A problem in probability theory about expected value and Fubini's theorem











up vote
1
down vote

favorite













Let ${X_{n}:ngeq1}$ be a seqeunce of square-integrable independent random variables on $(Omega,mathcal{F},mathbb{P})$ with $mathbb{E}[X_{n}]=0$ for every $ngeq1$. Set $S_{n}:=sum_{j=1}^{n}X_{j}$ for each $ngeq1$. Assume that $sum_{ngeq1}mathbb{E}[X_{n}^{2}]<infty$. Prove that, for every $p>1$,



i)$$big(mathbb{E}big[sup_{ngeq1}|S_{n}|^{2p}big]big)^{frac{1}{p}}leqfrac{p}{p-1}big(mathbb{E}big[|S|^{2p}big]big)^{frac{1}{p}}$$
Without loss of generality assume that $|S|^2 in L^p$ and you might first consider $min{big( sup_n|S_n|^2big),K}$ for any fixed $K>0.$



ii) If $Sin L^q$ for some $qin (2,infty)$, then $S_n to S$ also in $L^q$.




Using the said assumptions, it's known that that $S_{n}to S$ a.s. for some random variable S and due to completeness of $L^2(Omega,mathcal{F},mathbb{P})$, I know that $S in L^2$ and $S to S_n$ also in $L^2$. I also can prove that, for every $t>0$,



$$mathbb{P}big(sup_{ngeq1} |S_{n}|^2>tbig)leqfrac{1}{t}mathbb{E}big[S^2;sup_{ngeq1}|S_{n}|^2>tbig].$$



I think in order to prove the claim, I can combine the above statement with the following result



$$mathbb{E}[X^p]=pint_{0}^{infty}t^{p-1}mathbb{P}(X> t)dt = pint_{0}^{infty}t^{p-1}mathbb{P}(Xgeq t)dt.$$



My idea is to use Fubini's theorem to get the result, but I don't know how to connect them together.



I'd appreciate any help.










share|cite|improve this question


























    up vote
    1
    down vote

    favorite













    Let ${X_{n}:ngeq1}$ be a seqeunce of square-integrable independent random variables on $(Omega,mathcal{F},mathbb{P})$ with $mathbb{E}[X_{n}]=0$ for every $ngeq1$. Set $S_{n}:=sum_{j=1}^{n}X_{j}$ for each $ngeq1$. Assume that $sum_{ngeq1}mathbb{E}[X_{n}^{2}]<infty$. Prove that, for every $p>1$,



    i)$$big(mathbb{E}big[sup_{ngeq1}|S_{n}|^{2p}big]big)^{frac{1}{p}}leqfrac{p}{p-1}big(mathbb{E}big[|S|^{2p}big]big)^{frac{1}{p}}$$
    Without loss of generality assume that $|S|^2 in L^p$ and you might first consider $min{big( sup_n|S_n|^2big),K}$ for any fixed $K>0.$



    ii) If $Sin L^q$ for some $qin (2,infty)$, then $S_n to S$ also in $L^q$.




    Using the said assumptions, it's known that that $S_{n}to S$ a.s. for some random variable S and due to completeness of $L^2(Omega,mathcal{F},mathbb{P})$, I know that $S in L^2$ and $S to S_n$ also in $L^2$. I also can prove that, for every $t>0$,



    $$mathbb{P}big(sup_{ngeq1} |S_{n}|^2>tbig)leqfrac{1}{t}mathbb{E}big[S^2;sup_{ngeq1}|S_{n}|^2>tbig].$$



    I think in order to prove the claim, I can combine the above statement with the following result



    $$mathbb{E}[X^p]=pint_{0}^{infty}t^{p-1}mathbb{P}(X> t)dt = pint_{0}^{infty}t^{p-1}mathbb{P}(Xgeq t)dt.$$



    My idea is to use Fubini's theorem to get the result, but I don't know how to connect them together.



    I'd appreciate any help.










    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite












      Let ${X_{n}:ngeq1}$ be a seqeunce of square-integrable independent random variables on $(Omega,mathcal{F},mathbb{P})$ with $mathbb{E}[X_{n}]=0$ for every $ngeq1$. Set $S_{n}:=sum_{j=1}^{n}X_{j}$ for each $ngeq1$. Assume that $sum_{ngeq1}mathbb{E}[X_{n}^{2}]<infty$. Prove that, for every $p>1$,



      i)$$big(mathbb{E}big[sup_{ngeq1}|S_{n}|^{2p}big]big)^{frac{1}{p}}leqfrac{p}{p-1}big(mathbb{E}big[|S|^{2p}big]big)^{frac{1}{p}}$$
      Without loss of generality assume that $|S|^2 in L^p$ and you might first consider $min{big( sup_n|S_n|^2big),K}$ for any fixed $K>0.$



      ii) If $Sin L^q$ for some $qin (2,infty)$, then $S_n to S$ also in $L^q$.




      Using the said assumptions, it's known that that $S_{n}to S$ a.s. for some random variable S and due to completeness of $L^2(Omega,mathcal{F},mathbb{P})$, I know that $S in L^2$ and $S to S_n$ also in $L^2$. I also can prove that, for every $t>0$,



      $$mathbb{P}big(sup_{ngeq1} |S_{n}|^2>tbig)leqfrac{1}{t}mathbb{E}big[S^2;sup_{ngeq1}|S_{n}|^2>tbig].$$



      I think in order to prove the claim, I can combine the above statement with the following result



      $$mathbb{E}[X^p]=pint_{0}^{infty}t^{p-1}mathbb{P}(X> t)dt = pint_{0}^{infty}t^{p-1}mathbb{P}(Xgeq t)dt.$$



      My idea is to use Fubini's theorem to get the result, but I don't know how to connect them together.



      I'd appreciate any help.










      share|cite|improve this question














      Let ${X_{n}:ngeq1}$ be a seqeunce of square-integrable independent random variables on $(Omega,mathcal{F},mathbb{P})$ with $mathbb{E}[X_{n}]=0$ for every $ngeq1$. Set $S_{n}:=sum_{j=1}^{n}X_{j}$ for each $ngeq1$. Assume that $sum_{ngeq1}mathbb{E}[X_{n}^{2}]<infty$. Prove that, for every $p>1$,



      i)$$big(mathbb{E}big[sup_{ngeq1}|S_{n}|^{2p}big]big)^{frac{1}{p}}leqfrac{p}{p-1}big(mathbb{E}big[|S|^{2p}big]big)^{frac{1}{p}}$$
      Without loss of generality assume that $|S|^2 in L^p$ and you might first consider $min{big( sup_n|S_n|^2big),K}$ for any fixed $K>0.$



      ii) If $Sin L^q$ for some $qin (2,infty)$, then $S_n to S$ also in $L^q$.




      Using the said assumptions, it's known that that $S_{n}to S$ a.s. for some random variable S and due to completeness of $L^2(Omega,mathcal{F},mathbb{P})$, I know that $S in L^2$ and $S to S_n$ also in $L^2$. I also can prove that, for every $t>0$,



      $$mathbb{P}big(sup_{ngeq1} |S_{n}|^2>tbig)leqfrac{1}{t}mathbb{E}big[S^2;sup_{ngeq1}|S_{n}|^2>tbig].$$



      I think in order to prove the claim, I can combine the above statement with the following result



      $$mathbb{E}[X^p]=pint_{0}^{infty}t^{p-1}mathbb{P}(X> t)dt = pint_{0}^{infty}t^{p-1}mathbb{P}(Xgeq t)dt.$$



      My idea is to use Fubini's theorem to get the result, but I don't know how to connect them together.



      I'd appreciate any help.







      probability-theory convergence expected-value






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 15 at 16:27









      Weak Nullstellensatz

      163




      163






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          0
          down vote













          Observe that
          $$
          mathbb Eleft[S^2;sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright]
          =int_{0}^{+infty}Prleft(left{S^2gt sright}cap left{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}right)mathrm ds=int_0^{t/2}+int_{t/2}^{+infty}.
          $$

          Bound the first part by $t/2 Prleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}$ and the second one by $int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds$. We get
          $$
          tPrleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}leqslant 2int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds.
          $$






          share|cite|improve this answer





















            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: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            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%2f2999916%2fa-problem-in-probability-theory-about-expected-value-and-fubinis-theorem%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            0
            down vote













            Observe that
            $$
            mathbb Eleft[S^2;sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright]
            =int_{0}^{+infty}Prleft(left{S^2gt sright}cap left{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}right)mathrm ds=int_0^{t/2}+int_{t/2}^{+infty}.
            $$

            Bound the first part by $t/2 Prleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}$ and the second one by $int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds$. We get
            $$
            tPrleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}leqslant 2int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds.
            $$






            share|cite|improve this answer

























              up vote
              0
              down vote













              Observe that
              $$
              mathbb Eleft[S^2;sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright]
              =int_{0}^{+infty}Prleft(left{S^2gt sright}cap left{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}right)mathrm ds=int_0^{t/2}+int_{t/2}^{+infty}.
              $$

              Bound the first part by $t/2 Prleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}$ and the second one by $int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds$. We get
              $$
              tPrleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}leqslant 2int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds.
              $$






              share|cite|improve this answer























                up vote
                0
                down vote










                up vote
                0
                down vote









                Observe that
                $$
                mathbb Eleft[S^2;sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright]
                =int_{0}^{+infty}Prleft(left{S^2gt sright}cap left{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}right)mathrm ds=int_0^{t/2}+int_{t/2}^{+infty}.
                $$

                Bound the first part by $t/2 Prleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}$ and the second one by $int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds$. We get
                $$
                tPrleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}leqslant 2int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds.
                $$






                share|cite|improve this answer












                Observe that
                $$
                mathbb Eleft[S^2;sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright]
                =int_{0}^{+infty}Prleft(left{S^2gt sright}cap left{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}right)mathrm ds=int_0^{t/2}+int_{t/2}^{+infty}.
                $$

                Bound the first part by $t/2 Prleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}$ and the second one by $int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds$. We get
                $$
                tPrleft{sup_{ngeqslant 1}leftlvert S_nrightrvert^2gt tright}leqslant 2int_{t/2}^{+infty}Pr left{S^2gt sright}mathrm ds.
                $$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 16 at 16:12









                Davide Giraudo

                124k16149254




                124k16149254






























                     

                    draft saved


                    draft discarded



















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999916%2fa-problem-in-probability-theory-about-expected-value-and-fubinis-theorem%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Plaza Victoria

                    Puebla de Zaragoza

                    Musa