Concept

Tail bound

A statement of how much probability can lie beyond a given distance from the centre of a distribution. Its strength follows from its hypothesis: a variance gives a bound falling like one over the square of the distance, an exponential moment gives one falling exponentially.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

five values, unevenly weighted, and the mass outside 3 standard deviations. A distribution drawn as bars, with the windows one and a half, two and three standard deviations wide marked. The probability outside each window is summed and compared with the bound that knows only the variance.

How far from the average a thing can be

Knowing only an average and a spread — nothing about the shape, nothing about the number of outcomes, nothing about symmetry — the chance of landing three standard deviations out is at most one in nine. And there is a distribution that lands there exactly that often, so the bound cannot be improved.

probability · concentration

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

The 91 histograms 12 draws can produce. A triangle whose points are the possible histograms of a fixed number of draws over three faces, each drawn as a dot shaded by how far it is from the true distribution.

When the whole histogram deviates

The rung below priced a rare average. Ask instead for the chance that the whole tally of outcomes comes out wrong, and the exponent is no longer a function of one number — it is a distance between two distributions, and every rare-average rate is a shadow of it.

probability · central limit

Named alongside it

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

Central limit theorem Concentration inequality Heavy tails Large deviations Rate function Convergence rate Counting argument Empirical distribution Entropy Expectation Extremal example Legendre transform

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