Infinite divisibility (probability)

<h2 id="examples">Examples</h2>
Examples of continuous distributions that are infinitely divisible are the <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a>, the <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a>, the <a href="/facts/L%25C3%25A9vy_distribution/g3fVHgGz">Lévy distribution</a>, and all other members of the <a href="/facts/Stable_distribution/rvtRmhlz">stable distribution</a> family, as well as the <a href="/facts/Gamma_distribution/lczcdJmw">Gamma distribution</a>, the <a href="/facts/Chi-square_distribution/wYnWWgUe">chi-square distribution</a>, the <a href="/facts/Wald_distribution/mUPsu4EJ">Wald distribution</a>, the <a href="/facts/Log-normal_distribution/V2RXmmlE">Log-normal distribution</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> and the <a href="/facts/Student%2527s_t-distribution/DeT1SDqH">Student's t-distribution</a>.
Among the discrete distributions, examples are the <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a> and the <a href="/facts/Negative_binomial_distribution/znzj365Y">negative binomial distribution</a> (and hence the <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a> also). The <a href="/facts/One-point_distribution/EpsKKVRu">one-point distribution</a> whose only possible outcome is 0 is also (trivially) infinitely divisible.
The <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform distribution</a> and the <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a> are not infinitely divisible, nor are any other distributions with bounded <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> (≈ finite-sized <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>), other than the <a href="/facts/One-point_distribution/EpsKKVRu">one-point distribution</a> mentioned above.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The distribution of the <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocal</a> of a random variable having a Student's t-distribution is also not infinitely divisible.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Any <a href="/facts/Compound_Poisson_distribution/tLHJbhIP">compound Poisson distribution</a> is infinitely divisible; this follows immediately from the definition.

<h2 id="limit-theorem">Limit theorem</h2>
Infinitely divisible distributions appear in a broad generalization of the <a href="/facts/Central_limit_theorem/0gc2L0Wd">central limit theorem</a>: the limit as n → +∞ of the sum Sn = Xn1 + ... + Xnn of <a href="/facts/Statistical_independence/NUzQtnUL">independent</a> uniformly asymptotically negligible (u.a.n.) random variables within a triangular array

X
                  
                    11
                  
                
              
            
            
              
                
                  X
                  
                    21
                  
                
              
              
                
                  X
                  
                    22
                  
                
              
            
            
              
                
                  X
                  
                    31
                  
                
              
              
                
                  X
                  
                    32
                  
                
              
              
                
                  X
                  
                    33
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
              
                ⋱
              
            
          
        
      
    
    {\displaystyle {\begin{array}{cccc}X_{11}\\X_{21}&X_{22}\\X_{31}&X_{32}&X_{33}\\\vdots &\vdots &\vdots &\ddots \end{array}}}

approaches — in the <a href="/facts/Convergence_in_distribution/pWdatFlY">weak sense</a> — an infinitely divisible distribution. The uniformly asymptotically negligible (u.a.n.) condition is given by

lim
          
            n
            →
            ∞
          
        
        
        
          max
          
            1
            ≤
            k
            ≤
            n
          
        
        
        P
        (
        
          |
          
            X
            
              n
              k
            
          
          |
        
        >
        ε
        )
        =
        0
        
           for every 
        
        ε
        >
        0.
      
    
    {\displaystyle \lim _{n\to \infty }\,\max _{1\leq k\leq n}\;P(\left|X_{nk}\right|>\varepsilon )=0{\text{ for every }}\varepsilon >0.}

Thus, for example, if the uniform asymptotic negligibility (u.a.n.) condition is satisfied via an appropriate scaling of identically distributed random variables with finite <a href="/facts/Variance/ULBJKXD1">variance</a>, the weak convergence is to the <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a> in the classical version of the central limit theorem. More generally, if the u.a.n. condition is satisfied via a scaling of identically distributed random variables (with not necessarily finite second moment), then the weak convergence is to a <a href="/facts/Stable_distribution/rvtRmhlz">stable distribution</a>. On the other hand, for a <a href="/facts/Triangular_array/jP0w2YHG">triangular array</a> of independent (unscaled) <a href="/facts/Bernoulli_distribution/ChCtYyvs">Bernoulli random variables</a> where the u.a.n. condition is satisfied through

lim
          
            n
            →
            ∞
          
        
        n
        
          p
          
            n
          
        
        =
        λ
        ,
      
    
    {\displaystyle \lim _{n\rightarrow \infty }np_{n}=\lambda ,}

the weak convergence of the sum is to the Poisson distribution with mean λ as shown by the familiar proof of the <a href="/facts/Poisson_limit_theorem/s2F2nhBz">law of small numbers</a>.

<h2 id="lvy-process">Lévy process</h2>
Main article: <a href="/facts/L%25C3%25A9vy_process/m9A3YQW8">Lévy process</a>
Every infinitely divisible probability distribution corresponds in a natural way to a <a href="/facts/L%25C3%25A9vy_process/m9A3YQW8">Lévy process</a>. A Lévy process is a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a> { Lt : t ≥ 0 } with <a href="/facts/Stationary_increments/m9A3YQW8">stationary</a> <a href="/facts/Independent_increments/tlYU1KEB">independent increments</a>, where stationary means that for s < t, the <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> of Lt − Ls depends only on t − s and where independent increments means that that difference Lt − Ls is <a href="/facts/Statistical_independence/NUzQtnUL">independent</a> of the corresponding difference on any interval not overlapping with [s, t], and similarly for any finite number of mutually non-overlapping intervals.
If { Lt : t ≥ 0 } is a Lévy process then, for any t ≥ 0, the random variable Lt will be infinitely divisible: for any n, we can choose (Xn1, Xn2, ..., Xnn) = (Lt/n − L0, L2t/n − Lt/n, ..., Lt − L(n−1)t/n). Similarly, Lt − Ls is infinitely divisible for any s < t.
On the other hand, if F is an infinitely divisible distribution, we can construct a Lévy process { Lt : t ≥ 0 } from it. For any interval [s, t] where t − s > 0 equals a <a href="/facts/Rational_number/VJQmyY05">rational number</a> p/q, we can define Lt − Ls to have the same distribution as Xq1 + Xq2 + ... + Xqp. <a href="/facts/Irrational_number/Psv19TET">Irrational</a> values of t − s > 0 are handled via a continuity argument.

<h2 id="additive-process">Additive process</h2>
Main article: <a href="/facts/Additive_process/LY51m3UJ">Additive process</a>
An <a href="/facts/Additive_process/LY51m3UJ">additive process</a> 
 
 
 
 {
 
 X
 
 t
 
 
 
 }
 
 t
 ≥
 0
 
 
 
 
 {\displaystyle \{X_{t}\}_{t\geq 0}}
 
 (a <a href="/facts/Cadlag/R9Jz9QLh">cadlag</a>, <a href="/facts/Continuous_stochastic_process/GEmGTXYm">continuous in probability</a> stochastic process with <a href="/facts/Independent_increments/tlYU1KEB">independent increments</a>) has an infinitely divisible distribution for any 
 
 
 
 t
 ≥
 0
 
 
 {\displaystyle t\geq 0}
 
. Let 
 
 
 
 {
 
 μ
 
 t
 
 
 
 }
 
 t
 ≥
 0
 
 
 
 
 {\displaystyle \{\mu _{t}\}_{t\geq 0}}
 
 be its family of infinitely divisible distributions.

 
 
 
 {
 
 μ
 
 t
 
 
 
 }
 
 t
 ≥
 0
 
 
 
 
 {\displaystyle \{\mu _{t}\}_{t\geq 0}}
 
 satisfies a number of conditions of continuity and monotonicity. Moreover, if a family of infinitely divisible distributions 
 
 
 
 {
 
 μ
 
 t
 
 
 
 }
 
 t
 ≥
 0
 
 
 
 
 {\displaystyle \{\mu _{t}\}_{t\geq 0}}
 
 satisfies these continuity and monotonicity conditions, there exists (uniquely in law) an additive process 
 
 
 
 {
 
 μ
 
 t
 
 
 
 }
 
 t
 ≥
 0
 
 
 
 
 {\displaystyle \{\mu _{t}\}_{t\geq 0}}
 
 with this distribution.
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cram%25C3%25A9r%25E2%2580%2599s_decomposition_theorem/rYff3Awg">Cramér's theorem</a></li>
<li><a href="/facts/Indecomposable_distribution/AtSWJQA1">Indecomposable distribution</a></li>
<li><a href="/facts/Compound_Poisson_distribution/tLHJbhIP">Compound Poisson distribution</a></li></ul>
<h2 id="footnotes">Footnotes</h2>

<ul><li>Domínguez-Molina, J.A.; Rocha-Arteaga, A. (2007) "On the Infinite Divisibility of some Skewed Symmetric Distributions". Statistics and Probability Letters, 77 (6), 644–648 <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.spl.2006.09.014">10.1016/j.spl.2006.09.014</a></li>
<li>Steutel, F. W. (1979), "Infinite Divisibility in Theory and Practice" (with discussion), Scandinavian Journal of Statistics. 6, 57–64.</li>
<li>Steutel, F. W. and Van Harn, K. (2003), Infinite Divisibility of Probability Distributions on the Real Line (Marcel Dekker).</li></ul>

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

<ol>
<li id="fn:1">Lukacs, E. (1970) Characteristic Functions, Griffin, London. p. 107 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lukacs, E. (1970) Characteristic Functions, Griffin, London. p. 107 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Thorin, Olof (1977). "On the infinite divisibility of the lognormal distribution". Scandinavian Actuarial Journal. 1977 (3): 121–148. doi:10.1080/03461238.1977.10405635. ISSN 0346-1238. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Sato, Ken-iti (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. pp. 31, 148. ISBN 978-0-521-55302-5. <a href="978-0-521-55302-5" target="_blank">978-0-521-55302-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Johnson, N.L.; Kotz, S.; Balakrishnan, N. (1995). Continuous Univariate Distributions (2nd ed.). Wiley. volume 2, chapter 28, page 368. ISBN 0-471-58494-0. <a href="0-471-58494-0" target="_blank">0-471-58494-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Sato, Ken-Ito (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press. pp. 31–68. ISBN 9780521553025. <a href="9780521553025" target="_blank">9780521553025</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Infinite divisibility (probability) open-in-new

Infinite divisibility (probability)