Large deviations theory

<h2 id="introductory-examples">Introductory examples</h2>
<blockquote>Any large deviation is done in the least unlikely of all the unlikely ways!— Frank den Hollander, Large Deviations, p. 10</blockquote>
<h3>An elementary example</h3>
Consider a sequence of independent tosses of a fair coin. The possible outcomes could be heads or tails. Let us denote the possible outcome of the i-th trial by 
 
 
 
 
 X
 
 i
 
 
 
 
 {\displaystyle X_{i}}
 
, where we encode head as 1 and tail as 0. Now let 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 denote the mean value after 
 
 
 
 N
 
 
 {\displaystyle N}
 
 trials, namely

M
 
 N
 
 
 =
 
 
 1
 N
 
 
 
 ∑
 
 i
 =
 1
 
 
 N
 
 
 
 X
 
 i
 
 
 
 
 {\displaystyle M_{N}={\frac {1}{N}}\sum _{i=1}^{N}X_{i}}
 
.
Then 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 lies between 0 and 1. From the <a href="/facts/Law_of_large_numbers/X3Bjcy3v">law of large numbers</a> it follows that as N grows, the distribution of 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 converges to 
 
 
 
 0.5
 =
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle 0.5=\operatorname {E} [X]}
 
 (the expected value of a single coin toss).
Moreover, by the <a href="/facts/Central_limit_theorem/0gc2L0Wd">central limit theorem</a>, it follows that 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 is approximately normally distributed for large 
 
 
 
 N
 
 
 {\displaystyle N}
 
. The central limit theorem can provide more detailed information about the behavior of 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 than the law of large numbers. For example, we can approximately find a tail probability of 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 – the probability that 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 is greater than some value 
 
 
 
 x
 
 
 {\displaystyle x}
 
 – for a fixed value of 
 
 
 
 N
 
 
 {\displaystyle N}
 
. However, the approximation by the central limit theorem may not be accurate if 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is far from 
 
 
 
 E
 ⁡
 [
 
 X
 
 i
 
 
 ]
 
 
 {\displaystyle \operatorname {E} [X_{i}]}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is not sufficiently large. Also, it does not provide information about the convergence of the tail probabilities as 
 
 
 
 N
 →
 ∞
 
 
 {\displaystyle N\to \infty }
 
. However, the large deviation theory can provide answers for such problems.
Let us make this statement more precise. For a given value 
 
 
 
 0.5
 <
 x
 <
 1
 
 
 {\displaystyle 0.5<x<1}
 
, let us compute the tail probability 
 
 
 
 P
 (
 
 M
 
 N
 
 
 >
 x
 )
 
 
 {\displaystyle P(M_{N}>x)}
 
. Define

I
 (
 x
 )
 =
 x
 ln
 ⁡
 
 x
 
 +
 (
 1
 −
 x
 )
 ln
 ⁡
 (
 1
 −
 x
 )
 +
 ln
 ⁡
 
 2
 
 
 
 {\displaystyle I(x)=x\ln {x}+(1-x)\ln(1-x)+\ln {2}}
 
.
Note that the function 
 
 
 
 I
 (
 x
 )
 
 
 {\displaystyle I(x)}
 
 is a convex, nonnegative function that is zero at 
 
 
 
 x
 =
 
 
 
 1
 2
 
 
 
 
 
 {\displaystyle x={\tfrac {1}{2}}}
 
 and increases as 
 
 
 
 x
 
 
 {\displaystyle x}
 
 approaches 
 
 
 
 1
 
 
 {\displaystyle 1}
 
. It is the negative of the <a href="/facts/Bernoulli_entropy/s9cwzz4n">Bernoulli entropy</a> with 
 
 
 
 p
 =
 
 
 
 1
 2
 
 
 
 
 
 {\displaystyle p={\tfrac {1}{2}}}
 
; that it's appropriate for coin tosses follows from the <a href="/facts/Asymptotic_equipartition_property/Jcy9SLWc">asymptotic equipartition property</a> applied to a <a href="/facts/Bernoulli_trial/7fm8snCf">Bernoulli trial</a>. Then by <a href="/facts/Chernoff%2527s_inequality/RLC6VlRU">Chernoff's inequality</a>, it can be shown that 
 
 
 
 P
 (
 
 M
 
 N
 
 
 >
 x
 )
 <
 exp
 ⁡
 (
 −
 N
 I
 (
 x
 )
 )
 
 
 {\displaystyle P(M_{N}>x)<\exp(-NI(x))}
 
.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> This bound is rather sharp, in the sense that 
 
 
 
 I
 (
 x
 )
 
 
 {\displaystyle I(x)}
 
 cannot be replaced with a larger number which would yield a strict inequality for all positive 
 
 
 
 N
 
 
 {\displaystyle N}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> (However, the exponential bound can still be reduced by a subexponential factor on the order of 
 
 
 
 1
 
 /
 
 
 
 N
 
 
 
 
 {\displaystyle 1/{\sqrt {N}}}
 
; this follows from the <a href="/facts/Stirling_approximation/GfxV3BUH">Stirling approximation</a> applied to the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a> appearing in the <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli distribution</a>.) Hence, we obtain the following result:

P
 (
 
 M
 
 N
 
 
 >
 x
 )
 ≈
 exp
 ⁡
 (
 −
 N
 I
 (
 x
 )
 )
 
 
 {\displaystyle P(M_{N}>x)\approx \exp(-NI(x))}
 
.
The probability 
 
 
 
 P
 (
 
 M
 
 N
 
 
 >
 x
 )
 
 
 {\displaystyle P(M_{N}>x)}
 
 decays exponentially as 
 
 
 
 N
 →
 ∞
 
 
 {\displaystyle N\to \infty }
 
 at a rate depending on x. This formula approximates any tail probability of the sample mean of i.i.d. variables and gives its convergence as the number of samples increases.

<h3>Large deviations for sums of independent random variables</h3>
Main article: <a href="/facts/Cram%25C3%25A9r%2527s_theorem_(large_deviations)/QftHEKQ6">Cramér's theorem (large deviations)</a>
In the above example of coin-tossing we explicitly assumed that each toss is an
independent trial, and the probability of getting head or tail is always the same.
Let 
 
 
 
 X
 ,
 
 X
 
 1
 
 
 ,
 
 X
 
 2
 
 
 ,
 …
 
 
 {\displaystyle X,X_{1},X_{2},\ldots }
 
 be <a href="/facts/I.i.d./othIRaWt">independent and identically distributed</a> (i.i.d.) random variables whose common distribution satisfies a certain growth condition. Then the following limit exists:

lim
 
 N
 →
 ∞
 
 
 
 
 1
 N
 
 
 ln
 ⁡
 P
 (
 
 M
 
 N
 
 
 >
 x
 )
 =
 −
 I
 (
 x
 )
 
 
 {\displaystyle \lim _{N\to \infty }{\frac {1}{N}}\ln P(M_{N}>x)=-I(x)}
 
.
Here

M
 
 N
 
 
 =
 
 
 1
 N
 
 
 
 ∑
 
 i
 =
 1
 
 
 N
 
 
 
 X
 
 i
 
 
 
 
 {\displaystyle M_{N}={\frac {1}{N}}\sum _{i=1}^{N}X_{i}}
 
,
as before.
Function 
 
 
 
 I
 (
 ⋅
 )
 
 
 {\displaystyle I(\cdot )}
 
 is called the "<a href="/facts/Rate_function/vSuAI0Fv">rate function</a>" or "Cramér function" or sometimes the "entropy function".
The above-mentioned limit means that for large 
 
 
 
 N
 
 
 {\displaystyle N}
 
,

P
 (
 
 M
 
 N
 
 
 >
 x
 )
 ≈
 exp
 ⁡
 [
 −
 N
 I
 (
 x
 )
 ]
 
 
 {\displaystyle P(M_{N}>x)\approx \exp[-NI(x)]}
 
,
which is the basic result of large deviations theory.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
If we know the probability distribution of 
 
 
 
 X
 
 
 {\displaystyle X}
 
, an explicit expression for the rate function can be obtained. This is given by a <a href="/facts/Legendre%25E2%2580%2593Fenchel_transformation/DGDWBrDL">Legendre–Fenchel transformation</a>,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

I
 (
 x
 )
 =
 
 sup
 
 θ
 >
 0
 
 
 [
 θ
 x
 −
 λ
 (
 θ
 )
 ]
 
 
 {\displaystyle I(x)=\sup _{\theta >0}[\theta x-\lambda (\theta )]}
 
,
where

λ
        (
        θ
        )
        =
        ln
        ⁡
        E
        ⁡
        [
        exp
        ⁡
        (
        θ
        X
        )
        ]
      
    
    {\displaystyle \lambda (\theta )=\ln \operatorname {E} [\exp(\theta X)]}

is called the <a href="/facts/Cumulant_generating_function/crLC9QLE">cumulant generating function</a> (CGF) and 
 
 
 
 E
 
 
 {\displaystyle \operatorname {E} }
 
 denotes the <a href="/facts/Mathematical_expectation/1XV0JKL8">mathematical expectation</a>.
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 follows a <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a>, the rate function becomes a parabola with its apex at the mean of the normal distribution.
If 
 
 
 
 {
 
 X
 
 i
 
 
 }
 
 
 {\displaystyle \{X_{i}\}}
 
 is an irreducible and aperiodic <a href="/facts/Markov_chain/5oP5hntk">Markov chain</a>, the variant of the basic large deviations result stated above may hold.

<h3>Moderate deviations for sums of independent random variables</h3>
The previous example controlled the probability of the event 
 
 
 
 [
 
 M
 
 N
 
 
 >
 x
 ]
 
 
 {\displaystyle [M_{N}>x]}
 
, that is, the concentration of the law of 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 on the <a href="/facts/Compact_set/d0cgXJH7">compact set</a> 
 
 
 
 [
 −
 x
 ,
 x
 ]
 
 
 {\displaystyle [-x,x]}
 
. It is also possible to control the probability of the event 
 
 
 
 [
 
 M
 
 N
 
 
 >
 x
 
 a
 
 N
 
 
 ]
 
 
 {\displaystyle [M_{N}>xa_{N}]}
 
 for some sequence 
 
 
 
 
 a
 
 N
 
 
 →
 0
 
 
 {\displaystyle a_{N}\to 0}
 
. The following is an example of a moderate deviations principle:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

Theorem—Let 
 
 
 
 
 X
 
 1
 
 
 ,
 
 X
 
 2
 
 
 ,
 …
 
 
 {\displaystyle X_{1},X_{2},\dots }
 
 be a sequence of centered i.i.d variables with finite variance 
 
 
 
 
 σ
 
 2
 
 
 
 
 {\displaystyle \sigma ^{2}}
 
 such that 
 
 
 
 ∀
 λ
 ∈
 
 R
 
 ,
  
 ln
 ⁡
 
 E
 
 [
 
 e
 
 λ
 
 X
 
 1
 
 
 
 
 ]
 <
 ∞
 
 
 {\displaystyle \forall \lambda \in \mathbb {R} ,\ \ln \mathbb {E} [e^{\lambda X_{1}}]<\infty }
 
. Define 
 
 
 
 
 M
 
 N
 
 
 :=
 
 
 1
 N
 
 
 
 ∑
 
 n
 ≤
 N
 
 
 
 X
 
 N
 
 
 
 
 {\displaystyle M_{N}:={\frac {1}{N}}\sum \limits _{n\leq N}X_{N}}
 
. Then for any sequence 
 
 
 
 1
 ≪
 
 a
 
 N
 
 
 ≪
 
 
 N
 
 
 
 
 {\displaystyle 1\ll a_{N}\ll {\sqrt {N}}}
 
:

 
 
 
 
 lim
 
 N
 →
 +
 ∞
 
 
 
 
 
 a
 
 N
 
 
 2
 
 
 N
 
 
 ln
 ⁡
 
 P
 
 [
 
 a
 
 N
 
 
 
 M
 
 N
 
 
 ≥
 x
 ]
 =
 −
 
 
 
 x
 
 2
 
 
 
 2
 
 σ
 
 2
 
 
 
 
 
 
 
 {\displaystyle \lim \limits _{N\to +\infty }{\frac {a_{N}^{2}}{N}}\ln \mathbb {P} [a_{N}M_{N}\geq x]=-{\frac {x^{2}}{2\sigma ^{2}}}}

In particular, the limit case 
 
 
 
 
 a
 
 N
 
 
 =
 
 
 N
 
 
 
 
 {\displaystyle a_{N}={\sqrt {N}}}
 
 is the <a href="/facts/Central_limit_theorem/0gc2L0Wd">central limit theorem</a>.

<h2 id="formal-definition">Formal definition</h2>
Given a <a href="/facts/Polish_space/CkUFcGap">Polish space</a> 
 
 
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {X}}}
 
 let 
 
 
 
 {
 
 
 P
 
 
 N
 
 
 }
 
 
 {\displaystyle \{\mathbb {P} _{N}\}}
 
 be a sequence of <a href="/facts/Borel_algebra/jvSsfpkP">Borel</a> probability measures on 
 
 
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {X}}}
 
, let 
 
 
 
 {
 
 a
 
 N
 
 
 }
 
 
 {\displaystyle \{a_{N}\}}
 
 be a sequence of positive real numbers such that 
 
 
 
 
 lim
 
 N
 
 
 
 a
 
 N
 
 
 =
 ∞
 
 
 {\displaystyle \lim _{N}a_{N}=\infty }
 
, and finally let 
 
 
 
 I
 :
 
 
 X
 
 
 →
 [
 0
 ,
 ∞
 ]
 
 
 {\displaystyle I:{\mathcal {X}}\to [0,\infty ]}
 
 be a <a href="/facts/Lower_semicontinuous/4unJbWdW">lower semicontinuous</a> functional on 
 
 
 
 
 
 X
 
 
 .
 
 
 {\displaystyle {\mathcal {X}}.}
 
 The sequence 
 
 
 
 {
 
 
 P
 
 
 N
 
 
 }
 
 
 {\displaystyle \{\mathbb {P} _{N}\}}
 
 is said to satisfy a <a href="/facts/Large_deviation_principle/vSuAI0Fv">large deviation principle</a> with speed 
 
 
 
 {
 
 a
 
 n
 
 
 }
 
 
 {\displaystyle \{a_{n}\}}
 
 and rate 
 
 
 
 I
 
 
 {\displaystyle I}
 
 if, and only if, for each Borel <a href="/facts/Measurable_set/yCq7zlI4">measurable set</a> 
 
 
 
 E
 ⊂
 
 
 X
 
 
 
 
 {\displaystyle E\subset {\mathcal {X}}}
 
,

−
 
 inf
 
 x
 ∈
 
 E
 
 ∘
 
 
 
 
 I
 (
 x
 )
 ≤
 
 
 
 lim
 _
 
 
 
 N
 
 
 ⁡
 
 a
 
 N
 
 
 −
 1
 
 
 log
 ⁡
 (
 
 
 P
 
 
 N
 
 
 (
 E
 )
 )
 ≤
 
 
 
 lim
 ¯
 
 
 
 N
 
 
 ⁡
 
 a
 
 N
 
 
 −
 1
 
 
 log
 ⁡
 (
 
 
 P
 
 
 N
 
 
 (
 E
 )
 )
 ≤
 −
 
 inf
 
 x
 ∈
 
 
 E
 ¯
 
 
 
 
 I
 (
 x
 )
 
 
 {\displaystyle -\inf _{x\in E^{\circ }}I(x)\leq \varliminf _{N}a_{N}^{-1}\log(\mathbb {P} _{N}(E))\leq \varlimsup _{N}a_{N}^{-1}\log(\mathbb {P} _{N}(E))\leq -\inf _{x\in {\overline {E}}}I(x)}
 
,
where 
 
 
 
 
 
 E
 ¯
 
 
 
 
 {\displaystyle {\overline {E}}}
 
 and 
 
 
 
 
 E
 
 ∘
 
 
 
 
 {\displaystyle E^{\circ }}
 
 denote respectively the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> and <a href="/facts/Interior_(topology)/5sxGnQfY">interior</a> of 
 
 
 
 E
 
 
 {\displaystyle E}
 
.

<h2 id="brief-history">Brief history</h2>
The first rigorous results concerning large deviations are due to the Swedish mathematician <a href="/facts/Harald_Cram%25C3%25A9r/BWylq7t8">Harald Cramér</a>, who applied them to model the insurance business.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> From the point
of view of an insurance company, the earning is at a constant rate per month (the monthly premium) but the claims come randomly. For the company to be successful over a certain period of time (preferably many months), the total earning should exceed the total claim. Thus to estimate the premium you have to ask the following question: "What should we choose as the premium 
 
 
 
 q
 
 
 {\displaystyle q}
 
 such that over 
 
 
 
 N
 
 
 {\displaystyle N}
 
 months the total claim 
 
 
 
 C
 =
 Σ
 
 X
 
 i
 
 
 
 
 {\displaystyle C=\Sigma X_{i}}
 
 should be less than 
 
 
 
 N
 q
 
 
 {\displaystyle Nq}
 
?" This is clearly the same question asked by the large deviations theory. Cramér gave a solution to this question for i.i.d. <a href="/facts/Random_variable/TwTBXnLT">random variables</a>, where the rate function is expressed as a <a href="/facts/Power_series/vYh6sTps">power series</a>.
A very incomplete list of mathematicians who have made important advances would include <a href="/facts/Aleksei_Zinovyevich_Petrov/5PWxygkL">Petrov</a>,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> <a href="/facts/Sanov%2527s_theorem/Ich2HGVo">Sanov</a>,<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> <a href="/facts/S.R.S._Varadhan/4YGAz384">S.R.S. Varadhan</a> (who has won the Abel prize for his contribution to the theory), <a href="/facts/D._Ruelle/mjoK3TGl">D. Ruelle</a>, <a href="/facts/Oscar_Lanford/X6ewo3v6">O.E. Lanford</a>, <a href="/facts/Mark_Freidlin/QfQW3jIX">Mark Freidlin</a>, <a href="/facts/Alexander_D._Wentzell/GkStReMR">Alexander D. Wentzell</a>, <a href="/facts/Amir_Dembo/tndkGKpR">Amir Dembo</a>, and <a href="/facts/Ofer_Zeitouni/K0CcpRxX">Ofer Zeitouni</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="applications">Applications</h2>
Principles of large deviations may be effectively applied to gather information out of a probabilistic model. Thus, theory of large deviations finds its applications in <a href="/facts/Information_theory/qNLH3U5P">information theory</a> and <a href="/facts/Risk_management/xGhxtxdp">risk management</a>. In physics, the best known application of large deviations theory arise in <a href="/facts/Thermodynamics/q4hIx3yP">thermodynamics</a> and <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a> (in connection with relating <a href="/facts/Entropy/s52IXeFz">entropy</a> with rate function).<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h3>Large deviations and entropy</h3>
Main article: <a href="/facts/Asymptotic_equipartition_property/Jcy9SLWc">asymptotic equipartition property</a>
The rate function is related to the <a href="/facts/Entropy/s52IXeFz">entropy</a> in statistical mechanics. This can be heuristically seen in the following way. In statistical mechanics the entropy of a particular macro-state is related to the number of micro-states which corresponds to this macro-state. In our coin tossing example the mean value 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 could designate a particular macro-state. And the particular sequence of heads and tails which gives rise to a particular value of 
 
 
 
 
 M
 
 N
 
 
 
 
 {\displaystyle M_{N}}
 
 constitutes a particular micro-state. Loosely speaking a macro-state having a higher number of micro-states giving rise to it, has higher entropy. And a state with higher entropy has a higher chance of being realised in actual experiments. The macro-state with mean value of 1/2 (as many heads as tails) has the highest number of micro-states giving rise to it and it is indeed the state with the highest entropy. And in most practical situations we shall indeed obtain this macro-state for large numbers of trials. The "rate function" on the other hand measures the probability of appearance of a particular macro-state. The smaller the rate function the higher is the chance of a macro-state appearing. In our coin-tossing the value of the "rate function" for mean value equal to 1/2 is zero. In this way one can see the "rate function" as the negative of the "entropy".
There is a relation between the "rate function" in large deviations theory and the <a href="/facts/Kullback%25E2%2580%2593Leibler_divergence/nh7SjlPE">Kullback–Leibler divergence</a>, the connection is established by <a href="/facts/Sanov%2527s_theorem/Ich2HGVo">Sanov's theorem</a> (see Sanov<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> and 
Novak,<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> ch. 14.5).
In a special case, large deviations are closely related to the concept of <a href="/facts/Gromov%25E2%2580%2593Hausdorff_convergence/0Cy5VG6w">Gromov–Hausdorff limits</a>.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Large_deviation_principle/vSuAI0Fv">Large deviation principle</a></li>
<li><a href="/facts/Cram%25C3%25A9r%2527s_large_deviation_theorem/QftHEKQ6">Cramér's large deviation theorem</a></li>
<li><a href="/facts/Chernoff%2527s_inequality/RLC6VlRU">Chernoff's inequality</a></li>
<li><a href="/facts/Sanov%2527s_theorem/Ich2HGVo">Sanov's theorem</a></li>
<li><a href="/facts/Contraction_principle_(large_deviations_theory)/KbhJ1PVt">Contraction principle (large deviations theory)</a>, a result on how large deviations principles "<a href="/facts/Pushforward_measure/aFljwqr6">push forward</a>"</li>
<li><a href="/facts/Freidlin%25E2%2580%2593Wentzell_theorem/3Vkw9eoy">Freidlin–Wentzell theorem</a>, a large deviations principle for <a href="/facts/It%25C5%258D_diffusion/vlwAqWSU">Itō diffusions</a></li>
<li><a href="/facts/Legendre_transformation/P35nPen7">Legendre transformation</a>, Ensemble equivalence is based on this transformation.</li>
<li><a href="/facts/Laplace_principle_(large_deviations_theory)/24Al8Y46">Laplace principle</a>, a large deviations principle in Rd</li>
<li><a href="/facts/Laplace%2527s_method/Hr6YjG6o">Laplace's method</a></li>
<li><a href="/facts/Schilder%2527s_theorem/CTbRGJ70">Schilder's theorem</a>, a large deviations principle for <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a></li>
<li><a href="/facts/Varadhan%2527s_lemma/aYtrXloR">Varadhan's lemma</a></li>
<li><a href="/facts/Extreme_value_theory/17YICqaw">Extreme value theory</a></li>
<li><a href="/facts/Large_deviations_of_Gaussian_random_functions/T4SUrVYr">Large deviations of Gaussian random functions</a></li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="https://arxiv.org/abs/0804.2330v1">Special invited paper: Large deviations</a> by S. R. S. Varadhan The Annals of Probability 2008, Vol. 36, No. 2, 397–419 <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2F07-AOP348">10.1214/07-AOP348</a></li>
<li><a href="https://arxiv.org/abs/1106.4146">A basic introduction to large deviations: Theory, applications, simulations</a>, Hugo Touchette, arXiv:1106.4146.</li>
<li>Entropy, Large Deviations and Statistical Mechanics by R.S. Ellis, Springer Publication. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-29059-1</li>
<li>Large Deviations for Performance Analysis by Alan Weiss and Adam Shwartz. Chapman and Hall <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-412-06311-5</li>
<li>Large Deviations Techniques and Applications by Amir Dembo and Ofer Zeitouni. Springer <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98406-2</li>
<li>A course on large deviations with an introduction to Gibbs measures by Firas Rassoul-Agha and Timo Seppäläinen. Grad. Stud. Math., 162. American Mathematical Society <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-7578-0</li>
<li>Random Perturbations of Dynamical Systems by <a href="/facts/Mark_Freidlin/QfQW3jIX">M.I. Freidlin</a> and A.D. Wentzell. Springer <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98362-7</li>
<li>"Large Deviations for Two Dimensional Navier-Stokes Equation with Multiplicative Noise", S. S. Sritharan and P. Sundar, Stochastic Processes and Their Applications, Vol. 116 (2006) 1636–1659.<a href="http://www.nps.edu/Academics/Schools/GSEAS/SRI/R37.pdf">[2]</a></li>
<li>"Large Deviations for the Stochastic Shell Model of Turbulence", U. Manna, S. S. Sritharan and P. Sundar, NoDEA Nonlinear Differential Equations Appl. 16 (2009), no. 4, 493–521.<a href="http://www.nps.edu/Academics/Schools/GSEAS/SRI/R41.pdf">[3]</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">S.R.S. Varadhan, Asymptotic probability and differential equations, Comm. Pure Appl. Math. 19 (1966),261-286. <a href="/wiki/Communications_on_Pure_and_Applied_Mathematics" target="_blank">/wiki/Communications_on_Pure_and_Applied_Mathematics</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Large deviations for performance analysis: queues, communications, and computing", Shwartz, Adam, 1953- TN: 1228486 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Varadhan, S.R.S.,The Annals of Probability 2008, Vol. 36, No. 2, 397–419, [1] <a href="https://math.nyu.edu/faculty/varadhan/wald.pdf" target="_blank">https://math.nyu.edu/faculty/varadhan/wald.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"Large Deviations" (PDF). www.math.nyu.edu. 2 February 2012. Retrieved 11 June 2024. <a href="https://math.nyu.edu/~varadhan/Spring2012/Chapters1-2.pdf" target="_blank">https://math.nyu.edu/~varadhan/Spring2012/Chapters1-2.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">S.R.S. Varadhan, Large Deviations and Applications (SIAM, Philadelphia, 1984) <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
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<li id="fn:13">Gingrich, Todd R.; Horowitz, Jordan M.; Perunov, Nikolay; England, Jeremy L. (21 March 2016). "Dissipation Bounds All Steady-State Current Fluctuations". Physical Review Letters. 116 (12): 120601. arXiv:1512.02212. doi:10.1103/PhysRevLett.116.120601. <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.120601" target="_blank">https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.120601</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Harvey, Sarah E.; Lahiri, Subhaneil; Ganguli, Surya (7 July 2023). "Universal energy-accuracy tradeoffs in nonequilibrium cellular sensing". Physical Review E. 108 (1): 014403. arXiv:2002.10567. doi:10.1103/PhysRevE.108.014403. <a href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.014403" target="_blank">https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.014403</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
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</ol>

Large deviations theory open-in-new

Large deviations theory