What does $P(|X_n - X| \geq \epsilon)$ represent intuitively?

Question

I get that $P(|X_n - c| \geq \epsilon)$ represents the probability that the random variable $X_n$ is outside the interval of $(c - \epsilon, c + \epsilon)$ but I am not sure how it works with a random variable where $P(|X_n - X| \geq \epsilon)$. How can a random variable take only a single value like a constant? How would you look at this intuitively with the constant $c$ case?

Answer 1

I think this is a case where thinking of $X_n$ and $X$ as functions is helpful.

Suppose we had two functions $f,g : \mathbb R\to\mathbb R$ and maybe they look like this:

In that plot I marked on the x-axis the regions where $|f-g| > .75$, where $0.75$ was chosen arbitrarily. We can then think of the total length of the regions of disagreement as a sense of how different $f$ and $g$ are. If there are long intervals where $f$ and $g$ are more than $0.75$ apart, we'll have a big disagreement, while if there are only small regions of disagreement then $f$ and $g$ are pretty similar.

We are doing the exact same thing in $\newcommand{\e}{\varepsilon}P(|X_n-X|>\e)$: by definition, $$ P(|X_n-X|>\e) = P\left(\{\omega\in\Omega : |X_n(\omega) - X(\omega)| > \e\}\right). $$ Here, the sample space $\Omega$ is playing the role of the domain in my example with $f$ and $g$, and instead of just looking at the length of the region of disagreement (which would be using the Lebesgue measure), we measure it with $P$. But fundamentally the big idea is that we're measuring the region of the domain where $X_n$ and $X$ disagree by more than our threshold of $\e$.

If one of the functions happens to be constant it's the exact same idea, just simpler since only one thing is varying now.

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Here's the code for that plot if you want to experiment. I sampled $f$ and $g$ iid from a Gaussian process with a squared exponential kernel to get smooth arbitrary-looking functions.

set.seed(2)
n <- 1000
xseq <- seq(-5, 5, length=n)
K <- exp(-.5 * as.matrix(dist(xseq))^2)
sims <- MASS::mvrnorm(2, rep(0,n), K)
plot(sims[1,] ~ xseq, type="l", col=2, lwd=2, xlab="x", ylab="y",
     main="f and g with |f-g|>.75 marked", ylim=c(-2,2))
lines(sims[2,] ~ xseq, col=4, lwd=2)
legend("topleft", c("f", "g"), lwd=2, col=c(2,4), bty="n")

thresh <- .75
rug(xseq[abs(sims[1,] - sims[2,]) > thresh])