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Domain of attraction $F(x)=exp(-x-sin(x))$

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1

$begingroup$

I need to show that $F(x)=exp(-x-sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_k >0, b_k in mathbbR, k in mathbbN$ with $limlimits_k to infty fracU(kx)-b(k)a(k)=D(x)$ where $D(x)=G^leftarrow(e^frac1x),$ $G$ is a nondegenerate distribution function and $U=(frac11-F)^leftarrow$.
As a hint i know that the following holds $$ limlimits_k to infty U(n_kx)-log(n_k)= U_1(x)$$
where $U_1$ is the inverse of $exp(x+sin(x))$ and $n_k=[exp(2pi k)].$
Any approach?

probability-distributions stochastic-analysis extreme-value-theorem

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asked Mar 13 at 7:34

John DoeJohn Doe

183

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add a comment | 

1

$begingroup$

I need to show that $F(x)=exp(-x-sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_k >0, b_k in mathbbR, k in mathbbN$ with $limlimits_k to infty fracU(kx)-b(k)a(k)=D(x)$ where $D(x)=G^leftarrow(e^frac1x),$ $G$ is a nondegenerate distribution function and $U=(frac11-F)^leftarrow$.
As a hint i know that the following holds $$ limlimits_k to infty U(n_kx)-log(n_k)= U_1(x)$$
where $U_1$ is the inverse of $exp(x+sin(x))$ and $n_k=[exp(2pi k)].$
Any approach?

probability-distributions stochastic-analysis extreme-value-theorem

share|cite|improve this question

asked Mar 13 at 7:34

John DoeJohn Doe

183

$endgroup$

add a comment | 

$begingroup$

I need to show that $F(x)=exp(-x-sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_k >0, b_k in mathbbR, k in mathbbN$ with $limlimits_k to infty fracU(kx)-b(k)a(k)=D(x)$ where $D(x)=G^leftarrow(e^frac1x),$ $G$ is a nondegenerate distribution function and $U=(frac11-F)^leftarrow$.
As a hint i know that the following holds $$ limlimits_k to infty U(n_kx)-log(n_k)= U_1(x)$$
where $U_1$ is the inverse of $exp(x+sin(x))$ and $n_k=[exp(2pi k)].$
Any approach?

probability-distributions stochastic-analysis extreme-value-theorem

share|cite|improve this question

asked Mar 13 at 7:34

John DoeJohn Doe

183

$endgroup$

share|cite|improve this question

asked Mar 13 at 7:34

John DoeJohn Doe

183

share|cite|improve this question

asked Mar 13 at 7:34

John DoeJohn Doe

183

asked Mar 13 at 7:34

John DoeJohn Doe

183

asked Mar 13 at 7:34

John DoeJohn Doe

183