Convergence of the expected value of bounded random variables












0












$begingroup$


Let $X_n$ be a sequence of bounded random variable such that
$$
mathbb{E}left[X_nright]tomathbb{E}left[Xright]
$$
with $X$ a bounded random variable. Can I conclude that



$$
mathbb{E}left[X_n,Wright]tomathbb{E}left[X,Wright]
$$



for any bounded random variable $W$ ?






convergence random-variables expected-value







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  • $begingroup$
    @nicomezi in my counterexample we even have $X_nstackrel{d}{=}X$ for every $n$.
    $endgroup$
    – drhab
    Dec 21 '18 at 11:26










  • $begingroup$
    I meant $X_n overset{P} to X$ indeed, made a confusion. @drhab.
    $endgroup$
    – nicomezi
    Dec 21 '18 at 11:31
















0












$begingroup$


Let $X_n$ be a sequence of bounded random variable such that
$$
mathbb{E}left[X_nright]tomathbb{E}left[Xright]
$$
with $X$ a bounded random variable. Can I conclude that



$$
mathbb{E}left[X_n,Wright]tomathbb{E}left[X,Wright]
$$



for any bounded random variable $W$ ?






convergence random-variables expected-value







share|cite|improve this question









$endgroup$












  • $begingroup$
    @nicomezi in my counterexample we even have $X_nstackrel{d}{=}X$ for every $n$.
    $endgroup$
    – drhab
    Dec 21 '18 at 11:26










  • $begingroup$
    I meant $X_n overset{P} to X$ indeed, made a confusion. @drhab.
    $endgroup$
    – nicomezi
    Dec 21 '18 at 11:31














0












0








0





$begingroup$


Let $X_n$ be a sequence of bounded random variable such that
$$
mathbb{E}left[X_nright]tomathbb{E}left[Xright]
$$
with $X$ a bounded random variable. Can I conclude that



$$
mathbb{E}left[X_n,Wright]tomathbb{E}left[X,Wright]
$$



for any bounded random variable $W$ ?






convergence random-variables expected-value







share|cite|improve this question









$endgroup$




Let $X_n$ be a sequence of bounded random variable such that
$$
mathbb{E}left[X_nright]tomathbb{E}left[Xright]
$$
with $X$ a bounded random variable. Can I conclude that



$$
mathbb{E}left[X_n,Wright]tomathbb{E}left[X,Wright]
$$



for any bounded random variable $W$ ?






convergence random-variables expected-value




convergence random-variables expected-value






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 21 '18 at 11:13









AlmostSureUserAlmostSureUser

331418




331418












  • $begingroup$
    @nicomezi in my counterexample we even have $X_nstackrel{d}{=}X$ for every $n$.
    $endgroup$
    – drhab
    Dec 21 '18 at 11:26










  • $begingroup$
    I meant $X_n overset{P} to X$ indeed, made a confusion. @drhab.
    $endgroup$
    – nicomezi
    Dec 21 '18 at 11:31


















  • $begingroup$
    @nicomezi in my counterexample we even have $X_nstackrel{d}{=}X$ for every $n$.
    $endgroup$
    – drhab
    Dec 21 '18 at 11:26










  • $begingroup$
    I meant $X_n overset{P} to X$ indeed, made a confusion. @drhab.
    $endgroup$
    – nicomezi
    Dec 21 '18 at 11:31
















$begingroup$
@nicomezi in my counterexample we even have $X_nstackrel{d}{=}X$ for every $n$.
$endgroup$
– drhab
Dec 21 '18 at 11:26




$begingroup$
@nicomezi in my counterexample we even have $X_nstackrel{d}{=}X$ for every $n$.
$endgroup$
– drhab
Dec 21 '18 at 11:26












$begingroup$
I meant $X_n overset{P} to X$ indeed, made a confusion. @drhab.
$endgroup$
– nicomezi
Dec 21 '18 at 11:31




$begingroup$
I meant $X_n overset{P} to X$ indeed, made a confusion. @drhab.
$endgroup$
– nicomezi
Dec 21 '18 at 11:31










2 Answers
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$begingroup$

No.



Throw a fair coin and for every $n$ let $X_n=1$ if it lands on heads and $X_n=0$ otherwise.



Let $X=1$ if it lands on tails and $X=0$ otherwise.



Then let $W=X$ so that $X_nW=0$ and $XW=X$.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    Let $X$ have uniform distribution on $(-1,1)$, $X_n=-X$ for all $n$ and $W=X$. Then $EX_n W=-1/3$ for all $n$ and $EXW=1/3$ even though $EX_n =0 to 0=EX$.






    share|cite|improve this answer









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      2 Answers
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      2 Answers
      2






      active

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      2












      $begingroup$

      No.



      Throw a fair coin and for every $n$ let $X_n=1$ if it lands on heads and $X_n=0$ otherwise.



      Let $X=1$ if it lands on tails and $X=0$ otherwise.



      Then let $W=X$ so that $X_nW=0$ and $XW=X$.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        No.



        Throw a fair coin and for every $n$ let $X_n=1$ if it lands on heads and $X_n=0$ otherwise.



        Let $X=1$ if it lands on tails and $X=0$ otherwise.



        Then let $W=X$ so that $X_nW=0$ and $XW=X$.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          No.



          Throw a fair coin and for every $n$ let $X_n=1$ if it lands on heads and $X_n=0$ otherwise.



          Let $X=1$ if it lands on tails and $X=0$ otherwise.



          Then let $W=X$ so that $X_nW=0$ and $XW=X$.






          share|cite|improve this answer









          $endgroup$



          No.



          Throw a fair coin and for every $n$ let $X_n=1$ if it lands on heads and $X_n=0$ otherwise.



          Let $X=1$ if it lands on tails and $X=0$ otherwise.



          Then let $W=X$ so that $X_nW=0$ and $XW=X$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 21 '18 at 11:22









          drhabdrhab

          104k545136




          104k545136























              1












              $begingroup$

              Let $X$ have uniform distribution on $(-1,1)$, $X_n=-X$ for all $n$ and $W=X$. Then $EX_n W=-1/3$ for all $n$ and $EXW=1/3$ even though $EX_n =0 to 0=EX$.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                Let $X$ have uniform distribution on $(-1,1)$, $X_n=-X$ for all $n$ and $W=X$. Then $EX_n W=-1/3$ for all $n$ and $EXW=1/3$ even though $EX_n =0 to 0=EX$.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  Let $X$ have uniform distribution on $(-1,1)$, $X_n=-X$ for all $n$ and $W=X$. Then $EX_n W=-1/3$ for all $n$ and $EXW=1/3$ even though $EX_n =0 to 0=EX$.






                  share|cite|improve this answer









                  $endgroup$



                  Let $X$ have uniform distribution on $(-1,1)$, $X_n=-X$ for all $n$ and $W=X$. Then $EX_n W=-1/3$ for all $n$ and $EXW=1/3$ even though $EX_n =0 to 0=EX$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 21 '18 at 11:51









                  Kavi Rama MurthyKavi Rama Murthy

                  74.4k53270




                  74.4k53270






























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