Intuition

1. smoothed-out sensing (blurry vision analogy) imagine you are observing a probability distribution through a blurred lens. - if two distributions look the same under any level of blurring, they are weakly close. - weak convergence says that if you approximate a distribution with a sequence of others, their blurry perspectives should match in the limit. - you can’t detect small shifts in probability mass unless they significantly change smooth test functions. 2. movement of mass (sand pile analogy) think of  $\mu_n$  as describing how sand is distributed on a table. - weak convergence means that, from a distance, the sand piles look the same. - individual grains of sand might move around, but as long as the overall shape stays the same under any smooth measurement, the distributions are weakly converging. ## Note This is closely related to Convergence in Distribution which is pretty much weak convergence for Random Variables.