What is the distribution of the random sum?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












Let $X_n$ be iid $pm1$ with probability $1/2$. What is the distribution of $sum_n=1^infty dfraclog nnX_n$?



I can conclude the sum converges almost surely since $sum_n Var(dfraclog nnX_n)<infty$. Then I tried with the characteristic function but I am getting an infinite product of $cos(tdfraclog nn)$ and I don't know what happens to this product.




sequences-and-series probability-theory




share|cite|improve this question
























    up vote
    0
    down vote

    favorite












    Let $X_n$ be iid $pm1$ with probability $1/2$. What is the distribution of $sum_n=1^infty dfraclog nnX_n$?



    I can conclude the sum converges almost surely since $sum_n Var(dfraclog nnX_n)<infty$. Then I tried with the characteristic function but I am getting an infinite product of $cos(tdfraclog nn)$ and I don't know what happens to this product.




    sequences-and-series probability-theory




    share|cite|improve this question






















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $X_n$ be iid $pm1$ with probability $1/2$. What is the distribution of $sum_n=1^infty dfraclog nnX_n$?



      I can conclude the sum converges almost surely since $sum_n Var(dfraclog nnX_n)<infty$. Then I tried with the characteristic function but I am getting an infinite product of $cos(tdfraclog nn)$ and I don't know what happens to this product.




      sequences-and-series probability-theory




      share|cite|improve this question












      Let $X_n$ be iid $pm1$ with probability $1/2$. What is the distribution of $sum_n=1^infty dfraclog nnX_n$?



      I can conclude the sum converges almost surely since $sum_n Var(dfraclog nnX_n)<infty$. Then I tried with the characteristic function but I am getting an infinite product of $cos(tdfraclog nn)$ and I don't know what happens to this product.





      sequences-and-series probability-theory





      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 27 at 16:35









      Andrew Richards

      1139




      1139




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          0
          down vote













          The infinite product
          $$ prod_n=1^infty cosleft(t fraclog nnright) $$
          converges since $$cosleft(t fraclog nnright) approx 1 - fract^2 log^2 n2 n^2$$ and $sum_n 1/n^2$ converges. But I don't expect there to be a closed form of this product.






          share|cite|improve this answer




















          • That the infinite product converges is clear from the fact that the random sum converges almost surely and hence in distribution, so characteristic functions must converge. I need the form of the limit.
            – Andrew Richards
            Aug 27 at 17:10






          • 1




            What makes you think it has a "form"?
            – Robert Israel
            Aug 27 at 17:35










          • This is an exam question. At least can we say that the convergence does not happen to a Normal?
            – Andrew Richards
            Aug 28 at 16:25










          • It can't be normal because the characteristic function is $0$ at $pi n/(2 log n)$ for $n ge 2$.
            – Robert Israel
            Aug 28 at 17:30










          Your Answer




          StackExchange.ifUsing("editor", function ()
          return StackExchange.using("mathjaxEditing", function ()
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          );
          );
          , "mathjax-editing");

          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          convertImagesToLinks: true,
          noModals: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );













           

          draft saved


          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2896380%2fwhat-is-the-distribution-of-the-random-sum%23new-answer', 'question_page');

          );

          Post as a guest

















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          0
          down vote













          The infinite product
          $$ prod_n=1^infty cosleft(t fraclog nnright) $$
          converges since $$cosleft(t fraclog nnright) approx 1 - fract^2 log^2 n2 n^2$$ and $sum_n 1/n^2$ converges. But I don't expect there to be a closed form of this product.






          share|cite|improve this answer




















          • That the infinite product converges is clear from the fact that the random sum converges almost surely and hence in distribution, so characteristic functions must converge. I need the form of the limit.
            – Andrew Richards
            Aug 27 at 17:10






          • 1




            What makes you think it has a "form"?
            – Robert Israel
            Aug 27 at 17:35










          • This is an exam question. At least can we say that the convergence does not happen to a Normal?
            – Andrew Richards
            Aug 28 at 16:25










          • It can't be normal because the characteristic function is $0$ at $pi n/(2 log n)$ for $n ge 2$.
            – Robert Israel
            Aug 28 at 17:30














          up vote
          0
          down vote













          The infinite product
          $$ prod_n=1^infty cosleft(t fraclog nnright) $$
          converges since $$cosleft(t fraclog nnright) approx 1 - fract^2 log^2 n2 n^2$$ and $sum_n 1/n^2$ converges. But I don't expect there to be a closed form of this product.






          share|cite|improve this answer




















          • That the infinite product converges is clear from the fact that the random sum converges almost surely and hence in distribution, so characteristic functions must converge. I need the form of the limit.
            – Andrew Richards
            Aug 27 at 17:10






          • 1




            What makes you think it has a "form"?
            – Robert Israel
            Aug 27 at 17:35










          • This is an exam question. At least can we say that the convergence does not happen to a Normal?
            – Andrew Richards
            Aug 28 at 16:25










          • It can't be normal because the characteristic function is $0$ at $pi n/(2 log n)$ for $n ge 2$.
            – Robert Israel
            Aug 28 at 17:30












          up vote
          0
          down vote










          up vote
          0
          down vote









          The infinite product
          $$ prod_n=1^infty cosleft(t fraclog nnright) $$
          converges since $$cosleft(t fraclog nnright) approx 1 - fract^2 log^2 n2 n^2$$ and $sum_n 1/n^2$ converges. But I don't expect there to be a closed form of this product.






          share|cite|improve this answer












          The infinite product
          $$ prod_n=1^infty cosleft(t fraclog nnright) $$
          converges since $$cosleft(t fraclog nnright) approx 1 - fract^2 log^2 n2 n^2$$ and $sum_n 1/n^2$ converges. But I don't expect there to be a closed form of this product.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 27 at 16:56









          Robert Israel

          306k22201443




          306k22201443











          • That the infinite product converges is clear from the fact that the random sum converges almost surely and hence in distribution, so characteristic functions must converge. I need the form of the limit.
            – Andrew Richards
            Aug 27 at 17:10






          • 1




            What makes you think it has a "form"?
            – Robert Israel
            Aug 27 at 17:35










          • This is an exam question. At least can we say that the convergence does not happen to a Normal?
            – Andrew Richards
            Aug 28 at 16:25










          • It can't be normal because the characteristic function is $0$ at $pi n/(2 log n)$ for $n ge 2$.
            – Robert Israel
            Aug 28 at 17:30
















          • That the infinite product converges is clear from the fact that the random sum converges almost surely and hence in distribution, so characteristic functions must converge. I need the form of the limit.
            – Andrew Richards
            Aug 27 at 17:10






          • 1




            What makes you think it has a "form"?
            – Robert Israel
            Aug 27 at 17:35










          • This is an exam question. At least can we say that the convergence does not happen to a Normal?
            – Andrew Richards
            Aug 28 at 16:25










          • It can't be normal because the characteristic function is $0$ at $pi n/(2 log n)$ for $n ge 2$.
            – Robert Israel
            Aug 28 at 17:30















          That the infinite product converges is clear from the fact that the random sum converges almost surely and hence in distribution, so characteristic functions must converge. I need the form of the limit.
          – Andrew Richards
          Aug 27 at 17:10




          That the infinite product converges is clear from the fact that the random sum converges almost surely and hence in distribution, so characteristic functions must converge. I need the form of the limit.
          – Andrew Richards
          Aug 27 at 17:10




          1




          1




          What makes you think it has a "form"?
          – Robert Israel
          Aug 27 at 17:35




          What makes you think it has a "form"?
          – Robert Israel
          Aug 27 at 17:35












          This is an exam question. At least can we say that the convergence does not happen to a Normal?
          – Andrew Richards
          Aug 28 at 16:25




          This is an exam question. At least can we say that the convergence does not happen to a Normal?
          – Andrew Richards
          Aug 28 at 16:25












          It can't be normal because the characteristic function is $0$ at $pi n/(2 log n)$ for $n ge 2$.
          – Robert Israel
          Aug 28 at 17:30




          It can't be normal because the characteristic function is $0$ at $pi n/(2 log n)$ for $n ge 2$.
          – Robert Israel
          Aug 28 at 17:30

















           

          draft saved


          draft discarded















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2896380%2fwhat-is-the-distribution-of-the-random-sum%23new-answer', 'question_page');

          );

          Post as a guest
































































          -

          這個網誌中的熱門文章

          How to combine Bézier curves to a surface?

          圖片
          Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
          閱讀完整內容

          Mutual Information Always Non-negative

          圖片
          Clash Royale CLAN TAG #URR8PPP up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 '12 at 15:38 Cameron Buie 83.6k 7 71 154 asked Jun 17 '12 at 15:30 Omri 359 3 13 3 Convexity of the function $tmapsto tlog t$. – Did Jun 17 '12 at 15:33 In addition, the convexity properties require the coefficients in the linear combination sum 1. Then, as p(x,y) is a probability distribution, it fullfits such condition. – Francisco Javier Delgado Ceped Aug 10 at 18:44 add a comment  |  up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 ...
          閱讀完整內容

          Why am i infinitely getting the same tweet with the Twitter Search API?

          圖片
          up vote 0 down vote favorite My code workes perfectly for other queries, but for one query I am infinitely getting the same tweet. I can't understand what the problem is. API call: query='Allergic asthma OR Nonallergic asthma OR Occupational asthma OR EIB OR Exercise-induced bronchoconstriction OR Nocturnal asthma OR Cough-variant asthma' new_tweets = api.search(q=searchQuery, count=100, since_id=since_id, lang='en',tweet_mode='extended') if not new_tweets: print("No more tweets found") break for tweet in new_tweets: if True: print(jsonpickle.encode(tweet._json, unpicklable=False)) my_file = open('searchTweets.txt','a') my_file.write(jsonpickle.encode(tweet._json, unpicklable=False)+'n') my_file.close() else: continue Output in file: https://pastebin.com/JmRCsTeE These are the JSONs of the same Tweet. Other queries that work: Breast cancer OR Prostate cancer OR Colon cancer OR Lung cancer Type 2 dia...
          閱讀完整內容