Concept

Central limit theorem

The statement that a standardised sum of many independent contributions is approximately bell-shaped. It describes a window of width one over the square root of the count around the mean, and says nothing about probabilities further out than that.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

One set of sums, two scalings, two different limits. The exact distribution of a sum of n independent copies, scaled two ways. Divided by n it collapses onto the mean; divided by the square root of n it holds a fixed width and settles into a shape.

The average settles and the wobble does not

Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.

probability · central limit

The chance the average clears 0.75, against the number of draws. The exact probability that the average of n draws exceeds a fixed level, on a logarithmic scale, falling along a straight line whose slope is the rate function, with the normal approximation drawn beside it and diverging.

The tail is not a bell

The limit theorem describes a window of width one over the root of n around the mean; ask instead for the chance that an average lands a fixed distance away and the answer falls exponentially, at a rate computed from the summand before any n is chosen.

probability · central limit

Named alongside it

The objects these essays reach for when they reach for this one.

Concentration inequality Convergence Convergence rate Expectation Heavy tails Independence Large deviations Law of large numbers Legendre transform Moment generating function Normal distribution Rate function

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