Mixing
$X_n$ ~ $Gamma(n,n)$, find the limit in Law of $X_n$
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I am investigating the following idea.
Let $X_n$ ~ $Gamma(n,n)$. I want to find the limit in Law of this random variable.
I tried using Paul Levy theorem that says the following: If I find the limit of the characteristic function equal to some function $theta(t)$ continuous in 0, then there exists a random variable X such as $X_n to X$ with $theta$ as its characteristic function.
Following this idea I tried calculating the following limit:
$$ lim Big(frac{1}{1-int}Big)^n$$
However to me this goes to 0. But that is not possible because such a characteristic function cannot exist. Is this the right approach?
probability limits characteristic-functions
asked Jan 5 at 22:34
qcc101qcc101
548113
$endgroup$
add a comment |
$begingroup$
I am investigating the following idea.
Let $X_n$ ~ $Gamma(n,n)$. I want to find the limit in Law of this random variable.
I tried using Paul Levy theorem that says the following: If I find the limit of the characteristic function equal to some function $theta(t)$ continuous in 0, then there exists a random variable X such as $X_n to X$ with $theta$ as its characteristic function.
Following this idea I tried calculating the following limit:
$$ lim Big(frac{1}{1-int}Big)^n$$
However to me this goes to 0. But that is not possible because such a characteristic function cannot exist. Is this the right approach?
probability limits characteristic-functions
asked Jan 5 at 22:34
qcc101qcc101
548113
$endgroup$
2
$begingroup$
There is another parameterization of $Gamma$ distribution for which $$ varphi_n=(1-it/n)^nto e^{-it}. $$
$endgroup$
– d.k.o.
Jan 5 at 23:36
1
$begingroup$
Indeed: scale $n$ or $1/n$?
$endgroup$
– Did
Jan 6 at 0:34
$begingroup$
@Did What do you mean by scale? Anyway, such a characteristic function is only for the X = 1 constant random variable, am I right?
$endgroup$
– qcc101
Jan 6 at 7:19
$begingroup$
"What do you mean by scale?" One of the parameters of every gamma distribution. Please se the WP page, if refreshing your memories is needed.
$endgroup$
– Did
Jan 6 at 9:42
$begingroup$
I see. I used the following: $ Big( 1 - frac{it}{n} Big)^{-n}$, but I get $e^{it}$
$endgroup$
– qcc101
Jan 6 at 11:30
add a comment |
$begingroup$
I am investigating the following idea.
Let $X_n$ ~ $Gamma(n,n)$. I want to find the limit in Law of this random variable.
I tried using Paul Levy theorem that says the following: If I find the limit of the characteristic function equal to some function $theta(t)$ continuous in 0, then there exists a random variable X such as $X_n to X$ with $theta$ as its characteristic function.
Following this idea I tried calculating the following limit:
$$ lim Big(frac{1}{1-int}Big)^n$$
However to me this goes to 0. But that is not possible because such a characteristic function cannot exist. Is this the right approach?
probability limits characteristic-functions
asked Jan 5 at 22:34
qcc101qcc101
548113
$endgroup$
I am investigating the following idea.
Let $X_n$ ~ $Gamma(n,n)$. I want to find the limit in Law of this random variable.
I tried using Paul Levy theorem that says the following: If I find the limit of the characteristic function equal to some function $theta(t)$ continuous in 0, then there exists a random variable X such as $X_n to X$ with $theta$ as its characteristic function.
Following this idea I tried calculating the following limit:
$$ lim Big(frac{1}{1-int}Big)^n$$
However to me this goes to 0. But that is not possible because such a characteristic function cannot exist. Is this the right approach?
probability limits characteristic-functions
probability limits characteristic-functions
asked Jan 5 at 22:34
qcc101qcc101
548113
asked Jan 5 at 22:34
qcc101qcc101
548113
asked Jan 5 at 22:34
qcc101qcc101
548113
asked Jan 5 at 22:34
qcc101qcc101
548113
asked Jan 5 at 22:34
qcc101qcc101
548113
548113
2
$begingroup$
There is another parameterization of $Gamma$ distribution for which $$ varphi_n=(1-it/n)^nto e^{-it}. $$
$endgroup$
– d.k.o.
Jan 5 at 23:36
1
$begingroup$
Indeed: scale $n$ or $1/n$?
$endgroup$
– Did
Jan 6 at 0:34
$begingroup$
@Did What do you mean by scale? Anyway, such a characteristic function is only for the X = 1 constant random variable, am I right?
$endgroup$
– qcc101
Jan 6 at 7:19
$begingroup$
"What do you mean by scale?" One of the parameters of every gamma distribution. Please se the WP page, if refreshing your memories is needed.
$endgroup$
– Did
Jan 6 at 9:42
$begingroup$
I see. I used the following: $ Big( 1 - frac{it}{n} Big)^{-n}$, but I get $e^{it}$
$endgroup$
– qcc101
Jan 6 at 11:30
add a comment |
2
$begingroup$
There is another parameterization of $Gamma$ distribution for which $$ varphi_n=(1-it/n)^nto e^{-it}. $$
$endgroup$
– d.k.o.
Jan 5 at 23:36
1
$begingroup$
Indeed: scale $n$ or $1/n$?
$endgroup$
– Did
Jan 6 at 0:34
$begingroup$
@Did What do you mean by scale? Anyway, such a characteristic function is only for the X = 1 constant random variable, am I right?
$endgroup$
– qcc101
Jan 6 at 7:19
$begingroup$
"What do you mean by scale?" One of the parameters of every gamma distribution. Please se the WP page, if refreshing your memories is needed.
$endgroup$
– Did
Jan 6 at 9:42
$begingroup$
I see. I used the following: $ Big( 1 - frac{it}{n} Big)^{-n}$, but I get $e^{it}$
$endgroup$
– qcc101
Jan 6 at 11:30
2
2
$begingroup$
There is another parameterization of $Gamma$ distribution for which $$ varphi_n=(1-it/n)^nto e^{-it}. $$
$endgroup$
– d.k.o.
Jan 5 at 23:36
$begingroup$
There is another parameterization of $Gamma$ distribution for which $$ varphi_n=(1-it/n)^nto e^{-it}. $$
$endgroup$
– d.k.o.
Jan 5 at 23:36
1
1
$begingroup$
Indeed: scale $n$ or $1/n$?
$endgroup$
– Did
Jan 6 at 0:34
$begingroup$
Indeed: scale $n$ or $1/n$?
$endgroup$
– Did
Jan 6 at 0:34
$begingroup$
@Did What do you mean by scale? Anyway, such a characteristic function is only for the X = 1 constant random variable, am I right?
$endgroup$
– qcc101
Jan 6 at 7:19
$begingroup$
@Did What do you mean by scale? Anyway, such a characteristic function is only for the X = 1 constant random variable, am I right?
$endgroup$
– qcc101
Jan 6 at 7:19
$begingroup$
"What do you mean by scale?" One of the parameters of every gamma distribution. Please se the WP page, if refreshing your memories is needed.
$endgroup$
– Did
Jan 6 at 9:42
$begingroup$
"What do you mean by scale?" One of the parameters of every gamma distribution. Please se the WP page, if refreshing your memories is needed.
$endgroup$
– Did
Jan 6 at 9:42
$begingroup$
I see. I used the following: $ Big( 1 - frac{it}{n} Big)^{-n}$, but I get $e^{it}$
$endgroup$
– qcc101
Jan 6 at 11:30
$begingroup$
I see. I used the following: $ Big( 1 - frac{it}{n} Big)^{-n}$, but I get $e^{it}$
$endgroup$
– qcc101
Jan 6 at 11:30
add a comment |
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$begingroup$
There is another parameterization of $Gamma$ distribution for which $$ varphi_n=(1-it/n)^nto e^{-it}. $$
$endgroup$
– d.k.o.
Jan 5 at 23:36
1
$begingroup$
Indeed: scale $n$ or $1/n$?
$endgroup$
– Did
Jan 6 at 0:34
$begingroup$
@Did What do you mean by scale? Anyway, such a characteristic function is only for the X = 1 constant random variable, am I right?
$endgroup$
– qcc101
Jan 6 at 7:19
$begingroup$
"What do you mean by scale?" One of the parameters of every gamma distribution. Please se the WP page, if refreshing your memories is needed.
$endgroup$
– Did
Jan 6 at 9:42
$begingroup$
I see. I used the following: $ Big( 1 - frac{it}{n} Big)^{-n}$, but I get $e^{it}$
$endgroup$
– qcc101
Jan 6 at 11:30