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jat
jat
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I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. The median is easy to calculate, and it provides a natural My guess for the mean:

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

The reference I checked (Ledoux & Talagrand, 1991) helpfully told me that this calculation is "standard". The argument apparently depends on integration by parts, but I can't figure out the trick.

I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. The median is easy to calculate, and it provides a natural guess for the mean:

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

The reference I checked (Ledoux & Talagrand, 1991) helpfully told me that this calculation is "standard". The argument apparently depends on integration by parts, but I can't figure out the trick.

I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. My guess for the mean:

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

The reference I checked (Ledoux & Talagrand, 1991) helpfully told me that this calculation is "standard". The argument apparently depends on integration by parts, but I can't figure out the trick.

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jat
jat
  • 23
  • 3

I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. The answer appearsmedian is easy to becalculate, and it provides a natural guess for the mean:

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

Is there a simple way to do this? The referencesThe reference I checked (Ledoux & Talagrand, 1991) helpfully told me that thethis calculation is "standard". My approach is too awful to set in print The argument apparently depends on integration by parts, but I can't figure out the trick.

I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. The answer appears to be

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

Is there a simple way to do this? The references I checked helpfully told me that the calculation is "standard". My approach is too awful to set in print.

I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. The median is easy to calculate, and it provides a natural guess for the mean:

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

The reference I checked (Ledoux & Talagrand, 1991) helpfully told me that this calculation is "standard". The argument apparently depends on integration by parts, but I can't figure out the trick.

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jat
jat
  • 23
  • 3

Expectation of RVs with Poisson-type decay

I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. The answer appears to be

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

Is there a simple way to do this? The references I checked helpfully told me that the calculation is "standard". My approach is too awful to set in print.