Convergence in probability to a contant implication

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Suppose that a sequence of random variables $X_n$ converges in probability to $c$. Does it follow that
$$
lim_ntoinftyP(X_n<c)=0?
$$
Using the definition, I know that for any $epsilon>0$,
$$
lim_ntoinftyP(X_n<c-epsilon)=0.
$$
Can I take the limit w.r.t. $epsilon$?




probability-theory




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    up vote
    2
    down vote

    favorite












    Suppose that a sequence of random variables $X_n$ converges in probability to $c$. Does it follow that
    $$
    lim_ntoinftyP(X_n<c)=0?
    $$
    Using the definition, I know that for any $epsilon>0$,
    $$
    lim_ntoinftyP(X_n<c-epsilon)=0.
    $$
    Can I take the limit w.r.t. $epsilon$?




    probability-theory




    share|cite|improve this question





















      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Suppose that a sequence of random variables $X_n$ converges in probability to $c$. Does it follow that
      $$
      lim_ntoinftyP(X_n<c)=0?
      $$
      Using the definition, I know that for any $epsilon>0$,
      $$
      lim_ntoinftyP(X_n<c-epsilon)=0.
      $$
      Can I take the limit w.r.t. $epsilon$?




      probability-theory




      share|cite|improve this question











      Suppose that a sequence of random variables $X_n$ converges in probability to $c$. Does it follow that
      $$
      lim_ntoinftyP(X_n<c)=0?
      $$
      Using the definition, I know that for any $epsilon>0$,
      $$
      lim_ntoinftyP(X_n<c-epsilon)=0.
      $$
      Can I take the limit w.r.t. $epsilon$?





      probability-theory





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 27 at 8:48









      akm47

      184




      184




















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          Take $X_n=c-frac 1 n$. Then $X_n to c$ in probability (in fact almost surely) but $PX_n<c=1$ for all $n$.






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            active

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            1 Answer
            1






            active

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            active

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            up vote
            2
            down vote



            accepted










            Take $X_n=c-frac 1 n$. Then $X_n to c$ in probability (in fact almost surely) but $PX_n<c=1$ for all $n$.






            share|cite|improve this answer

























              up vote
              2
              down vote



              accepted










              Take $X_n=c-frac 1 n$. Then $X_n to c$ in probability (in fact almost surely) but $PX_n<c=1$ for all $n$.






              share|cite|improve this answer























                up vote
                2
                down vote



                accepted







                up vote
                2
                down vote



                accepted






                Take $X_n=c-frac 1 n$. Then $X_n to c$ in probability (in fact almost surely) but $PX_n<c=1$ for all $n$.






                share|cite|improve this answer













                Take $X_n=c-frac 1 n$. Then $X_n to c$ in probability (in fact almost surely) but $PX_n<c=1$ for all $n$.







                share|cite|improve this answer













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                share|cite|improve this answer











                answered Jul 27 at 8:57









                Kavi Rama Murthy

                20k2829




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