Glivenko–Cantelli theorem

In the theory of probability, the Glivenko–Cantelli theorem, named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows.[1] The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets.[2] The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.

Assume that are independent and identically-distributed random variables in with common cumulative distribution function . The empirical distribution function for is defined by

where is the indicator function of the set . For every (fixed) , is a sequence of random variables which converge to almost surely by the strong law of large numbers, that is, converges to pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of to .

Theorem

almost surely.[3]

This theorem originates with Valery Glivenko,[4] and Francesco Cantelli,[5] in 1933.

Remarks

Empirical measures

One can generalize the empirical distribution function by replacing the set by an arbitrary set C from a class of sets to obtain an empirical measure indexed by sets

Where is the indicator function of each set .

Further generalization is the map induced by on measurable real-valued functions f, which is given by

Then it becomes an important property of these classes that the strong law of large numbers holds uniformly on or .

Glivenko–Cantelli class

Consider a set with a sigma algebra of Borel subsets A and a probability measure P. For a class of subsets,

and a class of functions

define random variables

where is the empirical measure, is the corresponding map, and

, assuming that it exists.

Definitions

1. almost surely as .
2. in probability as .
3. , as (convergence in mean).
The Glivenko–Cantelli classes of functions are defined similarly.

Theorem (Vapnik and Chervonenkis, 1968)[7]

A class of sets is uniformly GC if and only if it is a Vapnik–Chervonenkis class.

Examples

, that is is uniformly Glivenko–Cantelli class.

See also

References

  1. Howard G.Tucker (1959). "A Generalization of the Glivenko-Cantelli Theorem". The Annals of Mathematical Statistics. 30: 828–830. doi:10.1214/aoms/1177706212.
  2. van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press. p. 279. ISBN 0-521-78450-6.
  3. van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press. p. 265. ISBN 0-521-78450-6.
  4. Glivenko, V. (1933). Sulla determinazione empirica della legge di probabilita. Giorn. Ist. Ital. Attuari 4, 92-99.
  5. Cantelli, F. P. (1933). Sulla determinazione empirica delle leggi di probabilita. Giorn. Ist. Ital. Attuari 4, 221-424.
  6. van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press. p. 268. ISBN 0-521-78450-6.
  7. Vapnik, V. N.; Chervonenkis, A. Ya (1971). "On uniform convergence of the frequencies of events to their probabilities". Theor. Prob. Appl. 16 (2): 264–280. doi:10.1137/1116025.

Further reading

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