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Fix typo in Weibull CDF
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Bullmoose
Bullmoose
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Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)$, pointwise convergence $\varphi_n(t)\rightarrow\varphi(t)$ implies convergence in distribution $X_n\xrightarrow{D}X$.

Are any converse results known for this?

Why?

I am interested in the moments of the maximum $M_n=\max (X_1,\ldots,X_n)$ where $X_1,\ldots,X_n$ are i.i.d. and have finite support $[x_l,x_u]$. Thus the distribution of $X_i$ belongs to the maximum domain of attraction of the Weibull distribution, meaning that, for appropriately chosen sequence $c_n$, $c_n^{-1}(M_n-x_u)\xrightarrow{D}W_\alpha$ where $W_\alpha$ is Weibull random variable with distribution function $\Psi_\alpha(x)=\exp[-(-x)^\alpha]$$\Psi_\alpha(x)=1-\exp[-(-x)^\alpha]$, $\alpha>0$. A some kind of a converse to Levy continuity theorem might allow me to use the Weibull characteristic function to obtain the desired moments. However, I would appreciate other suggestions -- perhaps there are more elegant ways to get there.

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)$, pointwise convergence $\varphi_n(t)\rightarrow\varphi(t)$ implies convergence in distribution $X_n\xrightarrow{D}X$.

Are any converse results known for this?

Why?

I am interested in the moments of the maximum $M_n=\max (X_1,\ldots,X_n)$ where $X_1,\ldots,X_n$ are i.i.d. and have finite support $[x_l,x_u]$. Thus the distribution of $X_i$ belongs to the maximum domain of attraction of the Weibull distribution, meaning that, for appropriately chosen sequence $c_n$, $c_n^{-1}(M_n-x_u)\xrightarrow{D}W_\alpha$ where $W_\alpha$ is Weibull random variable with distribution function $\Psi_\alpha(x)=\exp[-(-x)^\alpha]$, $\alpha>0$. A some kind of a converse to Levy continuity theorem might allow me to use the Weibull characteristic function to obtain the desired moments. However, I would appreciate other suggestions -- perhaps there are more elegant ways to get there.

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)$, pointwise convergence $\varphi_n(t)\rightarrow\varphi(t)$ implies convergence in distribution $X_n\xrightarrow{D}X$.

Are any converse results known for this?

Why?

I am interested in the moments of the maximum $M_n=\max (X_1,\ldots,X_n)$ where $X_1,\ldots,X_n$ are i.i.d. and have finite support $[x_l,x_u]$. Thus the distribution of $X_i$ belongs to the maximum domain of attraction of the Weibull distribution, meaning that, for appropriately chosen sequence $c_n$, $c_n^{-1}(M_n-x_u)\xrightarrow{D}W_\alpha$ where $W_\alpha$ is Weibull random variable with distribution function $\Psi_\alpha(x)=1-\exp[-(-x)^\alpha]$, $\alpha>0$. A some kind of a converse to Levy continuity theorem might allow me to use the Weibull characteristic function to obtain the desired moments. However, I would appreciate other suggestions -- perhaps there are more elegant ways to get there.

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Bullmoose
Bullmoose
  • 917
  • 6
  • 16

Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)$, pointwise convergence $\varphi_n(t)\rightarrow\varphi(t)$ implies convergence in distribution $X_n\xrightarrow{D}X$.

Are any converse results known for this?

Why?

I am interested in the moments of the maximum $M_n=\max (X_1,\ldots,X_n)$ where $X_1,\ldots,X_n$ are i.i.d. and have finite support $[x_l,x_u]$. Thus the distribution of $X_i$ belongs to the maximum domain of attraction of the Weibull distribution, meaning that, for appropriately chosen sequence $c_n$, $c_n^{-1}(M_n-x_u)\xrightarrow{D}W_\alpha$ where $W_\alpha$ is Weibull random variable with distribution function $\Psi_\alpha(x)=\exp[-(-x)^\alpha]$, $\alpha>0$. A some kind of a converse to Levy continuity theorem might allow me to use the Weibull characteristic function to obtain the desired moments. However, I would appreciate other suggestions -- perhaps there are more elegant ways to get there.