lim_{n \to \infty} y_n = \infty$
Fix $x \in \mathbb{R}$
Define event $\large{{A}_{n}}$ $= (X\leq x \ \cap Y\leq y_n), n\geq 1$
Since $y_n \uparrow$, it follows that $\large A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots$
$$\therefore y_n \nearrow \infty \implies \ \huge \cup_{\small{{n=1}}}^{\small{\infty}}\large A_n =\normalsize(X\leq x \ \cap Y< \infty) $$
This can be briefed by writing
$\large A_n$ $\nearrow \underbrace{(X\leq x \ \cap Y\leq \infty)}_{\text{let's call this event }A}\\ \therefore \large A_n \nearrow A \\ \implies \large \mathbb{P} (A_n) \nearrow \mathbb{P}(A) \\ \implies \normalsize \mathbb{P}(X\leq x \ \cap Y< y_n) \nearrow \normalsize \mathbb{P}(X\leq x)$
We can now write the univariate cdf of the random variable $X$ as
$$\large\mathbf{F}_{X}{(x)} \\ = \mathbb{P}(\mathrm{X}\leq x) \\ = \mathbb{P}(\mathrm{X}\leq x \cap \mathrm{Y}<\infty) \\ = \mathbb{P}(\lim_{y \nearrow \infty} \{ \mathrm{X}\leq x \cap \mathrm{Y}\leq y\})\\ = \mathbb{P}(\lim_{n \to \infty} \{ \mathrm{X}\leq x \cap \mathrm{Y}\leq y_{n}\}), \ \textrm{ for any sequence }\ y_{n}\uparrow \infty \\ = \lim_{n \to \infty} \mathbb{P}( \mathrm{X}\leq x \cap \mathrm{Y}\leq y_{n}) \\ = \lim_{y \nearrow \infty} \mathbb{P}( \mathrm{X}\leq x \cap \mathrm{Y}\leq y ) \\ = \lim_{y \to \infty} \mathbb{P}( \mathrm{X}\leq x \cap \mathrm{Y}\leq y ) \\ = \large \lim_{y \to \infty} \mathbf{F}_{\mathrm{X}}( {x,y} ) $$
Similarly, taking any sequence $x_n \nearrow \infty$, fixing $y \in \mathbb{R}$ and defining another event $\large B_n$ $=(X \leq x_n \ \cap Y \leq y), n\geq 1$, we can show that $ x_n \nearrow \infty \implies \ \huge \cup_{\small{{n=1}}}^{\small{\infty}}\large B_n =\normalsize(X\leq x_n \ \cap Y\leq y) $ and $\normalsize \mathbb{P}(X\leq x_n \ \cap Y\leq y) \nearrow \normalsize \mathbb{P}(Y\leq y)$ and we can write the cdf of the random variable $Y$ as $\large \mathbf{F}_{Y}{(y)}= \large \lim_{x \to \infty} \mathbf{F}_{\mathrm{Y}}( {x,y} )$.
So in both cases whenever we represent the univariate cdf of $X$ and $Y$, we see that the sequences $x_n$ and $y_n$ are going to infinity.
Also, by the properties of bivariate cdf, we have (for r.v.s $X,Y$),
- $\forall a_1, a_2, b_1, b_2 \in \mathbb{R} \ \text{ with } \ a_1<a_2 \ \text{ and } \ b_1<b_2$,
$\mathbf{F}(a_2,b_2) - \mathbf{F}(a_2,b_1) - \mathbf{F}(a_1,b_2) + \mathbf{F}(a_1,b_1) $
$\forall x\in \mathbb{R} \ \lim_{y \to -\infty}\mathbf{F}_{X,Y}(x,y) = 0 \\ \forall y\in \mathbb{R} \ \lim_{x \to -\infty}\mathbf{F}_{X,Y}(x,y) = 0$
$\lim_{\\{x \to \infty \\ y \to \infty}} \mathbf{F}_{X,Y}(x,y) = 1 $
$\mathbf{F}_{X,Y}(x,y)$ is right continuous in both co-ordinates, i.e. if $x_n \searrow x$ and $y_n \searrow y$, then $\mathbf{F}(x_{n},y_{n}) \searrow \mathbf{F}(x,y)$
• So my questions are
a. Why $x_n$ or $y_n$ has to decrease to $x$ or $y$ to prove a cdf is right continuous?
b. What happens if they increase to $x$ or $y$? Will they become left continuous?
c. Can we say if $x_n \searrow x$ and $y_n\nearrow y$ (or vice-versa) then $\mathbf{F}_{X,Y}(x,y)$ will have jump discontinuity?
Question b and c are out of my curiosity regarding the behavior of cdf in context of right continuity.
Also the following post uses a decreasing sequence of events in order to prove the right continuity of a cdf, which I specifically do not understand why.
Any help, explanations in this regards are welcome, valuable and highly appreciated.
For a., that's just a definition of being right-continuous in both coordinates. Another way of saying "$f$ is right-continuous in both coordinates" is "if $x_n \searrow x$ and $y_n \searrow y$, then $\lim_{n \rightarrow \infty} f(x_n,y_n) = f(x,y)$." That doesn't mean this is the only situation where $\lim_{n \rightarrow \infty} f(x_n,y_n) = f(x,y)$. As you pointed out, there may be some non-decreasing sequences $x_n \rightarrow x$ and $y_n \rightarrow y$ such that $\lim_{n \rightarrow \infty} f(x_n,y_n) = f(x,y)$, but that is entirely separate from right-continuity.
The statement here is "Any cdf $F$ is right-continuous in both coordinates." To prove that, we can show the equivalent statement "if $x_n \searrow x$ and $y_n \searrow y$, then $\lim_{n \rightarrow \infty} F(x_n,y_n) = F(x,y)$," which is why we take arbitrary sequences $x_n \searrow x$ and $y_n \searrow y$.
For b., if we took any $x_n \nearrow x$ and $y_n \nearrow y$ and were able to show $\lim_{n \rightarrow \infty} F(x_n,y_n) = F(x,y)$, that would indeed show $F$ is left-continuous. The issue is that some cdfs aren't left-continuous, so we wouldn't be able to show that whenever $x_n \nearrow x$ and $y_n \nearrow y$, we have $\lim_{n \rightarrow \infty} F(x_n,y_n) = F(x,y)$. We can probably find some sequences like that, and some cdfs that are left-continuous, but we couldn't show it for every sequence and every cdf.
For c., I'm not really sure what you are asking. I don't see how the sequences $x_n$ and $y_n$ relate to $F$. For right-continuity, we take $x_n \searrow x$ and $y_n \searrow y$ and conclude $\lim_{n \rightarrow \infty} F(x_n,y_n) = F(x,y)$, so there is a relation between the sequences and $F$. Here, I don't see any such relation we're assuming (or trying to show).