Slowly varying function

<h2 id="basic-definitions">Basic definitions</h2>
Definition 1. A <a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> L : (0, +∞) → (0, +∞) is called slowly varying (at infinity) if for all a > 0,

lim
          
            x
            →
            ∞
          
        
        
          
            
              L
              (
              a
              x
              )
            
            
              L
              (
              x
              )
            
          
        
        =
        1.
      
    
    {\displaystyle \lim _{x\to \infty }{\frac {L(ax)}{L(x)}}=1.}

Definition 2. Let L : (0, +∞) → (0, +∞). Then L is a regularly varying function if and only if 
 
 
 
 ∀
 a
 >
 0
 ,
 
 g
 
 L
 
 
 (
 a
 )
 =
 
 lim
 
 x
 →
 ∞
 
 
 
 
 
 L
 (
 a
 x
 )
 
 
 L
 (
 x
 )
 
 
 
 ∈
 
 
 R
 
 
 +
 
 
 
 
 {\displaystyle \forall a>0,g_{L}(a)=\lim _{x\to \infty }{\frac {L(ax)}{L(x)}}\in \mathbb {R} ^{+}}
 
. In particular, the <a href="/facts/Limit_of_a_function/HFbQ69e5">limit</a> must be finite.
These definitions are due to <a href="/facts/Jovan_Karamata/AFhBkUXm">Jovan Karamata</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="basic-properties">Basic properties</h2>
Regularly varying functions have some important properties:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by Bingham, Goldie & Teugels (1987).

<h3>Uniformity of the limiting behaviour</h3>
Theorem 1. The limit in definitions 1 and 2 is <a href="/facts/Uniform_convergence/ecz9eNWn">uniform</a> if a is restricted to a compact <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a>.

<h3>Karamata's characterization theorem</h3>
Theorem 2. Every regularly varying function f : (0, +∞) → (0, +∞) is of the form

f
        (
        x
        )
        =
        
          x
          
            β
          
        
        L
        (
        x
        )
      
    
    {\displaystyle f(x)=x^{\beta }L(x)}

where 

<ul><li>β is a <a href="/facts/Real_number/R02gw5Pb">real number</a>,</li>
<li>L is a slowly varying function.</li></ul>
Note. This implies that the function g(a) in definition 2 has necessarily to be of the following form

g
        (
        a
        )
        =
        
          a
          
            ρ
          
        
      
    
    {\displaystyle g(a)=a^{\rho }}

where the real number ρ is called the index of regular variation.

<h3>Karamata representation theorem</h3>
Theorem 3. A function L is slowly varying if and only if there exists B > 0 such that for all x ≥ B the function can be written in the form

L
        (
        x
        )
        =
        exp
        ⁡
        
          (
          
            η
            (
            x
            )
            +
            
              ∫
              
                B
              
              
                x
              
            
            
              
                
                  ε
                  (
                  t
                  )
                
                t
              
            
            
            d
            t
          
          )
        
      
    
    {\displaystyle L(x)=\exp \left(\eta (x)+\int _{B}^{x}{\frac {\varepsilon (t)}{t}}\,dt\right)}

where 

<ul><li>η(x) is a <a href="/facts/Bounded_function/whVjbtuk">bounded</a> <a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> of a real variable converging to a finite number as x goes to infinity</li>
<li>ε(x) is a bounded measurable function of a real variable converging to zero as x goes to infinity.</li></ul>
<h2 id="examples">Examples</h2>
<ul><li>If L is a measurable function and has a limit</li></ul>

lim
          
            x
            →
            ∞
          
        
        L
        (
        x
        )
        =
        b
        ∈
        (
        0
        ,
        ∞
        )
        ,
      
    
    {\displaystyle \lim _{x\to \infty }L(x)=b\in (0,\infty ),}

then L is a slowly varying function.
<ul><li>For any β ∈ R, the function L(x) = log β x is slowly varying.</li>
<li>The function L(x) = x is not slowly varying, nor is L(x) = x β for any real β ≠ 0. However, these functions are regularly varying.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Analytic_number_theory/mrHbWwrx">Analytic number theory</a></li>
<li><a href="/facts/Hardy%25E2%2580%2593Littlewood_tauberian_theorem/1bwpzTsN">Hardy–Littlewood tauberian theorem</a> and its treatment by Karamata</li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Bingham, N.H. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Karamata_theory&oldid=25937">"Karamata theory"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987), <a href="https://archive.org/details/regularvariation0000bing">Regular Variation</a>, Encyclopedia of Mathematics and its Applications, vol. 27, <a href="/facts/Cambridge/gNRdbmwK">Cambridge</a>: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-30787-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0898871">0898871</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0617.26001">0617.26001</a></li>
<li>Galambos, J.; <a href="/facts/Eugene_Seneta/LTkDlQyv">Seneta, E.</a> (1973), "Regularly Varying Sequences", <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>, 41 (1): 110–116, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2038824">10.2307/2038824</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9939">0002-9939</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2038824">2038824</a>.</li></ul>

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

<ol>
<li id="fn:1">See (Galambos & Seneta 1973) - Galambos, J.; Seneta, E. (1973), "Regularly Varying Sequences", Proceedings of the American Mathematical Society, 41 (1): 110–116, doi:10.2307/2038824, ISSN 0002-9939, JSTOR 2038824 <a href="https://doi.org/10.2307%2F2038824" target="_blank">https://doi.org/10.2307%2F2038824</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">See (Bingham, Goldie & Teugels 1987). - Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987), Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge: Cambridge University Press, ISBN 0-521-30787-2, MR 0898871, Zbl 0617.26001 <a href="https://archive.org/details/regularvariation0000bing" target="_blank">https://archive.org/details/regularvariation0000bing</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">See (Galambos & Seneta 1973) - Galambos, J.; Seneta, E. (1973), "Regularly Varying Sequences", Proceedings of the American Mathematical Society, 41 (1): 110–116, doi:10.2307/2038824, ISSN 0002-9939, JSTOR 2038824 <a href="https://doi.org/10.2307%2F2038824" target="_blank">https://doi.org/10.2307%2F2038824</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">See (Bingham, Goldie & Teugels 1987). - Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987), Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge: Cambridge University Press, ISBN 0-521-30787-2, MR 0898871, Zbl 0617.26001 <a href="https://archive.org/details/regularvariation0000bing" target="_blank">https://archive.org/details/regularvariation0000bing</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">See (Galambos & Seneta 1973) - Galambos, J.; Seneta, E. (1973), "Regularly Varying Sequences", Proceedings of the American Mathematical Society, 41 (1): 110–116, doi:10.2307/2038824, ISSN 0002-9939, JSTOR 2038824 <a href="https://doi.org/10.2307%2F2038824" target="_blank">https://doi.org/10.2307%2F2038824</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Slowly varying function open-in-new

Slowly varying function