Asymptotics or it is also know as the large sample theory, refers to behavior of estimator as the sample goes to infinity. It is generic framework for assesment of properties of estimators and statistical tests. It is assumed that the sample size n grows indefinitely, and the properties of statistical procedures are evaluated in the limit as n → ∞

It is similar to law of large number, that states that for a sequence of iid (independent identical distributed) random variables X1, X2,,,the sample average X converge to the population mean , as sample size n → ∞ .

Lets, plot some charts in r and explain this, assume there are 10000 standand normals variables and try to plot the cummulative means and see the outcome.

n <- 10000
means <- cumsum(rnorm(n))/(1:n)
plot(means,type = "l", lty=2)
abline(h=0,col="red",lty=2)

So for example, the average could be the result of n coin flips, the sample proportion of heads.As we flip a fair coin over and over, it eventually converges to the true probability of a head. Here, the command is exactly flipping a coin 1,0000 times where 0 is a tail and 1 is a head. Take the cumulative sum, and then dividing it by 1 to n to get the cumulative means. When we plot the cumulative means, as the number of coin flips going into the sample proportion goes to infinity it converges to the true value, which is 0.5, which is right there.

n =10000
means <- cumsum(sample(0:1, n, replace = TRUE))/(1:n)
plot(means,type = "l", lty=2)
abline(h=0.5,col="red",lty=2)