Weak law of large numbers for reciprocal of normal
In two different journal articles:
The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense
and
The first negative moment in the sense of the Cauchy principal value
the concept of the principal valued first moment is presented in the case where the first moment of a r.v. does not exist. The principal valued first moment (a.k.a. principal valued first negative moment FNM) for a continuous r.v. is defined by
begin{equation}
PV!E(Y^{-1}) = lim_{epsilonto0^+}left(int_{-infty}^{-epsilon}+int_{epsilon}^inftyright)frac{1}{y}f_Y(y),mathrm dy.
end{equation}
The author makes the following assertion in both articles:
Hence, through different viewpoints
of the integral of FNM, the concept of the (Cauchy) principal
value, widely used in the probability theory such as the
weak law of large numbers [11], can be used to avoid the
nonexistence of the FNM in the usual sense.
The same citation is used both times which is Feller's An Introduction to Probability Theory and its Applications vol. 2.
My question is simple: How is the FNM used in the weak law of large numbers? Is there a version of the WLLN for r.v's without moments that makes use of the FNM?
I found the following generalization of the WLLN in Feller's book (VII.7) for sequences of i.i.d. r.v's
begin{equation}
frac{S_n-nmathrm EXI{|X|leq n}}{n}overset{p}{to}0 text{as} ntoinftyiff x,mathsf P(|X|>x)to 0 text{as} xtoinfty.
end{equation}
I checked this condition for the reciprocal normal, i.e. $X=Y^{-1}$ for $Ysimmathcal N(mu,sigma^2)$ and found that
begin{equation}
lim_{xtoinfty}x,mathsf P(|X|>x)=2f_Y(0)neq 0.
end{equation}
So it would seem that Feller's WLLN does not apply. Can someone help me understand the claim made in the papers? Where is the FNM used in the WLLN?
probability-theory law-of-large-numbers cauchy-principal-value
asked Nov 28 at 17:46
Aaron Hendrickson
563410
add a comment |
In two different journal articles:
The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense
and
The first negative moment in the sense of the Cauchy principal value
the concept of the principal valued first moment is presented in the case where the first moment of a r.v. does not exist. The principal valued first moment (a.k.a. principal valued first negative moment FNM) for a continuous r.v. is defined by
begin{equation}
PV!E(Y^{-1}) = lim_{epsilonto0^+}left(int_{-infty}^{-epsilon}+int_{epsilon}^inftyright)frac{1}{y}f_Y(y),mathrm dy.
end{equation}
The author makes the following assertion in both articles:
Hence, through different viewpoints
of the integral of FNM, the concept of the (Cauchy) principal
value, widely used in the probability theory such as the
weak law of large numbers [11], can be used to avoid the
nonexistence of the FNM in the usual sense.
The same citation is used both times which is Feller's An Introduction to Probability Theory and its Applications vol. 2.
My question is simple: How is the FNM used in the weak law of large numbers? Is there a version of the WLLN for r.v's without moments that makes use of the FNM?
I found the following generalization of the WLLN in Feller's book (VII.7) for sequences of i.i.d. r.v's
begin{equation}
frac{S_n-nmathrm EXI{|X|leq n}}{n}overset{p}{to}0 text{as} ntoinftyiff x,mathsf P(|X|>x)to 0 text{as} xtoinfty.
end{equation}
I checked this condition for the reciprocal normal, i.e. $X=Y^{-1}$ for $Ysimmathcal N(mu,sigma^2)$ and found that
begin{equation}
lim_{xtoinfty}x,mathsf P(|X|>x)=2f_Y(0)neq 0.
end{equation}
So it would seem that Feller's WLLN does not apply. Can someone help me understand the claim made in the papers? Where is the FNM used in the WLLN?
probability-theory law-of-large-numbers cauchy-principal-value
asked Nov 28 at 17:46
Aaron Hendrickson
563410
add a comment |
In two different journal articles:
The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense
and
The first negative moment in the sense of the Cauchy principal value
the concept of the principal valued first moment is presented in the case where the first moment of a r.v. does not exist. The principal valued first moment (a.k.a. principal valued first negative moment FNM) for a continuous r.v. is defined by
begin{equation}
PV!E(Y^{-1}) = lim_{epsilonto0^+}left(int_{-infty}^{-epsilon}+int_{epsilon}^inftyright)frac{1}{y}f_Y(y),mathrm dy.
end{equation}
The author makes the following assertion in both articles:
Hence, through different viewpoints
of the integral of FNM, the concept of the (Cauchy) principal
value, widely used in the probability theory such as the
weak law of large numbers [11], can be used to avoid the
nonexistence of the FNM in the usual sense.
The same citation is used both times which is Feller's An Introduction to Probability Theory and its Applications vol. 2.
My question is simple: How is the FNM used in the weak law of large numbers? Is there a version of the WLLN for r.v's without moments that makes use of the FNM?
I found the following generalization of the WLLN in Feller's book (VII.7) for sequences of i.i.d. r.v's
begin{equation}
frac{S_n-nmathrm EXI{|X|leq n}}{n}overset{p}{to}0 text{as} ntoinftyiff x,mathsf P(|X|>x)to 0 text{as} xtoinfty.
end{equation}
I checked this condition for the reciprocal normal, i.e. $X=Y^{-1}$ for $Ysimmathcal N(mu,sigma^2)$ and found that
begin{equation}
lim_{xtoinfty}x,mathsf P(|X|>x)=2f_Y(0)neq 0.
end{equation}
So it would seem that Feller's WLLN does not apply. Can someone help me understand the claim made in the papers? Where is the FNM used in the WLLN?
probability-theory law-of-large-numbers cauchy-principal-value
asked Nov 28 at 17:46
Aaron Hendrickson
563410
In two different journal articles:
The First Negative Moment of Skew-t and Generalized Student's t-Distributions in the Principal Value Sense
and
The first negative moment in the sense of the Cauchy principal value
the concept of the principal valued first moment is presented in the case where the first moment of a r.v. does not exist. The principal valued first moment (a.k.a. principal valued first negative moment FNM) for a continuous r.v. is defined by
begin{equation}
PV!E(Y^{-1}) = lim_{epsilonto0^+}left(int_{-infty}^{-epsilon}+int_{epsilon}^inftyright)frac{1}{y}f_Y(y),mathrm dy.
end{equation}
The author makes the following assertion in both articles:
Hence, through different viewpoints
of the integral of FNM, the concept of the (Cauchy) principal
value, widely used in the probability theory such as the
weak law of large numbers [11], can be used to avoid the
nonexistence of the FNM in the usual sense.
The same citation is used both times which is Feller's An Introduction to Probability Theory and its Applications vol. 2.
My question is simple: How is the FNM used in the weak law of large numbers? Is there a version of the WLLN for r.v's without moments that makes use of the FNM?
I found the following generalization of the WLLN in Feller's book (VII.7) for sequences of i.i.d. r.v's
begin{equation}
frac{S_n-nmathrm EXI{|X|leq n}}{n}overset{p}{to}0 text{as} ntoinftyiff x,mathsf P(|X|>x)to 0 text{as} xtoinfty.
end{equation}
I checked this condition for the reciprocal normal, i.e. $X=Y^{-1}$ for $Ysimmathcal N(mu,sigma^2)$ and found that
begin{equation}
lim_{xtoinfty}x,mathsf P(|X|>x)=2f_Y(0)neq 0.
end{equation}
So it would seem that Feller's WLLN does not apply. Can someone help me understand the claim made in the papers? Where is the FNM used in the WLLN?
probability-theory law-of-large-numbers cauchy-principal-value
probability-theory law-of-large-numbers cauchy-principal-value
asked Nov 28 at 17:46
Aaron Hendrickson
563410
asked Nov 28 at 17:46
Aaron Hendrickson
563410
edited Nov 28 at 17:51
asked Nov 28 at 17:46
Aaron Hendrickson
563410
asked Nov 28 at 17:46
Aaron Hendrickson
563410
asked Nov 28 at 17:46
Aaron Hendrickson
563410
563410
add a comment |
add a comment |
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