<p>Let <span class="math-container">$X_1,X_2,\ldots,X_n$</span> be <span class="math-container">$n$</span> independent Poisson random variables with <span class="math-container">$\mu(X_i) = \lambda_i$</span>. I am interested in upper tail bounds for the random variable <span class="math-container">$Z = \sum_{i} a_i X_i$</span> where <span class="math-container">$a_i \in [0,1]$</span> for <span class="math-container">$1\le i \le n$</span>. For instance if each <span class="math-container">$a_i =  1$</span> then <span class="math-container">$Z$</span> is simply a Poisson rv with mean <span class="math-container">$\sum_i \lambda_i$</span> and one would get tail inequalities such as the one I found in a note <a href="http://www.cs.columbia.edu/%7Eccanonne/files/misc/2017-poissonconcentration.pdf" rel="nofollow noreferrer">here</a>. For the general weighted case Chernoff style tail bounds should apply but I am not finding an easy reference. Any help would be appreciated.</p>

CC BY-SA 4.0
2020-07-08T15:50:58.930Z
One can replace $X_i$ by a sum of many binomial random variables and then apply Chernoff  bounds to that sum so tail bounds should apply but I am trying to avoid formalizing all the messy details and cite some source.
Chandra Chekuri
CC BY-SA 4.0
2020-07-08T21:04:35.907Z
Your link states that Poisson's are sub-exponential. Sub-exponential random variables are closed under taking independent sums, and have tail inequalities. If you compute the precise sub-exponential parameter of the Poisson, then the results from [here](https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf) should make the rest quite simple.
Mark Schultz-Wu
<bounds>
Tail bounds for sum of weighted Poisson random variables
Chandra Chekuri
# Tail bounds for sum of weighted Poisson random variables

(Asked by: username_0 on 2020-07-08)

Let $X\_1,X\_2,\ldots,X\_n$ be $n$ independent Poisson random variables with $\mu(X\_i) = \lambda\_i$. I am interested in upper tail bounds for the random variable $Z = \sum\_{i} a\_i X\_i$ where $a\_i \in [0,1]$ for $1\le i \le n$. For instance if each $a\_i = 1$ then $Z$ is simply a Poisson rv with mean $\sum\_i \lambda\_i$ and one would get tail inequalities such as the one I found in a note [here](http://www.cs.columbia.edu/%7Eccanonne/files/misc/2017-poissonconcentration.pdf). For the general weighted case Chernoff style tail bounds should apply but I am not finding an easy reference. Any help would be appreciated.

**Tags:** bounds
> **Comments:**
> **username_0**: One can replace $X_i$ by a sum of many binomial random variables and then apply Chernoff  bounds to that sum so tail bounds should apply but I am trying to avoid formalizing all the messy details and cite some source.  
> **username_1**: Your link states that Poisson's are sub-exponential. Sub-exponential random variables are closed under taking independent sums, and have tail inequalities. If you compute the precise sub-exponential parameter of the Poisson, then the results from [here](https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf) should make the rest quite simple.  

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