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Limit of characteristic functions
Martingale that converges almost surely to $-infty$.Help with a random walk problem.How to show the following characteristic function is positive definiteShow $Pxi_1+dots+xi_n=1=(sum_i=1^n lambda_i)Delta + O(Delta^2)$ (Shiryaev's Probability page 44)Functions of a random walk and martingalesConsistency in the definition of cross cumulantsLevy's theorem for characteristic functions.Is the running maximum of random walk a martingale?Approximating a multinomial as $p(xi_1,ldots,xi_N)proptoexpleft(-fracn2sum_i=1^Nfrac(xi_i-p_i)^2p_iright)$Proving Chebyshev inequality
$begingroup$
Let $xi_1 ... , xi_n$ be iid with $E xi_i^2 < infty $
what is $$ lim _nrightarrow infty varphi_barxi$$
where $varphi$ is the caracteristic function and $barxi$ the mean of all the $xi_i$.
I don't know how to solve this, can someone help me please.
solutioin (?): $varphi_barxi(t) = varphi_xi(t/n)^n = left[ 1 + i mu fractn + O(fractn)right]^n = e^itmu$. By taylor expansion where $mu$ is the mean of $xi_i$.
Have I missed something?
probability-theory characteristic-functions probability-limit-theorems
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
asked Jun 14 '13 at 12:18
MartaMarta
62
$endgroup$
add a comment |
$begingroup$
Let $xi_1 ... , xi_n$ be iid with $E xi_i^2 < infty $
what is $$ lim _nrightarrow infty varphi_barxi$$
where $varphi$ is the caracteristic function and $barxi$ the mean of all the $xi_i$.
I don't know how to solve this, can someone help me please.
solutioin (?): $varphi_barxi(t) = varphi_xi(t/n)^n = left[ 1 + i mu fractn + O(fractn)right]^n = e^itmu$. By taylor expansion where $mu$ is the mean of $xi_i$.
Have I missed something?
probability-theory characteristic-functions probability-limit-theorems
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
asked Jun 14 '13 at 12:18
MartaMarta
62
$endgroup$
$begingroup$
Do you know the central limit theorem?
$endgroup$
– Nate Eldredge
Jun 14 '13 at 13:20
add a comment |
$begingroup$
Let $xi_1 ... , xi_n$ be iid with $E xi_i^2 < infty $
what is $$ lim _nrightarrow infty varphi_barxi$$
where $varphi$ is the caracteristic function and $barxi$ the mean of all the $xi_i$.
I don't know how to solve this, can someone help me please.
solutioin (?): $varphi_barxi(t) = varphi_xi(t/n)^n = left[ 1 + i mu fractn + O(fractn)right]^n = e^itmu$. By taylor expansion where $mu$ is the mean of $xi_i$.
Have I missed something?
probability-theory characteristic-functions probability-limit-theorems
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
asked Jun 14 '13 at 12:18
MartaMarta
62
$endgroup$
Let $xi_1 ... , xi_n$ be iid with $E xi_i^2 < infty $
what is $$ lim _nrightarrow infty varphi_barxi$$
where $varphi$ is the caracteristic function and $barxi$ the mean of all the $xi_i$.
I don't know how to solve this, can someone help me please.
solutioin (?): $varphi_barxi(t) = varphi_xi(t/n)^n = left[ 1 + i mu fractn + O(fractn)right]^n = e^itmu$. By taylor expansion where $mu$ is the mean of $xi_i$.
Have I missed something?
probability-theory characteristic-functions probability-limit-theorems
probability-theory characteristic-functions probability-limit-theorems
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
asked Jun 14 '13 at 12:18
MartaMarta
62
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
asked Jun 14 '13 at 12:18
MartaMarta
62
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
edited Jun 14 '13 at 21:39
Davide Giraudo
128k17156268
128k17156268
asked Jun 14 '13 at 12:18
MartaMarta
62
asked Jun 14 '13 at 12:18
MartaMarta
62
asked Jun 14 '13 at 12:18
MartaMarta
62
62
$begingroup$
Do you know the central limit theorem?
$endgroup$
– Nate Eldredge
Jun 14 '13 at 13:20
add a comment |
$begingroup$
Do you know the central limit theorem?
$endgroup$
– Nate Eldredge
Jun 14 '13 at 13:20
$begingroup$
Do you know the central limit theorem?
$endgroup$
– Nate Eldredge
Jun 14 '13 at 13:20
$begingroup$
Do you know the central limit theorem?
$endgroup$
– Nate Eldredge
Jun 14 '13 at 13:20
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, you didn't miss anything. What you proved is that $barxi_n$ converges in law to the constant $mu$. We can deduce from that that there is convergence in probability to this constant, which is nothing but weak law of large numbers.
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
votes
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votes
$begingroup$
No, you didn't miss anything. What you proved is that $barxi_n$ converges in law to the constant $mu$. We can deduce from that that there is convergence in probability to this constant, which is nothing but weak law of large numbers.
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
$endgroup$
add a comment |
$begingroup$
No, you didn't miss anything. What you proved is that $barxi_n$ converges in law to the constant $mu$. We can deduce from that that there is convergence in probability to this constant, which is nothing but weak law of large numbers.
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
$endgroup$
add a comment |
$begingroup$
No, you didn't miss anything. What you proved is that $barxi_n$ converges in law to the constant $mu$. We can deduce from that that there is convergence in probability to this constant, which is nothing but weak law of large numbers.
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
$endgroup$
No, you didn't miss anything. What you proved is that $barxi_n$ converges in law to the constant $mu$. We can deduce from that that there is convergence in probability to this constant, which is nothing but weak law of large numbers.
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
answered Jun 14 '13 at 21:38
Davide GiraudoDavide Giraudo
128k17156268
128k17156268
add a comment |
add a comment |
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$begingroup$
Do you know the central limit theorem?
$endgroup$
– Nate Eldredge
Jun 14 '13 at 13:20