Prove that almost surely $limsup_{ntoinfty}frac{X_n}{ln n}=frac{1}{lambda}$












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Let ${X_n}$, n=1 to infinity, be independent random variables distributed $Exp(lambda)$.



Prove that almost surely $$limsup_{ntoinfty}frac{X_n}{ln n}=frac{1}{lambda}$$
My idea was to look at the $$sum_{k=1}^{infty}Pleft(left|frac{X_k}{ln k}-frac{1}{lambda}right|>epsilonright)$$ and prove it is finite. But I think it is not.
$$Pleft(X_k<left(-epsilon+frac{1}{lambda}right)ln kright)=frac{1}{k}(1-k^{lambdaepsilon})$$ and we can take such espilon value so it is divergent as $ktoinfty$, which means sum does not converge.










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    0












    $begingroup$


    Let ${X_n}$, n=1 to infinity, be independent random variables distributed $Exp(lambda)$.



    Prove that almost surely $$limsup_{ntoinfty}frac{X_n}{ln n}=frac{1}{lambda}$$
    My idea was to look at the $$sum_{k=1}^{infty}Pleft(left|frac{X_k}{ln k}-frac{1}{lambda}right|>epsilonright)$$ and prove it is finite. But I think it is not.
    $$Pleft(X_k<left(-epsilon+frac{1}{lambda}right)ln kright)=frac{1}{k}(1-k^{lambdaepsilon})$$ and we can take such espilon value so it is divergent as $ktoinfty$, which means sum does not converge.










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Let ${X_n}$, n=1 to infinity, be independent random variables distributed $Exp(lambda)$.



      Prove that almost surely $$limsup_{ntoinfty}frac{X_n}{ln n}=frac{1}{lambda}$$
      My idea was to look at the $$sum_{k=1}^{infty}Pleft(left|frac{X_k}{ln k}-frac{1}{lambda}right|>epsilonright)$$ and prove it is finite. But I think it is not.
      $$Pleft(X_k<left(-epsilon+frac{1}{lambda}right)ln kright)=frac{1}{k}(1-k^{lambdaepsilon})$$ and we can take such espilon value so it is divergent as $ktoinfty$, which means sum does not converge.










      share|cite|improve this question











      $endgroup$




      Let ${X_n}$, n=1 to infinity, be independent random variables distributed $Exp(lambda)$.



      Prove that almost surely $$limsup_{ntoinfty}frac{X_n}{ln n}=frac{1}{lambda}$$
      My idea was to look at the $$sum_{k=1}^{infty}Pleft(left|frac{X_k}{ln k}-frac{1}{lambda}right|>epsilonright)$$ and prove it is finite. But I think it is not.
      $$Pleft(X_k<left(-epsilon+frac{1}{lambda}right)ln kright)=frac{1}{k}(1-k^{lambdaepsilon})$$ and we can take such espilon value so it is divergent as $ktoinfty$, which means sum does not converge.







      probability probability-distributions convergence exponential-distribution almost-everywhere






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      edited Dec 5 '18 at 14:06









      the_candyman

      8,84622045




      8,84622045










      asked Dec 5 '18 at 14:02









      ryszard egginkryszard eggink

      315110




      315110






















          1 Answer
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          $begingroup$

          Observe that for $c>0$, we have that
          $$
          sum_{n=0}^infty P(X_n>clog n)=sum_{n=0}^inftyfrac{1}{n^{lambda c}}=begin{cases}
          <infty&text{if }c>1/lambda\
          infty&text{if }c<1/lambda
          end{cases}quad (1)
          $$

          where we used the fact that $P(X>x)=e^{-lambda x}$ for $xgeq 0$ if $Xsim text{exp}(lambda)$. It follows by Borel-Cantelli that for each $varepsilon>0$
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}+varepsilonquad text{i.o}right)=0
          $$

          whence
          $$
          limsup frac{X_n}{log n}leq lambda^{-1}.
          $$

          with probability $1$. Similarly for each $varepsilon>0$,
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}-varepsilonquad text{i.o}right)=1
          $$

          whence
          $$
          limsup frac{X_n}{log n}geq lambda^{-1}.$$

          with probability $1$.






          share|cite|improve this answer









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          • $begingroup$
            thanks! :) nice work
            $endgroup$
            – ryszard eggink
            Dec 5 '18 at 14:26











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          1 Answer
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          1 Answer
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          active

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          active

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          3












          $begingroup$

          Observe that for $c>0$, we have that
          $$
          sum_{n=0}^infty P(X_n>clog n)=sum_{n=0}^inftyfrac{1}{n^{lambda c}}=begin{cases}
          <infty&text{if }c>1/lambda\
          infty&text{if }c<1/lambda
          end{cases}quad (1)
          $$

          where we used the fact that $P(X>x)=e^{-lambda x}$ for $xgeq 0$ if $Xsim text{exp}(lambda)$. It follows by Borel-Cantelli that for each $varepsilon>0$
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}+varepsilonquad text{i.o}right)=0
          $$

          whence
          $$
          limsup frac{X_n}{log n}leq lambda^{-1}.
          $$

          with probability $1$. Similarly for each $varepsilon>0$,
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}-varepsilonquad text{i.o}right)=1
          $$

          whence
          $$
          limsup frac{X_n}{log n}geq lambda^{-1}.$$

          with probability $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks! :) nice work
            $endgroup$
            – ryszard eggink
            Dec 5 '18 at 14:26
















          3












          $begingroup$

          Observe that for $c>0$, we have that
          $$
          sum_{n=0}^infty P(X_n>clog n)=sum_{n=0}^inftyfrac{1}{n^{lambda c}}=begin{cases}
          <infty&text{if }c>1/lambda\
          infty&text{if }c<1/lambda
          end{cases}quad (1)
          $$

          where we used the fact that $P(X>x)=e^{-lambda x}$ for $xgeq 0$ if $Xsim text{exp}(lambda)$. It follows by Borel-Cantelli that for each $varepsilon>0$
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}+varepsilonquad text{i.o}right)=0
          $$

          whence
          $$
          limsup frac{X_n}{log n}leq lambda^{-1}.
          $$

          with probability $1$. Similarly for each $varepsilon>0$,
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}-varepsilonquad text{i.o}right)=1
          $$

          whence
          $$
          limsup frac{X_n}{log n}geq lambda^{-1}.$$

          with probability $1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks! :) nice work
            $endgroup$
            – ryszard eggink
            Dec 5 '18 at 14:26














          3












          3








          3





          $begingroup$

          Observe that for $c>0$, we have that
          $$
          sum_{n=0}^infty P(X_n>clog n)=sum_{n=0}^inftyfrac{1}{n^{lambda c}}=begin{cases}
          <infty&text{if }c>1/lambda\
          infty&text{if }c<1/lambda
          end{cases}quad (1)
          $$

          where we used the fact that $P(X>x)=e^{-lambda x}$ for $xgeq 0$ if $Xsim text{exp}(lambda)$. It follows by Borel-Cantelli that for each $varepsilon>0$
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}+varepsilonquad text{i.o}right)=0
          $$

          whence
          $$
          limsup frac{X_n}{log n}leq lambda^{-1}.
          $$

          with probability $1$. Similarly for each $varepsilon>0$,
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}-varepsilonquad text{i.o}right)=1
          $$

          whence
          $$
          limsup frac{X_n}{log n}geq lambda^{-1}.$$

          with probability $1$.






          share|cite|improve this answer









          $endgroup$



          Observe that for $c>0$, we have that
          $$
          sum_{n=0}^infty P(X_n>clog n)=sum_{n=0}^inftyfrac{1}{n^{lambda c}}=begin{cases}
          <infty&text{if }c>1/lambda\
          infty&text{if }c<1/lambda
          end{cases}quad (1)
          $$

          where we used the fact that $P(X>x)=e^{-lambda x}$ for $xgeq 0$ if $Xsim text{exp}(lambda)$. It follows by Borel-Cantelli that for each $varepsilon>0$
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}+varepsilonquad text{i.o}right)=0
          $$

          whence
          $$
          limsup frac{X_n}{log n}leq lambda^{-1}.
          $$

          with probability $1$. Similarly for each $varepsilon>0$,
          $$
          Pleft(frac{X_n}{log n}>lambda^{-1}-varepsilonquad text{i.o}right)=1
          $$

          whence
          $$
          limsup frac{X_n}{log n}geq lambda^{-1}.$$

          with probability $1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 5 '18 at 14:22









          Foobaz JohnFoobaz John

          21.8k41352




          21.8k41352












          • $begingroup$
            thanks! :) nice work
            $endgroup$
            – ryszard eggink
            Dec 5 '18 at 14:26


















          • $begingroup$
            thanks! :) nice work
            $endgroup$
            – ryszard eggink
            Dec 5 '18 at 14:26
















          $begingroup$
          thanks! :) nice work
          $endgroup$
          – ryszard eggink
          Dec 5 '18 at 14:26




          $begingroup$
          thanks! :) nice work
          $endgroup$
          – ryszard eggink
          Dec 5 '18 at 14:26


















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