Holder Continuity of 1/f (Pink) noise

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For $omega=omega(t)$ which is 1/f noise, does anyone know the supremum of all its possible Holder exponents i.e. supremum of all $alpha$ such that



$$|omega(t)-omega(s)|leq C|t-s|^alpha $$ for some $C$ and $t,sin J$ where $J$ is some time interval.



I was thinking maybe $frac12$ would be an $alpha$ that works because the Fourier transform of 1/f noise decays like $mathcalO(frac1sqrtn)$ but I am not sure if this is the supremum or even a reasonable guess. I do think the Fourier transform would be a good angle of approach though.



Please let me know your thoughts! Thanks!




stochastic-processes fourier-analysis uniform-continuity holder-spaces noise




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    For $omega=omega(t)$ which is 1/f noise, does anyone know the supremum of all its possible Holder exponents i.e. supremum of all $alpha$ such that



    $$|omega(t)-omega(s)|leq C|t-s|^alpha $$ for some $C$ and $t,sin J$ where $J$ is some time interval.



    I was thinking maybe $frac12$ would be an $alpha$ that works because the Fourier transform of 1/f noise decays like $mathcalO(frac1sqrtn)$ but I am not sure if this is the supremum or even a reasonable guess. I do think the Fourier transform would be a good angle of approach though.



    Please let me know your thoughts! Thanks!




    stochastic-processes fourier-analysis uniform-continuity holder-spaces noise




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      For $omega=omega(t)$ which is 1/f noise, does anyone know the supremum of all its possible Holder exponents i.e. supremum of all $alpha$ such that



      $$|omega(t)-omega(s)|leq C|t-s|^alpha $$ for some $C$ and $t,sin J$ where $J$ is some time interval.



      I was thinking maybe $frac12$ would be an $alpha$ that works because the Fourier transform of 1/f noise decays like $mathcalO(frac1sqrtn)$ but I am not sure if this is the supremum or even a reasonable guess. I do think the Fourier transform would be a good angle of approach though.



      Please let me know your thoughts! Thanks!




      stochastic-processes fourier-analysis uniform-continuity holder-spaces noise




      share|cite|improve this question











      For $omega=omega(t)$ which is 1/f noise, does anyone know the supremum of all its possible Holder exponents i.e. supremum of all $alpha$ such that



      $$|omega(t)-omega(s)|leq C|t-s|^alpha $$ for some $C$ and $t,sin J$ where $J$ is some time interval.



      I was thinking maybe $frac12$ would be an $alpha$ that works because the Fourier transform of 1/f noise decays like $mathcalO(frac1sqrtn)$ but I am not sure if this is the supremum or even a reasonable guess. I do think the Fourier transform would be a good angle of approach though.



      Please let me know your thoughts! Thanks!





      stochastic-processes fourier-analysis uniform-continuity holder-spaces noise





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      asked Jul 27 at 16:18









      Aakash Lakshmanan

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