Can someone explain what plim is?

Question

In my Introductory Econometrics class we discussed a concept of "plim" or "probability limit. I'm not sure what this means though and my professor doesn't explain it well at all. Can someone tell me what this is if you have heard of it? It seems to be used in the same way we would use a regular limit but I just don't understand it. Thanks!!

Answer 1

According to this Wikipedia article

If we have a sequence of real numbers $x_1, x_2, x_3, \ldots$, then we have a precise meaning for the statement

\begin{equation} \lim_{n \to \infty} x_n = x. \end{equation}

In particular, for any $\varepsilon > 0$, there is an $N$ such that $|x_n - x| < \varepsilon$ whenever $n \geq N$.

Now suppose that I have a sequence of random variables $X_1, X_2, X_3, \ldots$. What do we mean when we talk about convergence of such a sequence? In fact, unlike the case with real numbers, there are many things that we could mean.

Perhaps we mean that the value of the random variables gets close to a real number $x$ in the sense that the probability that $X_n$ is very different from $x$ (i.e., $|X_n - x|$ is large) gets very small as $n$ gets big. Perhaps we mean that the distribution of $X_n$ gets very close to the distribution of some other random variable $Y$ as $n$ gets large (then we would need a definition for the distance between distributions).

So here is the definition of a probability limit.

Definition: Let $X_1, X_2, X_3, \ldots$ be a sequences of random variables and let $X$ be a random variable. $X_n \to X$ in probability if for every $\varepsilon > 0$ we have

\begin{equation} \lim_{n \to \infty} P(|X_n - X| \geq \varepsilon) = 0. \end{equation}

Because probabilities are real numbers between $0$ and $1$ the limit is a standard calculus style limit. The definition says that $X$ is the probability limit of $X_n$ if the probability that the real number $|X_n - X|$ is bigger than any positive $\varepsilon$ gets very small as $n$ gets large.

Example: Consider repeatedly throwing a fair coin. Let $X_n$ be a random variable that is $0$ if the $n$-th toss is tails and $1$ if it is heads. Let $S_n/n$ be the mean of $X_1, X_2, \ldots, X_n$. You probably know that the mean converges to $0.5$ if you flip the coin enough times. What we mean is that for any $\varepsilon > 0$ we have

\begin{equation} \lim_{n \to \infty} P\left(\left|\frac{S_n}{n} - 0.5 \right| \geq \varepsilon \right) = 0. \end{equation}