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Let $X_1,X_2,\ldots,X_n$ be an i.i.d. sequence of $n$ positive random variables with mean $E[X_1]=\mu_X<\infty$ and the second moment $E[X_1^2]=\infty$.

I am interested in upper-bounding $P\left(|\frac{1}{n}\sum_{i=1}^nX_i-\mu_X|\geq x\right)$ (though a lower bound on $P\left(\frac{1}{n}\sum_{i=1}^nX_i-\mu_X \geq x\right)$ would be great as well). My go-to method for this is Chebyshev's inequality, however, the second moment for $X_1$'s (and, hence, the variance) is infinite so that's of no use.

However, I do have an expression for the tail probability of individual member of this i.i.d. sequence, $P(X_1>x)$ in a (more-or-less) nice form. Can I use it to get the desired bound(s) on the sequence average?

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    $\begingroup$ If you cannot use Chebyshev’s inequality (because of the infinite variance) you can always consider applying “truncated Chebyshev inequality” – see e.g. books.google.pl/… $\endgroup$
    – Waldemar
    Commented Nov 20, 2013 at 8:18
  • $\begingroup$ How about the following for a simple lower bound: $$P\left(\frac{1}{n}X_i - \mu_X > x\right)\geq \left[P(X_i - \mu_X > x)\right]^n.$$ This turns out to be sufficiently tight in certain cases, for instance, when $P(X_1 > x)$ goes down exponentially in $x$. $\endgroup$
    – Skoro
    Commented Nov 23, 2013 at 17:45
  • $\begingroup$ @Waldemar Been working with your suggestion before going away for a few days... Haven't gotten the result I need yet. Didn't know about the truncated Cheybyshev's inequality before, it's neat. Thanks for the pointer to the book -- I do realize it's one of the first results when one searches for "truncated Chebyshev's inequality" but it's quite good. $\endgroup$
    – Bullmoose
    Commented Nov 28, 2013 at 4:07
  • $\begingroup$ @Skoro Are there random variables whose tails decay exponentially that have infinite (or undefined) variance? $\endgroup$
    – Bullmoose
    Commented Nov 28, 2013 at 4:08

2 Answers 2

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You probably need some kind of Hoffmann-Jorgensen inequality. Google that name, or look at

Hitczenko, Paweł(1-DREX); Montgomery-Smith, Stephen(1-MO) Measuring the magnitude of sums of independent random variables. (English summary) Ann. Probab. 29 (2001), no. 1, 447–466 or

Montgomery-Smith, Stephen J.(1-MO); Pruss, Alexander R.(1-PITT-Q) A comparison inequality for sums of independent random variables. (English summary) J. Math. Anal. Appl. 254 (2001), no. 1, 35–42

or references therein.

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How about using Nagaev's review paper http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aop/1176994938 You didn't specify why the results there are of no use to you. Specifically, Theorem 1.1.

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  • $\begingroup$ Hmmm... looks useful. Thanks for pointing me to this paper (again). $\endgroup$
    – Bullmoose
    Commented Nov 28, 2013 at 4:30

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