Convergence related to normal cdf
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Consider the following problem:
For each $x>0$, let $yinmathbb R$ solve the equation
$$int_y^{y+x}[Phi(z)-r]dz=0,$$
where $rin (0,1)$, and $Phi(cdot)$ is the normal cdf. We can show that for each $x>0$ there is a unique $y$ satisfying the equation, and so this defines a function $y(x)$. Moreover, we can show that
$$lim_{xto+infty}frac{y(x)}{x}=-(1-r).$$
The question is: Can we show that $y(x)+(1-r)xto 0$ as $xto +infty$? In fact, I think we can show that $y(x)+(1-r)x$ is convergent. But I am not sure whether the limit is $0$.
Any hints or suggestions on this will be highly appreciated!
calculus real-analysis probability normal-distribution
asked Nov 20 at 10:12
user146512
536
add a comment |
up vote
0
down vote
favorite
Consider the following problem:
For each $x>0$, let $yinmathbb R$ solve the equation
$$int_y^{y+x}[Phi(z)-r]dz=0,$$
where $rin (0,1)$, and $Phi(cdot)$ is the normal cdf. We can show that for each $x>0$ there is a unique $y$ satisfying the equation, and so this defines a function $y(x)$. Moreover, we can show that
$$lim_{xto+infty}frac{y(x)}{x}=-(1-r).$$
The question is: Can we show that $y(x)+(1-r)xto 0$ as $xto +infty$? In fact, I think we can show that $y(x)+(1-r)x$ is convergent. But I am not sure whether the limit is $0$.
Any hints or suggestions on this will be highly appreciated!
calculus real-analysis probability normal-distribution
asked Nov 20 at 10:12
user146512
536
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the following problem:
For each $x>0$, let $yinmathbb R$ solve the equation
$$int_y^{y+x}[Phi(z)-r]dz=0,$$
where $rin (0,1)$, and $Phi(cdot)$ is the normal cdf. We can show that for each $x>0$ there is a unique $y$ satisfying the equation, and so this defines a function $y(x)$. Moreover, we can show that
$$lim_{xto+infty}frac{y(x)}{x}=-(1-r).$$
The question is: Can we show that $y(x)+(1-r)xto 0$ as $xto +infty$? In fact, I think we can show that $y(x)+(1-r)x$ is convergent. But I am not sure whether the limit is $0$.
Any hints or suggestions on this will be highly appreciated!
calculus real-analysis probability normal-distribution
asked Nov 20 at 10:12
user146512
536
Consider the following problem:
For each $x>0$, let $yinmathbb R$ solve the equation
$$int_y^{y+x}[Phi(z)-r]dz=0,$$
where $rin (0,1)$, and $Phi(cdot)$ is the normal cdf. We can show that for each $x>0$ there is a unique $y$ satisfying the equation, and so this defines a function $y(x)$. Moreover, we can show that
$$lim_{xto+infty}frac{y(x)}{x}=-(1-r).$$
The question is: Can we show that $y(x)+(1-r)xto 0$ as $xto +infty$? In fact, I think we can show that $y(x)+(1-r)x$ is convergent. But I am not sure whether the limit is $0$.
Any hints or suggestions on this will be highly appreciated!
calculus real-analysis probability normal-distribution
calculus real-analysis probability normal-distribution
asked Nov 20 at 10:12
user146512
536
asked Nov 20 at 10:12
user146512
536
asked Nov 20 at 10:12
user146512
536
asked Nov 20 at 10:12
user146512
536
asked Nov 20 at 10:12
user146512
536
536
add a comment |
add a comment |
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