If $Φ$ is the cdf of the standard normal distribution and $φ_i:ℝ→ℝ$ is linear, are the 1st and 2nd...












0












$begingroup$


Let





  • $dinmathbb N$ with $d>1$


  • $lambda^d$ denote the Lebesuge measure on $mathcal Bleft(mathbb R^dright)$


  • $fin C^2(mathbb R)$ be positive and $$pi(x):=prod_{i=1}^df(x_i);;;text{for }xinmathbb R^d$$


  • $Phi$ denote the cdf of the standard normal distribution


  • $ell>0$ and $$sigma^2:=frac{ell^2}{d-1}$$

  • $xinmathbb R^d$


  • $(Omega,mathcal A,operatorname P)$ be a probability space


  • $Y$ be a $mathbb R^d$-valued random variable on $(Omega,mathcal A,operatorname P)$ with $Ysimmathcal N(0,sigma I)$ (where $I$ denotes the $d$-dimensional identity matrix)


Now, let $$g(z):=frac1{d-1}sum_{i=2}^dleft|frac{f'(z_i)}{f(z_i)}right|^2;;;text{for }zinmathbb R^d,$$ $varepsilon:=g-g(x_1)$ and $$M(z):=Phileft(frac1{{g(x)}^{1/2}}left(frac zell-frac{ell g(x)}2right)right)+e^zPhileft(-frac z{ell{{g(x)}^{1/2}}}-frac{ell{g(x)}^{1/2}}2right);;;text{for }zinmathbb R.$$ Note that there is a real-valued random variable $W$ on $(Omega,mathcal A,operatorname P)$ with $$W(omega)inleft{(1-t)x_1+tY_1(omega):tin[0,1]right}tag1$$ and $$M(varepsilon(Y_1))=M(0)+frac12M(0)g'(x_1)(Y_1-x_1)+frac12left(T(W)right)(Y_1-x_1)^2tag2,$$ where $$T(w):=g''(w)M'(varepsilon(w))+{g'(w)}^2M''(varepsilon(w));;;text{for }winmathbb R.$$




How can we show that $T(W)$ is uniformly bounded with respect to $x$, if the domain of $x$ is restricted to a compact set?




Most probably, the question is highly overcomplicated. I've read that the claim follows by noting that $M'$ and $M''$ are bounded. However, if I'm not terribly wrong, $M'$ and $M''$ are unbounded. And even if they would be bounded, $g'$ and $g''$ (which also occur in the definition of $T$) should be unbounded ... So, I don't see what we need here.










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$endgroup$

















    0












    $begingroup$


    Let





    • $dinmathbb N$ with $d>1$


    • $lambda^d$ denote the Lebesuge measure on $mathcal Bleft(mathbb R^dright)$


    • $fin C^2(mathbb R)$ be positive and $$pi(x):=prod_{i=1}^df(x_i);;;text{for }xinmathbb R^d$$


    • $Phi$ denote the cdf of the standard normal distribution


    • $ell>0$ and $$sigma^2:=frac{ell^2}{d-1}$$

    • $xinmathbb R^d$


    • $(Omega,mathcal A,operatorname P)$ be a probability space


    • $Y$ be a $mathbb R^d$-valued random variable on $(Omega,mathcal A,operatorname P)$ with $Ysimmathcal N(0,sigma I)$ (where $I$ denotes the $d$-dimensional identity matrix)


    Now, let $$g(z):=frac1{d-1}sum_{i=2}^dleft|frac{f'(z_i)}{f(z_i)}right|^2;;;text{for }zinmathbb R^d,$$ $varepsilon:=g-g(x_1)$ and $$M(z):=Phileft(frac1{{g(x)}^{1/2}}left(frac zell-frac{ell g(x)}2right)right)+e^zPhileft(-frac z{ell{{g(x)}^{1/2}}}-frac{ell{g(x)}^{1/2}}2right);;;text{for }zinmathbb R.$$ Note that there is a real-valued random variable $W$ on $(Omega,mathcal A,operatorname P)$ with $$W(omega)inleft{(1-t)x_1+tY_1(omega):tin[0,1]right}tag1$$ and $$M(varepsilon(Y_1))=M(0)+frac12M(0)g'(x_1)(Y_1-x_1)+frac12left(T(W)right)(Y_1-x_1)^2tag2,$$ where $$T(w):=g''(w)M'(varepsilon(w))+{g'(w)}^2M''(varepsilon(w));;;text{for }winmathbb R.$$




    How can we show that $T(W)$ is uniformly bounded with respect to $x$, if the domain of $x$ is restricted to a compact set?




    Most probably, the question is highly overcomplicated. I've read that the claim follows by noting that $M'$ and $M''$ are bounded. However, if I'm not terribly wrong, $M'$ and $M''$ are unbounded. And even if they would be bounded, $g'$ and $g''$ (which also occur in the definition of $T$) should be unbounded ... So, I don't see what we need here.










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let





      • $dinmathbb N$ with $d>1$


      • $lambda^d$ denote the Lebesuge measure on $mathcal Bleft(mathbb R^dright)$


      • $fin C^2(mathbb R)$ be positive and $$pi(x):=prod_{i=1}^df(x_i);;;text{for }xinmathbb R^d$$


      • $Phi$ denote the cdf of the standard normal distribution


      • $ell>0$ and $$sigma^2:=frac{ell^2}{d-1}$$

      • $xinmathbb R^d$


      • $(Omega,mathcal A,operatorname P)$ be a probability space


      • $Y$ be a $mathbb R^d$-valued random variable on $(Omega,mathcal A,operatorname P)$ with $Ysimmathcal N(0,sigma I)$ (where $I$ denotes the $d$-dimensional identity matrix)


      Now, let $$g(z):=frac1{d-1}sum_{i=2}^dleft|frac{f'(z_i)}{f(z_i)}right|^2;;;text{for }zinmathbb R^d,$$ $varepsilon:=g-g(x_1)$ and $$M(z):=Phileft(frac1{{g(x)}^{1/2}}left(frac zell-frac{ell g(x)}2right)right)+e^zPhileft(-frac z{ell{{g(x)}^{1/2}}}-frac{ell{g(x)}^{1/2}}2right);;;text{for }zinmathbb R.$$ Note that there is a real-valued random variable $W$ on $(Omega,mathcal A,operatorname P)$ with $$W(omega)inleft{(1-t)x_1+tY_1(omega):tin[0,1]right}tag1$$ and $$M(varepsilon(Y_1))=M(0)+frac12M(0)g'(x_1)(Y_1-x_1)+frac12left(T(W)right)(Y_1-x_1)^2tag2,$$ where $$T(w):=g''(w)M'(varepsilon(w))+{g'(w)}^2M''(varepsilon(w));;;text{for }winmathbb R.$$




      How can we show that $T(W)$ is uniformly bounded with respect to $x$, if the domain of $x$ is restricted to a compact set?




      Most probably, the question is highly overcomplicated. I've read that the claim follows by noting that $M'$ and $M''$ are bounded. However, if I'm not terribly wrong, $M'$ and $M''$ are unbounded. And even if they would be bounded, $g'$ and $g''$ (which also occur in the definition of $T$) should be unbounded ... So, I don't see what we need here.










      share|cite|improve this question









      $endgroup$




      Let





      • $dinmathbb N$ with $d>1$


      • $lambda^d$ denote the Lebesuge measure on $mathcal Bleft(mathbb R^dright)$


      • $fin C^2(mathbb R)$ be positive and $$pi(x):=prod_{i=1}^df(x_i);;;text{for }xinmathbb R^d$$


      • $Phi$ denote the cdf of the standard normal distribution


      • $ell>0$ and $$sigma^2:=frac{ell^2}{d-1}$$

      • $xinmathbb R^d$


      • $(Omega,mathcal A,operatorname P)$ be a probability space


      • $Y$ be a $mathbb R^d$-valued random variable on $(Omega,mathcal A,operatorname P)$ with $Ysimmathcal N(0,sigma I)$ (where $I$ denotes the $d$-dimensional identity matrix)


      Now, let $$g(z):=frac1{d-1}sum_{i=2}^dleft|frac{f'(z_i)}{f(z_i)}right|^2;;;text{for }zinmathbb R^d,$$ $varepsilon:=g-g(x_1)$ and $$M(z):=Phileft(frac1{{g(x)}^{1/2}}left(frac zell-frac{ell g(x)}2right)right)+e^zPhileft(-frac z{ell{{g(x)}^{1/2}}}-frac{ell{g(x)}^{1/2}}2right);;;text{for }zinmathbb R.$$ Note that there is a real-valued random variable $W$ on $(Omega,mathcal A,operatorname P)$ with $$W(omega)inleft{(1-t)x_1+tY_1(omega):tin[0,1]right}tag1$$ and $$M(varepsilon(Y_1))=M(0)+frac12M(0)g'(x_1)(Y_1-x_1)+frac12left(T(W)right)(Y_1-x_1)^2tag2,$$ where $$T(w):=g''(w)M'(varepsilon(w))+{g'(w)}^2M''(varepsilon(w));;;text{for }winmathbb R.$$




      How can we show that $T(W)$ is uniformly bounded with respect to $x$, if the domain of $x$ is restricted to a compact set?




      Most probably, the question is highly overcomplicated. I've read that the claim follows by noting that $M'$ and $M''$ are bounded. However, if I'm not terribly wrong, $M'$ and $M''$ are unbounded. And even if they would be bounded, $g'$ and $g''$ (which also occur in the definition of $T$) should be unbounded ... So, I don't see what we need here.







      probability-theory normal-distribution






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      asked Dec 11 '18 at 15:08









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