Almost complete convergence












0












$begingroup$


In the book nonparametric functionnal data analysis, page 232, i don't understand why:
The almost complete convergence of $Y_n$ to $lne 0$ implies that there
exists some $delta > 0$ (choose for instance $delta = l/2$) such that



$$sum_{ige 0}mathsf{P}(|Y_n| le delta) < infty.$$
Cordially.










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$endgroup$








  • 1




    $begingroup$
    What does almost complete convergence mean?
    $endgroup$
    – Kavi Rama Murthy
    Dec 14 '18 at 8:52










  • $begingroup$
    Thank you very much
    $endgroup$
    – estimation
    Dec 14 '18 at 13:00










  • $begingroup$
    almost complete convergence of (Yn) to Y MEAN for each δ > 0: $$∑n∈N P(|Yn-Y | >δ)<∞.$$
    $endgroup$
    – estimation
    Dec 14 '18 at 13:06












  • $begingroup$
    the sum over n∈N, of P(|Yn−Y|>δ) is finite.
    $endgroup$
    – estimation
    Dec 14 '18 at 13:22


















0












$begingroup$


In the book nonparametric functionnal data analysis, page 232, i don't understand why:
The almost complete convergence of $Y_n$ to $lne 0$ implies that there
exists some $delta > 0$ (choose for instance $delta = l/2$) such that



$$sum_{ige 0}mathsf{P}(|Y_n| le delta) < infty.$$
Cordially.










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What does almost complete convergence mean?
    $endgroup$
    – Kavi Rama Murthy
    Dec 14 '18 at 8:52










  • $begingroup$
    Thank you very much
    $endgroup$
    – estimation
    Dec 14 '18 at 13:00










  • $begingroup$
    almost complete convergence of (Yn) to Y MEAN for each δ > 0: $$∑n∈N P(|Yn-Y | >δ)<∞.$$
    $endgroup$
    – estimation
    Dec 14 '18 at 13:06












  • $begingroup$
    the sum over n∈N, of P(|Yn−Y|>δ) is finite.
    $endgroup$
    – estimation
    Dec 14 '18 at 13:22
















0












0








0





$begingroup$


In the book nonparametric functionnal data analysis, page 232, i don't understand why:
The almost complete convergence of $Y_n$ to $lne 0$ implies that there
exists some $delta > 0$ (choose for instance $delta = l/2$) such that



$$sum_{ige 0}mathsf{P}(|Y_n| le delta) < infty.$$
Cordially.










share|cite|improve this question











$endgroup$




In the book nonparametric functionnal data analysis, page 232, i don't understand why:
The almost complete convergence of $Y_n$ to $lne 0$ implies that there
exists some $delta > 0$ (choose for instance $delta = l/2$) such that



$$sum_{ige 0}mathsf{P}(|Y_n| le delta) < infty.$$
Cordially.







probability-theory statistics measure-theory






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share|cite|improve this question













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share|cite|improve this question








edited Dec 14 '18 at 9:33









d.k.o.

9,520628




9,520628










asked Dec 14 '18 at 8:49









estimationestimation

12




12








  • 1




    $begingroup$
    What does almost complete convergence mean?
    $endgroup$
    – Kavi Rama Murthy
    Dec 14 '18 at 8:52










  • $begingroup$
    Thank you very much
    $endgroup$
    – estimation
    Dec 14 '18 at 13:00










  • $begingroup$
    almost complete convergence of (Yn) to Y MEAN for each δ > 0: $$∑n∈N P(|Yn-Y | >δ)<∞.$$
    $endgroup$
    – estimation
    Dec 14 '18 at 13:06












  • $begingroup$
    the sum over n∈N, of P(|Yn−Y|>δ) is finite.
    $endgroup$
    – estimation
    Dec 14 '18 at 13:22
















  • 1




    $begingroup$
    What does almost complete convergence mean?
    $endgroup$
    – Kavi Rama Murthy
    Dec 14 '18 at 8:52










  • $begingroup$
    Thank you very much
    $endgroup$
    – estimation
    Dec 14 '18 at 13:00










  • $begingroup$
    almost complete convergence of (Yn) to Y MEAN for each δ > 0: $$∑n∈N P(|Yn-Y | >δ)<∞.$$
    $endgroup$
    – estimation
    Dec 14 '18 at 13:06












  • $begingroup$
    the sum over n∈N, of P(|Yn−Y|>δ) is finite.
    $endgroup$
    – estimation
    Dec 14 '18 at 13:22










1




1




$begingroup$
What does almost complete convergence mean?
$endgroup$
– Kavi Rama Murthy
Dec 14 '18 at 8:52




$begingroup$
What does almost complete convergence mean?
$endgroup$
– Kavi Rama Murthy
Dec 14 '18 at 8:52












$begingroup$
Thank you very much
$endgroup$
– estimation
Dec 14 '18 at 13:00




$begingroup$
Thank you very much
$endgroup$
– estimation
Dec 14 '18 at 13:00












$begingroup$
almost complete convergence of (Yn) to Y MEAN for each δ > 0: $$∑n∈N P(|Yn-Y | >δ)<∞.$$
$endgroup$
– estimation
Dec 14 '18 at 13:06






$begingroup$
almost complete convergence of (Yn) to Y MEAN for each δ > 0: $$∑n∈N P(|Yn-Y | >δ)<∞.$$
$endgroup$
– estimation
Dec 14 '18 at 13:06














$begingroup$
the sum over n∈N, of P(|Yn−Y|>δ) is finite.
$endgroup$
– estimation
Dec 14 '18 at 13:22






$begingroup$
the sum over n∈N, of P(|Yn−Y|>δ) is finite.
$endgroup$
– estimation
Dec 14 '18 at 13:22












1 Answer
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$begingroup$

Suppose that $l>0$. Since
begin{align}
mathsf{P}(|Y_n|le l/2)&=mathsf{P}(-l/2le Y_nle l/2)\
&le mathsf{P}(Y_nle l-l/2)le mathsf{P}(|Y_n-l|ge l/2),
end{align}



$$
sum_{nge 1}mathsf{P}(|Y_n|le l/2)le sum_{nge 1}mathsf{P}(|Y_n-l|ge l/2)<infty.
$$






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    $begingroup$

    Suppose that $l>0$. Since
    begin{align}
    mathsf{P}(|Y_n|le l/2)&=mathsf{P}(-l/2le Y_nle l/2)\
    &le mathsf{P}(Y_nle l-l/2)le mathsf{P}(|Y_n-l|ge l/2),
    end{align}



    $$
    sum_{nge 1}mathsf{P}(|Y_n|le l/2)le sum_{nge 1}mathsf{P}(|Y_n-l|ge l/2)<infty.
    $$






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Suppose that $l>0$. Since
      begin{align}
      mathsf{P}(|Y_n|le l/2)&=mathsf{P}(-l/2le Y_nle l/2)\
      &le mathsf{P}(Y_nle l-l/2)le mathsf{P}(|Y_n-l|ge l/2),
      end{align}



      $$
      sum_{nge 1}mathsf{P}(|Y_n|le l/2)le sum_{nge 1}mathsf{P}(|Y_n-l|ge l/2)<infty.
      $$






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Suppose that $l>0$. Since
        begin{align}
        mathsf{P}(|Y_n|le l/2)&=mathsf{P}(-l/2le Y_nle l/2)\
        &le mathsf{P}(Y_nle l-l/2)le mathsf{P}(|Y_n-l|ge l/2),
        end{align}



        $$
        sum_{nge 1}mathsf{P}(|Y_n|le l/2)le sum_{nge 1}mathsf{P}(|Y_n-l|ge l/2)<infty.
        $$






        share|cite|improve this answer









        $endgroup$



        Suppose that $l>0$. Since
        begin{align}
        mathsf{P}(|Y_n|le l/2)&=mathsf{P}(-l/2le Y_nle l/2)\
        &le mathsf{P}(Y_nle l-l/2)le mathsf{P}(|Y_n-l|ge l/2),
        end{align}



        $$
        sum_{nge 1}mathsf{P}(|Y_n|le l/2)le sum_{nge 1}mathsf{P}(|Y_n-l|ge l/2)<infty.
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 14 '18 at 9:31









        d.k.o.d.k.o.

        9,520628




        9,520628






























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