Probability of large deviations

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A probability of the type


is a sequence of independent random variables, and and are two numerical sequences such that , and in probability.

If the random variables have the same distribution with mathematical expectation zero and finite variance , one may write and , where as . Cramér's theorem and strengthened versions of it are particularly important in this connection (cf. Cramér theorem).

To obtain guaranteed bounds for the probability of large deviations one uses inequalities of the type of the Chebyshev inequality in probability theory; these provide the so-called exponential bounds for the probability of large deviations. For instance, if the random variables are independent, , , with probability one, and , then the estimate

the right-hand side of which decreases exponentially with increasing , is valid for all .


[1] M. Loève, "Probability theory" , Springer (1977)
[2] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
[3] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)
[4] Yu.V. Prokhorov, "Multidimensional distributions: inequalities and limit theorems" J. Soviet Math. , 2 : 5 (1976) pp. 475–488 Itogi Nauk. i Tekhn. , 10 (1972) pp. 5–24
[5] V.V. Yurinskii, "Exponential bounds for large deviations" Theory Probab. Appl. , 19 : 1 (1974) pp. 154–159 Teor. Veroyatnost. i Primenen. , 19 : 1 (1974) pp. 152–153


There are substantial new developments which link the rate of exponential decay to entropy. These developments find widespread use in statistical physics and in statistics. Cf. Limit theorems and [a1], [a2].

A second recent development concerns the development of limit theorems and large deviation theory for stochastic processes instead of sums of independent random variables, cf. [a3].


[a1] R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985)
[a2] D.W. Stroock, "An introduction to the theory of large deviations" , Springer (1984)
[a3] A.D. [A.D. Ventsel'] Wentzell, "Limit theorems on large deviations for Markov stochastic processes" , Kluwer (1990) (Translated from Russian)
[a4] H. Cramér, "Sur un nouveau théorème-limite de la théorie des probabilités" , Act. Sci. et Ind. , 736 , Hermann (1938) pp. 5–24
[a5] P. Groeneboom, J. Oosterhoff, F.H. Ruymgaart, "Large deviation theorems for empirical probability measures" Ann. Probl. , 7 (1979) pp. 553–586
How to Cite This Entry:
Probability of large deviations. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.V. PetrovV.V. Yurinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article