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General Central Limit Theorem for Binomial Random Variables
$begingroup$
Question
Let $(X_n)_{ngeq 1}$ be a sequence of arbitrary binomial random variables such that $EX_nto infty$ and $text{Var}(X_n)/EX_n^2to 0$ as $nto infty$. Then
show that
$$
Z_n=frac{X_n-EX_n}{sqrt{text{Var}(X_n)}}stackrel{text{d}}{to } N(0,1).
$$
My attempt
The idea of the proof is to find the characteristic function $varphi_{n}$
of $Z_n$ and show that $varphi_{n}(t)to exp(-t^2/2)$.
To this end, let $X_nsim text{Bin}(m_n,p_n)$. Since $varphi_{X_n}(t)=(1-p_n+p_ne^{it})^{m_n}$ we have that
$$
varphi_{n}(t)=varphi_{Z_n}(t)=expleft(
frac{-itm_np_n}{sqrt{m_np_n(1-p_n)}}
right)
left(1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)right)^{m_n}.
$$
At this point I tried to taylor expand the exponential terms, mimic the proof of classical clt and use the fact that if $c_jto 0$, $a_jto infty$, $a_jc_jto lambda$, then $(1+c_j)^{a_j}to e^{lambda}$ as $jto infty$. Doing so yields for example that
$$
1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)=1+p_nfrac{it}{sqrt{m_np_n(1-p_n)}}-p_nfrac{t^2}{2(m_np_n(1-p_n))}+dotsb
$$
and
$$
expleft(-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}right)=1-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}-frac{t^2m_n^2p_n^2}{2m_np_n(1-p_n)}+dotsb.
$$
In the first of these we have something that resembles the term $-t^2/2$ but I can't proceed from here.
Any help is appreciated.
probability probability-theory central-limit-theorem characteristic-functions
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
$endgroup$
add a comment |
$begingroup$
Question
Let $(X_n)_{ngeq 1}$ be a sequence of arbitrary binomial random variables such that $EX_nto infty$ and $text{Var}(X_n)/EX_n^2to 0$ as $nto infty$. Then
show that
$$
Z_n=frac{X_n-EX_n}{sqrt{text{Var}(X_n)}}stackrel{text{d}}{to } N(0,1).
$$
My attempt
The idea of the proof is to find the characteristic function $varphi_{n}$
of $Z_n$ and show that $varphi_{n}(t)to exp(-t^2/2)$.
To this end, let $X_nsim text{Bin}(m_n,p_n)$. Since $varphi_{X_n}(t)=(1-p_n+p_ne^{it})^{m_n}$ we have that
$$
varphi_{n}(t)=varphi_{Z_n}(t)=expleft(
frac{-itm_np_n}{sqrt{m_np_n(1-p_n)}}
right)
left(1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)right)^{m_n}.
$$
At this point I tried to taylor expand the exponential terms, mimic the proof of classical clt and use the fact that if $c_jto 0$, $a_jto infty$, $a_jc_jto lambda$, then $(1+c_j)^{a_j}to e^{lambda}$ as $jto infty$. Doing so yields for example that
$$
1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)=1+p_nfrac{it}{sqrt{m_np_n(1-p_n)}}-p_nfrac{t^2}{2(m_np_n(1-p_n))}+dotsb
$$
and
$$
expleft(-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}right)=1-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}-frac{t^2m_n^2p_n^2}{2m_np_n(1-p_n)}+dotsb.
$$
In the first of these we have something that resembles the term $-t^2/2$ but I can't proceed from here.
Any help is appreciated.
probability probability-theory central-limit-theorem characteristic-functions
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
$endgroup$
$begingroup$
Unless you are required to use the characteristic function, invoking Lindeberg's condition is probably the most straightforward.
$endgroup$
– user321627
Jan 28 at 1:03
$begingroup$
I need to use characteristic functions to solve the problem. I can't invoke Lindeberg's condition. We are covering cfs in class.
$endgroup$
– Foobaz John
Jan 28 at 1:19
$begingroup$
I can only think of how to prove it using MGF's and the same $n$ and $p$ across the entire sequence.
$endgroup$
– user321627
Jan 28 at 2:06
1
$begingroup$
Try working on $ln varphi_{n}(t)$, then Taylor expanding the logarithm, and then Taylor expanding the exponential that is left.
$endgroup$
– user52227
Jan 28 at 14:33
add a comment |
$begingroup$
Question
Let $(X_n)_{ngeq 1}$ be a sequence of arbitrary binomial random variables such that $EX_nto infty$ and $text{Var}(X_n)/EX_n^2to 0$ as $nto infty$. Then
show that
$$
Z_n=frac{X_n-EX_n}{sqrt{text{Var}(X_n)}}stackrel{text{d}}{to } N(0,1).
$$
My attempt
The idea of the proof is to find the characteristic function $varphi_{n}$
of $Z_n$ and show that $varphi_{n}(t)to exp(-t^2/2)$.
To this end, let $X_nsim text{Bin}(m_n,p_n)$. Since $varphi_{X_n}(t)=(1-p_n+p_ne^{it})^{m_n}$ we have that
$$
varphi_{n}(t)=varphi_{Z_n}(t)=expleft(
frac{-itm_np_n}{sqrt{m_np_n(1-p_n)}}
right)
left(1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)right)^{m_n}.
$$
At this point I tried to taylor expand the exponential terms, mimic the proof of classical clt and use the fact that if $c_jto 0$, $a_jto infty$, $a_jc_jto lambda$, then $(1+c_j)^{a_j}to e^{lambda}$ as $jto infty$. Doing so yields for example that
$$
1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)=1+p_nfrac{it}{sqrt{m_np_n(1-p_n)}}-p_nfrac{t^2}{2(m_np_n(1-p_n))}+dotsb
$$
and
$$
expleft(-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}right)=1-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}-frac{t^2m_n^2p_n^2}{2m_np_n(1-p_n)}+dotsb.
$$
In the first of these we have something that resembles the term $-t^2/2$ but I can't proceed from here.
Any help is appreciated.
probability probability-theory central-limit-theorem characteristic-functions
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
$endgroup$
Question
Let $(X_n)_{ngeq 1}$ be a sequence of arbitrary binomial random variables such that $EX_nto infty$ and $text{Var}(X_n)/EX_n^2to 0$ as $nto infty$. Then
show that
$$
Z_n=frac{X_n-EX_n}{sqrt{text{Var}(X_n)}}stackrel{text{d}}{to } N(0,1).
$$
My attempt
The idea of the proof is to find the characteristic function $varphi_{n}$
of $Z_n$ and show that $varphi_{n}(t)to exp(-t^2/2)$.
To this end, let $X_nsim text{Bin}(m_n,p_n)$. Since $varphi_{X_n}(t)=(1-p_n+p_ne^{it})^{m_n}$ we have that
$$
varphi_{n}(t)=varphi_{Z_n}(t)=expleft(
frac{-itm_np_n}{sqrt{m_np_n(1-p_n)}}
right)
left(1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)right)^{m_n}.
$$
At this point I tried to taylor expand the exponential terms, mimic the proof of classical clt and use the fact that if $c_jto 0$, $a_jto infty$, $a_jc_jto lambda$, then $(1+c_j)^{a_j}to e^{lambda}$ as $jto infty$. Doing so yields for example that
$$
1-p_n+p_nexpleft(frac{it}{sqrt{m_np_n(1-p_n)}}right)=1+p_nfrac{it}{sqrt{m_np_n(1-p_n)}}-p_nfrac{t^2}{2(m_np_n(1-p_n))}+dotsb
$$
and
$$
expleft(-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}right)=1-frac{itm_np_n}{sqrt{m_np_n(1-p_n)}}-frac{t^2m_n^2p_n^2}{2m_np_n(1-p_n)}+dotsb.
$$
In the first of these we have something that resembles the term $-t^2/2$ but I can't proceed from here.
Any help is appreciated.
probability probability-theory central-limit-theorem characteristic-functions
probability probability-theory central-limit-theorem characteristic-functions
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
asked Jan 28 at 0:01
Foobaz JohnFoobaz John
22.8k41452
22.8k41452
$begingroup$
Unless you are required to use the characteristic function, invoking Lindeberg's condition is probably the most straightforward.
$endgroup$
– user321627
Jan 28 at 1:03
$begingroup$
I need to use characteristic functions to solve the problem. I can't invoke Lindeberg's condition. We are covering cfs in class.
$endgroup$
– Foobaz John
Jan 28 at 1:19
$begingroup$
I can only think of how to prove it using MGF's and the same $n$ and $p$ across the entire sequence.
$endgroup$
– user321627
Jan 28 at 2:06
1
$begingroup$
Try working on $ln varphi_{n}(t)$, then Taylor expanding the logarithm, and then Taylor expanding the exponential that is left.
$endgroup$
– user52227
Jan 28 at 14:33
add a comment |
$begingroup$
Unless you are required to use the characteristic function, invoking Lindeberg's condition is probably the most straightforward.
$endgroup$
– user321627
Jan 28 at 1:03
$begingroup$
I need to use characteristic functions to solve the problem. I can't invoke Lindeberg's condition. We are covering cfs in class.
$endgroup$
– Foobaz John
Jan 28 at 1:19
$begingroup$
I can only think of how to prove it using MGF's and the same $n$ and $p$ across the entire sequence.
$endgroup$
– user321627
Jan 28 at 2:06
1
$begingroup$
Try working on $ln varphi_{n}(t)$, then Taylor expanding the logarithm, and then Taylor expanding the exponential that is left.
$endgroup$
– user52227
Jan 28 at 14:33
$begingroup$
Unless you are required to use the characteristic function, invoking Lindeberg's condition is probably the most straightforward.
$endgroup$
– user321627
Jan 28 at 1:03
$begingroup$
Unless you are required to use the characteristic function, invoking Lindeberg's condition is probably the most straightforward.
$endgroup$
– user321627
Jan 28 at 1:03
$begingroup$
I need to use characteristic functions to solve the problem. I can't invoke Lindeberg's condition. We are covering cfs in class.
$endgroup$
– Foobaz John
Jan 28 at 1:19
$begingroup$
I need to use characteristic functions to solve the problem. I can't invoke Lindeberg's condition. We are covering cfs in class.
$endgroup$
– Foobaz John
Jan 28 at 1:19
$begingroup$
I can only think of how to prove it using MGF's and the same $n$ and $p$ across the entire sequence.
$endgroup$
– user321627
Jan 28 at 2:06
$begingroup$
I can only think of how to prove it using MGF's and the same $n$ and $p$ across the entire sequence.
$endgroup$
– user321627
Jan 28 at 2:06
1
1
$begingroup$
Try working on $ln varphi_{n}(t)$, then Taylor expanding the logarithm, and then Taylor expanding the exponential that is left.
$endgroup$
– user52227
Jan 28 at 14:33
$begingroup$
Try working on $ln varphi_{n}(t)$, then Taylor expanding the logarithm, and then Taylor expanding the exponential that is left.
$endgroup$
– user52227
Jan 28 at 14:33
add a comment |
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$begingroup$
Unless you are required to use the characteristic function, invoking Lindeberg's condition is probably the most straightforward.
$endgroup$
– user321627
Jan 28 at 1:03
$begingroup$
I need to use characteristic functions to solve the problem. I can't invoke Lindeberg's condition. We are covering cfs in class.
$endgroup$
– Foobaz John
Jan 28 at 1:19
$begingroup$
I can only think of how to prove it using MGF's and the same $n$ and $p$ across the entire sequence.
$endgroup$
– user321627
Jan 28 at 2:06
1
$begingroup$
Try working on $ln varphi_{n}(t)$, then Taylor expanding the logarithm, and then Taylor expanding the exponential that is left.
$endgroup$
– user52227
Jan 28 at 14:33