convergence in distribution
Suppose you have a whole sequence of random variables — X_1, X_2, X_3, and so on — and you want to say they 'settle down' toward some limiting variable X. The gentlest, most modest way to say this is: the SHAPE of their distribution settles down. You do not promise that the individual numbers get close to anything; you only promise that the histogram of X_n starts to look like the histogram of X. That is convergence in distribution.
Precisely: X_n converges in distribution to X if the cumulative distribution functions agree in the limit — F_n(x) = P(X_n <= x) approaches F(x) = P(X <= x) at every point x where F is continuous (we forgive disagreements at jump points of F). The picture is purely about the pile of probability, not about which outcome produced which number. To check it you can compare CDFs, or equivalently show that E[g(X_n)] approaches E[g(X)] for every bounded continuous function g, or — the favourite engine — show the characteristic functions converge.
This is the weakest of the four standard modes, and it is exactly what the central limit theorem delivers: the standardized sum of many iid variables converges in distribution to a standard normal. Be honest about what it does NOT say. It says nothing about the variables living on the same outcome — X_n and X need not even be defined on the same probability space, and X_n need not get numerically close to X for any particular outcome. It is a statement about laws, not about points.
Let X_n be uniform on the finite set {1/n, 2/n, ..., n/n}. As n grows, this discrete ladder of points fills in the interval [0, 1], and F_n(x) approaches x for 0 <= x <= 1 — so X_n converges in distribution to a continuous Uniform(0, 1) variable, even though every X_n is discrete.
Only the shape of the distribution converges; the variables need not share outcomes.
Convergence in distribution is about laws, not points: X_n need not approach X for any individual outcome, and the variables need not even live on the same probability space.