Limitation on the difference of characteristic functions











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Let $ X, Y $ be independent with characteristic functions $varphi_X(t) , varphi_Y(t) $.



Show that: $$ sup_{t in mathbb{R}}mid varphi_x(t) - varphi_Y(t) mid le 2P(X neq Y) $$



I would appreciate any tips or hints.






probability random-variables







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  • That's a nice question. Looking forward to an answer.
    – Hendrra
    8 hours ago















up vote
2
down vote

favorite
1












Let $ X, Y $ be independent with characteristic functions $varphi_X(t) , varphi_Y(t) $.



Show that: $$ sup_{t in mathbb{R}}mid varphi_x(t) - varphi_Y(t) mid le 2P(X neq Y) $$



I would appreciate any tips or hints.






probability random-variables







share|cite|improve this question
























  • That's a nice question. Looking forward to an answer.
    – Hendrra
    8 hours ago













up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





Let $ X, Y $ be independent with characteristic functions $varphi_X(t) , varphi_Y(t) $.



Show that: $$ sup_{t in mathbb{R}}mid varphi_x(t) - varphi_Y(t) mid le 2P(X neq Y) $$



I would appreciate any tips or hints.






probability random-variables







share|cite|improve this question















Let $ X, Y $ be independent with characteristic functions $varphi_X(t) , varphi_Y(t) $.



Show that: $$ sup_{t in mathbb{R}}mid varphi_x(t) - varphi_Y(t) mid le 2P(X neq Y) $$



I would appreciate any tips or hints.






probability random-variables




probability random-variables






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 8 hours ago

























asked 8 hours ago









Wywana

535




535












  • That's a nice question. Looking forward to an answer.
    – Hendrra
    8 hours ago


















  • That's a nice question. Looking forward to an answer.
    – Hendrra
    8 hours ago
















That's a nice question. Looking forward to an answer.
– Hendrra
8 hours ago




That's a nice question. Looking forward to an answer.
– Hendrra
8 hours ago










2 Answers
2






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4
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accepted










Note that for all $t,x,yin mathbb{R}$, it holds
$$
|e^{itx}-e^{ity}|leq 2 cdot 1_{{xneq y}}.
$$ Thus, for any $tin mathbb{R}$, we have
$$
|varphi_X(t)-varphi_Y(t)|leq E[|e^{itX}-e^{itY}|]leq 2E[1_{{Xneq Y}}]=2P(Xneq Y).
$$ Now, take supremum over $tin mathbb{R}$ to get
$$
sup_{tinmathbb{R}}|varphi_X(t)-varphi_Y(t)|leq 2P(Xneq Y),
$$as desired.






share|cite|improve this answer





















  • That's a tidy solution!
    – Hendrra
    5 hours ago


















up vote
1
down vote













What sort of distributions do $X$ and $Y$ have? If the distributions have no discrete terms, $P(Xne Y)=1$ and the inequality always hold trivially, since all $|phi(t)|le 1.$






share|cite|improve this answer





















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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

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    active

    oldest

    votes








    up vote
    4
    down vote



    accepted










    Note that for all $t,x,yin mathbb{R}$, it holds
    $$
    |e^{itx}-e^{ity}|leq 2 cdot 1_{{xneq y}}.
    $$     Thus, for any $tin mathbb{R}$, we have
    $$
    |varphi_X(t)-varphi_Y(t)|leq E[|e^{itX}-e^{itY}|]leq 2E[1_{{Xneq Y}}]=2P(Xneq Y).
    $$     Now, take supremum over $tin mathbb{R}$ to get
    $$
    sup_{tinmathbb{R}}|varphi_X(t)-varphi_Y(t)|leq 2P(Xneq Y),
    $$    as desired.






    share|cite|improve this answer





















    • That's a tidy solution!
      – Hendrra
      5 hours ago















    up vote
    4
    down vote



    accepted










    Note that for all $t,x,yin mathbb{R}$, it holds
    $$
    |e^{itx}-e^{ity}|leq 2 cdot 1_{{xneq y}}.
    $$     Thus, for any $tin mathbb{R}$, we have
    $$
    |varphi_X(t)-varphi_Y(t)|leq E[|e^{itX}-e^{itY}|]leq 2E[1_{{Xneq Y}}]=2P(Xneq Y).
    $$     Now, take supremum over $tin mathbb{R}$ to get
    $$
    sup_{tinmathbb{R}}|varphi_X(t)-varphi_Y(t)|leq 2P(Xneq Y),
    $$    as desired.






    share|cite|improve this answer





















    • That's a tidy solution!
      – Hendrra
      5 hours ago













    up vote
    4
    down vote



    accepted







    up vote
    4
    down vote



    accepted






    Note that for all $t,x,yin mathbb{R}$, it holds
    $$
    |e^{itx}-e^{ity}|leq 2 cdot 1_{{xneq y}}.
    $$     Thus, for any $tin mathbb{R}$, we have
    $$
    |varphi_X(t)-varphi_Y(t)|leq E[|e^{itX}-e^{itY}|]leq 2E[1_{{Xneq Y}}]=2P(Xneq Y).
    $$     Now, take supremum over $tin mathbb{R}$ to get
    $$
    sup_{tinmathbb{R}}|varphi_X(t)-varphi_Y(t)|leq 2P(Xneq Y),
    $$    as desired.






    share|cite|improve this answer












    Note that for all $t,x,yin mathbb{R}$, it holds
    $$
    |e^{itx}-e^{ity}|leq 2 cdot 1_{{xneq y}}.
    $$     Thus, for any $tin mathbb{R}$, we have
    $$
    |varphi_X(t)-varphi_Y(t)|leq E[|e^{itX}-e^{itY}|]leq 2E[1_{{Xneq Y}}]=2P(Xneq Y).
    $$     Now, take supremum over $tin mathbb{R}$ to get
    $$
    sup_{tinmathbb{R}}|varphi_X(t)-varphi_Y(t)|leq 2P(Xneq Y),
    $$    as desired.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 6 hours ago









    Song

    3,435315




    3,435315












    • That's a tidy solution!
      – Hendrra
      5 hours ago


















    • That's a tidy solution!
      – Hendrra
      5 hours ago
















    That's a tidy solution!
    – Hendrra
    5 hours ago




    That's a tidy solution!
    – Hendrra
    5 hours ago










    up vote
    1
    down vote













    What sort of distributions do $X$ and $Y$ have? If the distributions have no discrete terms, $P(Xne Y)=1$ and the inequality always hold trivially, since all $|phi(t)|le 1.$






    share|cite|improve this answer

























      up vote
      1
      down vote













      What sort of distributions do $X$ and $Y$ have? If the distributions have no discrete terms, $P(Xne Y)=1$ and the inequality always hold trivially, since all $|phi(t)|le 1.$






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        What sort of distributions do $X$ and $Y$ have? If the distributions have no discrete terms, $P(Xne Y)=1$ and the inequality always hold trivially, since all $|phi(t)|le 1.$






        share|cite|improve this answer












        What sort of distributions do $X$ and $Y$ have? If the distributions have no discrete terms, $P(Xne Y)=1$ and the inequality always hold trivially, since all $|phi(t)|le 1.$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 6 hours ago









        herb steinberg

        2,4432310




        2,4432310






























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