Cdf of a discrete random variable and convergence of distributions

[Artusartos]

In the page that I attached, it says "...while at the continuity points x of $F_x$ (i.e., $x \not= 0$), $lim F_{X_n}(x) = F_X(x)$." But we know that the graph of $F_X(x)$ is a straight line y=0, with only x=0 at y=1, right? But then all the points to the right of zero should not be equal to the limit of $F_{X_n}(x)$, right? Because $F_X(x)$ is always zero at those points, but $F_X(x)$ is 1? So how do I make sense of that?

Thanks in advance

 

[Stephen Tashi]

But we know that the graph of $F_X(x)$ is a straight line y=0, with only x=0 at y=1, right?

No, I think $F_X(x)$ is the cumulative distribution, not a density function.

 

[Artusartos]

No, I think $F_X(x)$ is the cumulative distribution, not a density function.

Oh, ok...

But it's still confusing. What if n=4 (for example)? Then $$F_{X_n} = 1$$ if $$x \geq 1/4$$, and $$F_{X_n}=0$$, when $$x < 1/4$$, right? So for any x between 0 and 1/4, the limit at those points is 0, but the limit of $$F_X$$ at those points is 1...so the limits are not equal, are they?

 

[Stephen Tashi]

So for any x between 0 and 1/4, the limit at those points is 0,

What limit are you talking about? Something like $lim_{x \rightarrow 1/8} F_{X_4}(x)$ ? I see nothing in the discussion in the book that dealt with that sort of limit. The limits under consideration involve $n \rightarrow \infty$.

 

[blue_raver22]

for your question and for attaching the page for reference. The statement you referenced is discussing the convergence of distributions, which is a concept in probability theory that deals with the behavior of a sequence of random variables as the number of variables in the sequence increases. This is different from the cumulative distribution function (CDF), which is a function that maps the probability of a random variable being less than or equal to a certain value.

In the statement, the limit refers to the behavior of the CDF of the sequence of random variables (F_{X_n}(x)) as the number of variables (n) increases. The CDF of a discrete random variable is a step function, where the jumps occur at the values of the random variable. So, at the continuity points (x \not= 0), the limit of F_{X_n}(x) as n increases equals the value of F_X(x). This means that as the number of variables in the sequence increases, the CDF of the sequence approaches the CDF of the original random variable at those points.

You are correct in saying that the graph of F_X(x) is a straight line with a value of 1 at x=0 and a value of 0 at all other points. However, this is for the CDF of a specific random variable, not a sequence of random variables. The statement is discussing the behavior of the CDF of a sequence of random variables, which may have different CDFs at different points.

In summary, the statement is discussing the convergence of distributions, which is a concept in probability theory that deals with the behavior of a sequence of random variables. The limit mentioned is referring to the behavior of the CDF of the sequence as the number of variables in the sequence increases, and it does not necessarily equal the CDF of a specific random variable at all points. I hope this helps clarify the concept for you.