Tail Asymptotics of a Sum of Random Variables
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Let $Xsim P$ be a non-negative random variable and let $X_1,dots,X_n$ be i.i.d. copies of $X$. Furthermore, define the right tail function of $X$ as $G_1(x) := P(X>x)$ and the right tail function of $sum_{i=1}^{n} X_i$ as $G_n(x) := P(sum_{i=1}^{n} X_i>x)$.
Is there any result that relates the two tail functions $G_1$ and $G_n$ with each other - or an upper-bound which holds? For example what if $X$ is assumed to be Pareto- or exponentially-distributed?
Best regards,
Carl
probability probability-theory statistics
asked Jan 16 at 13:25
CarlCarl
11311
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add a comment |
$begingroup$
Let $Xsim P$ be a non-negative random variable and let $X_1,dots,X_n$ be i.i.d. copies of $X$. Furthermore, define the right tail function of $X$ as $G_1(x) := P(X>x)$ and the right tail function of $sum_{i=1}^{n} X_i$ as $G_n(x) := P(sum_{i=1}^{n} X_i>x)$.
Is there any result that relates the two tail functions $G_1$ and $G_n$ with each other - or an upper-bound which holds? For example what if $X$ is assumed to be Pareto- or exponentially-distributed?
Best regards,
Carl
probability probability-theory statistics
asked Jan 16 at 13:25
CarlCarl
11311
$endgroup$
1
$begingroup$
See this post, in particular the answer, for a rather loose inequality that applies in general.
$endgroup$
– Lee David Chung Lin
Jan 29 at 23:58
add a comment |
$begingroup$
Let $Xsim P$ be a non-negative random variable and let $X_1,dots,X_n$ be i.i.d. copies of $X$. Furthermore, define the right tail function of $X$ as $G_1(x) := P(X>x)$ and the right tail function of $sum_{i=1}^{n} X_i$ as $G_n(x) := P(sum_{i=1}^{n} X_i>x)$.
Is there any result that relates the two tail functions $G_1$ and $G_n$ with each other - or an upper-bound which holds? For example what if $X$ is assumed to be Pareto- or exponentially-distributed?
Best regards,
Carl
probability probability-theory statistics
asked Jan 16 at 13:25
CarlCarl
11311
$endgroup$
Let $Xsim P$ be a non-negative random variable and let $X_1,dots,X_n$ be i.i.d. copies of $X$. Furthermore, define the right tail function of $X$ as $G_1(x) := P(X>x)$ and the right tail function of $sum_{i=1}^{n} X_i$ as $G_n(x) := P(sum_{i=1}^{n} X_i>x)$.
Is there any result that relates the two tail functions $G_1$ and $G_n$ with each other - or an upper-bound which holds? For example what if $X$ is assumed to be Pareto- or exponentially-distributed?
Best regards,
Carl
probability probability-theory statistics
probability probability-theory statistics
asked Jan 16 at 13:25
CarlCarl
11311
asked Jan 16 at 13:25
CarlCarl
11311
asked Jan 16 at 13:25
CarlCarl
11311
asked Jan 16 at 13:25
CarlCarl
11311
asked Jan 16 at 13:25
CarlCarl
11311
11311
1
$begingroup$
See this post, in particular the answer, for a rather loose inequality that applies in general.
$endgroup$
– Lee David Chung Lin
Jan 29 at 23:58
add a comment |
1
$begingroup$
See this post, in particular the answer, for a rather loose inequality that applies in general.
$endgroup$
– Lee David Chung Lin
Jan 29 at 23:58
1
1
$begingroup$
See this post, in particular the answer, for a rather loose inequality that applies in general.
$endgroup$
– Lee David Chung Lin
Jan 29 at 23:58
$begingroup$
See this post, in particular the answer, for a rather loose inequality that applies in general.
$endgroup$
– Lee David Chung Lin
Jan 29 at 23:58
add a comment |
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1
$begingroup$
See this post, in particular the answer, for a rather loose inequality that applies in general.
$endgroup$
– Lee David Chung Lin
Jan 29 at 23:58