Large-n limit of the distribution of the normalized sum of Cauchy random variablesLimit of a rescaled random sum of i.i.d. random variablesNon-normality of limit of random variablesOn the sum of uniform independent random variablesSum of independent random variablesHow to obtain the probability distribution of a sum of dependent discrete random variables more efficientlyRate of convergence of a test statistic towards a Gaussian random variableCalculating the expectation of a sum of dependent random variablesVariance of sum of $m$ dependent random variablesSum of random variables are equal in distributionA lower bound on the expected sum of Bernoulli random variables given a constraint on its distribution
Large-n limit of the distribution of the normalized sum of Cauchy random variables
Limit of a rescaled random sum of i.i.d. random variablesNon-normality of limit of random variablesOn the sum of uniform independent random variablesSum of independent random variablesHow to obtain the probability distribution of a sum of dependent discrete random variables more efficientlyRate of convergence of a test statistic towards a Gaussian random variableCalculating the expectation of a sum of dependent random variablesVariance of sum of $m$ dependent random variablesSum of random variables are equal in distributionA lower bound on the expected sum of Bernoulli random variables given a constraint on its distribution
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What is the large-n limit of a distribution of the following sample statistic:$$xequivdisplaystylefracsum^nX_i,sqrt,sum^nX_i^2,,$$ when sampling the Cauchy(0,1) distribution? Monte Carlo simulation indicates that convergence to this limit is quite fast, and that the resulting (symmetric) PDF has very sharp cusps at -1 and +1, but of course does not yield an analytic expression for this PDF - would anyone be able to find it?
Here is how the positive half of the PDF looks like:
pr.probability probability-distributions limit-theorems
edited Jun 16 at 15:43
Carlo Beenakker
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asked Jun 16 at 14:37
HonzaHonza
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add a comment |
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What is the large-n limit of a distribution of the following sample statistic:$$xequivdisplaystylefracsum^nX_i,sqrt,sum^nX_i^2,,$$ when sampling the Cauchy(0,1) distribution? Monte Carlo simulation indicates that convergence to this limit is quite fast, and that the resulting (symmetric) PDF has very sharp cusps at -1 and +1, but of course does not yield an analytic expression for this PDF - would anyone be able to find it?
Here is how the positive half of the PDF looks like:
pr.probability probability-distributions limit-theorems
edited Jun 16 at 15:43
Carlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
asked Jun 16 at 14:37
HonzaHonza
686 bronze badges
New contributor
Honza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
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A somewhat random remark: the limiting distribution is also the distribution of $X_1 / langle Xrangle_1$ if $X_t$ is the Cauchy process and $langle Xrangle_t$ is the quadratic variation process, a $tfrac12$-stable subordinator. I bet someone has studied the joint law of $X_t$ and $langle Xrangle_t$, but unfortunately I do not have time now to search for the reference.
$endgroup$
– Mateusz Kwaśnicki
Jun 16 at 20:17
add a comment |
$begingroup$
What is the large-n limit of a distribution of the following sample statistic:$$xequivdisplaystylefracsum^nX_i,sqrt,sum^nX_i^2,,$$ when sampling the Cauchy(0,1) distribution? Monte Carlo simulation indicates that convergence to this limit is quite fast, and that the resulting (symmetric) PDF has very sharp cusps at -1 and +1, but of course does not yield an analytic expression for this PDF - would anyone be able to find it?
Here is how the positive half of the PDF looks like:
pr.probability probability-distributions limit-theorems
edited Jun 16 at 15:43
Carlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
asked Jun 16 at 14:37
HonzaHonza
686 bronze badges
New contributor
Honza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
What is the large-n limit of a distribution of the following sample statistic:$$xequivdisplaystylefracsum^nX_i,sqrt,sum^nX_i^2,,$$ when sampling the Cauchy(0,1) distribution? Monte Carlo simulation indicates that convergence to this limit is quite fast, and that the resulting (symmetric) PDF has very sharp cusps at -1 and +1, but of course does not yield an analytic expression for this PDF - would anyone be able to find it?
Here is how the positive half of the PDF looks like:
pr.probability probability-distributions limit-theorems
pr.probability probability-distributions limit-theorems
edited Jun 16 at 15:43
Carlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
asked Jun 16 at 14:37
HonzaHonza
686 bronze badges
New contributor
Honza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jun 16 at 15:43
Carlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
asked Jun 16 at 14:37
HonzaHonza
686 bronze badges
New contributor
Honza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jun 16 at 15:43
Carlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
edited Jun 16 at 15:43
Carlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
edited Jun 16 at 15:43
Carlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
84k9 gold badges199 silver badges303 bronze badges
asked Jun 16 at 14:37
HonzaHonza
686 bronze badges
New contributor
Honza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jun 16 at 14:37
HonzaHonza
686 bronze badges
asked Jun 16 at 14:37
HonzaHonza
686 bronze badges
686 bronze badges
New contributor
Honza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Honza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
A somewhat random remark: the limiting distribution is also the distribution of $X_1 / langle Xrangle_1$ if $X_t$ is the Cauchy process and $langle Xrangle_t$ is the quadratic variation process, a $tfrac12$-stable subordinator. I bet someone has studied the joint law of $X_t$ and $langle Xrangle_t$, but unfortunately I do not have time now to search for the reference.
$endgroup$
– Mateusz Kwaśnicki
Jun 16 at 20:17
add a comment |
$begingroup$
A somewhat random remark: the limiting distribution is also the distribution of $X_1 / langle Xrangle_1$ if $X_t$ is the Cauchy process and $langle Xrangle_t$ is the quadratic variation process, a $tfrac12$-stable subordinator. I bet someone has studied the joint law of $X_t$ and $langle Xrangle_t$, but unfortunately I do not have time now to search for the reference.
$endgroup$
– Mateusz Kwaśnicki
Jun 16 at 20:17
$begingroup$
A somewhat random remark: the limiting distribution is also the distribution of $X_1 / langle Xrangle_1$ if $X_t$ is the Cauchy process and $langle Xrangle_t$ is the quadratic variation process, a $tfrac12$-stable subordinator. I bet someone has studied the joint law of $X_t$ and $langle Xrangle_t$, but unfortunately I do not have time now to search for the reference.
$endgroup$
– Mateusz Kwaśnicki
Jun 16 at 20:17
$begingroup$
A somewhat random remark: the limiting distribution is also the distribution of $X_1 / langle Xrangle_1$ if $X_t$ is the Cauchy process and $langle Xrangle_t$ is the quadratic variation process, a $tfrac12$-stable subordinator. I bet someone has studied the joint law of $X_t$ and $langle Xrangle_t$, but unfortunately I do not have time now to search for the reference.
$endgroup$
– Mateusz Kwaśnicki
Jun 16 at 20:17
add a comment |
1 Answer
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This desired large-$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Self-normalized Sums (1973). The Cauchy distribution is the case $alpha=1$ on page 798. The singularity in $lim_nrightarrowinftyP_n(x)$ at $x=pm 1$ is logarithmic
$$P_n(x)rightarrow-pi^-2log|1-x^2|,$$
see equation (5.12) and figure 4 (reproduced below), while for large $|x|$ the decay is Gaussian.
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
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Thanks for your help; the article you mention is as exhaustive as it could be - one has to accept that the asymptotic PDF is not a "pretty" function!
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– Honza
Jun 16 at 22:29
add a comment |
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1 Answer
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1 Answer
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$begingroup$
This desired large-$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Self-normalized Sums (1973). The Cauchy distribution is the case $alpha=1$ on page 798. The singularity in $lim_nrightarrowinftyP_n(x)$ at $x=pm 1$ is logarithmic
$$P_n(x)rightarrow-pi^-2log|1-x^2|,$$
see equation (5.12) and figure 4 (reproduced below), while for large $|x|$ the decay is Gaussian.
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
$endgroup$
$begingroup$
Thanks for your help; the article you mention is as exhaustive as it could be - one has to accept that the asymptotic PDF is not a "pretty" function!
$endgroup$
– Honza
Jun 16 at 22:29
add a comment |
$begingroup$
This desired large-$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Self-normalized Sums (1973). The Cauchy distribution is the case $alpha=1$ on page 798. The singularity in $lim_nrightarrowinftyP_n(x)$ at $x=pm 1$ is logarithmic
$$P_n(x)rightarrow-pi^-2log|1-x^2|,$$
see equation (5.12) and figure 4 (reproduced below), while for large $|x|$ the decay is Gaussian.
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
$endgroup$
$begingroup$
Thanks for your help; the article you mention is as exhaustive as it could be - one has to accept that the asymptotic PDF is not a "pretty" function!
$endgroup$
– Honza
Jun 16 at 22:29
add a comment |
$begingroup$
This desired large-$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Self-normalized Sums (1973). The Cauchy distribution is the case $alpha=1$ on page 798. The singularity in $lim_nrightarrowinftyP_n(x)$ at $x=pm 1$ is logarithmic
$$P_n(x)rightarrow-pi^-2log|1-x^2|,$$
see equation (5.12) and figure 4 (reproduced below), while for large $|x|$ the decay is Gaussian.
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
$endgroup$
This desired large-$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Self-normalized Sums (1973). The Cauchy distribution is the case $alpha=1$ on page 798. The singularity in $lim_nrightarrowinftyP_n(x)$ at $x=pm 1$ is logarithmic
$$P_n(x)rightarrow-pi^-2log|1-x^2|,$$
see equation (5.12) and figure 4 (reproduced below), while for large $|x|$ the decay is Gaussian.
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
answered Jun 16 at 15:37
Carlo BeenakkerCarlo Beenakker
84k9 gold badges199 silver badges303 bronze badges
84k9 gold badges199 silver badges303 bronze badges
$begingroup$
Thanks for your help; the article you mention is as exhaustive as it could be - one has to accept that the asymptotic PDF is not a "pretty" function!
$endgroup$
– Honza
Jun 16 at 22:29
add a comment |
$begingroup$
Thanks for your help; the article you mention is as exhaustive as it could be - one has to accept that the asymptotic PDF is not a "pretty" function!
$endgroup$
– Honza
Jun 16 at 22:29
$begingroup$
Thanks for your help; the article you mention is as exhaustive as it could be - one has to accept that the asymptotic PDF is not a "pretty" function!
$endgroup$
– Honza
Jun 16 at 22:29
$begingroup$
Thanks for your help; the article you mention is as exhaustive as it could be - one has to accept that the asymptotic PDF is not a "pretty" function!
$endgroup$
– Honza
Jun 16 at 22:29
add a comment |
Honza is a new contributor. Be nice, and check out our Code of Conduct.
Honza is a new contributor. Be nice, and check out our Code of Conduct.
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A somewhat random remark: the limiting distribution is also the distribution of $X_1 / langle Xrangle_1$ if $X_t$ is the Cauchy process and $langle Xrangle_t$ is the quadratic variation process, a $tfrac12$-stable subordinator. I bet someone has studied the joint law of $X_t$ and $langle Xrangle_t$, but unfortunately I do not have time now to search for the reference.
$endgroup$
– Mateusz Kwaśnicki
Jun 16 at 20:17