View source for Probability of large deviations

{{MSC|60F10}}

[[Category:Limit theorems]]

A probability of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749401.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749402.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749403.png" /> is a sequence of independent random variables, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749405.png" /> are two numerical sequences such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749406.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749407.png" /> in probability.

If the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749408.png" /> have the same distribution with mathematical expectation zero and finite variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p0749409.png" />, one may write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494012.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494013.png" />. Cramér's theorem and strengthened versions of it are particularly important in this connection (cf. [[Cramér theorem|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|Chebyshev inequality in probability theory]]; these provide the so-called exponential bounds for the probability of large deviations. For instance, if the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494014.png" /> are independent, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494017.png" /> with probability one, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494019.png" />, then the estimate

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494020.png" /></td> </tr></table>

the right-hand side of which decreases exponentially with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494021.png" />, is valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074940/p07494022.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Loève, "Probability theory" , Springer (1977) {{MR|0651017}} {{MR|0651018}} {{ZBL|0359.60001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) {{MR|0322926}} {{ZBL|0219.60027}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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</TD></TR></table>



====Comments====
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|Limit theorems]] and [[#References|[a1]]], [[#References|[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. [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) {{MR|0793553}} {{ZBL|0566.60097}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.W. Stroock, "An introduction to the theory of large deviations" , Springer (1984) {{MR|0755154}} {{MR|0758258}} {{ZBL|0552.60022}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.D. [A.D. Ventsel'] Wentzell, "Limit theorems on large deviations for Markov stochastic processes" , Kluwer (1990) (Translated from Russian) {{MR|1135113}} {{ZBL|0743.60029}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Cramér, "Sur un nouveau théorème-limite de la théorie des probabilités" , ''Act. Sci. et Ind.'' , '''736''' , Hermann (1938) pp. 5–24 {{MR|}} {{ZBL|64.0529.01}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Groeneboom, J. Oosterhoff, F.H. Ruymgaart, "Large deviation theorems for empirical probability measures" ''Ann. Probl.'' , '''7''' (1979) pp. 553–586 {{MR|0537208}} {{ZBL|0425.60021}} </TD></TR></table>