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I wonder wheather there exsits a probability inequality for the sum of independent normal random variables ([itex]X_i[/itex] are i.i.d. normal random varianble with mean [itex]\mu[/itex] and variance [itex]\sigma^2[/itex]):

[itex]

P\left(\frac{1}{n}\sum_{i=1}^n X_i - \mu> \epsilon\right)\leq

f(\epsilon, \sigma^2,n) \right).

[/itex]

We know that Bernstein inequality is for the sum of bounded random variables:

[itex]

P\left(\frac{1}{n}\sum_{i=1}^n X_i -\mu > \epsilon\right)\leq

\exp\left(-\frac{n\epsilon^2}{2\sigma^2+ 2c\epsilon/3} \right).

[/itex]

I wonder whether there is some similar inequality for normal variables.

Thanks!

Phonic

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# Probability inequality for the sum of independent normal random variables

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