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Question 1:
Is there a bound on P( y_N > 2 s), the probability that y_N will be more than 2 standard deviation above the mean?
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The
Chebyshev (multiple spellings) inequality is a quick, coarse estimate: The probability that a random variable is n standard deviations away from the mean is at most 1/n^2. A one-tailed version is slightly better: The probability that a random variable is n standard deviations above the mean is at most 1/(1+n^2). Here that is 1/5.
This is not sharp for n>1, but there isn't a bound depending on N and not the random variables that approaches the result for a normally distributed random variable. For example, consider a Poisson distribution such that 1 is 2 standard deviations above the mean. The mean is 3-2sqrt(2)=.1716. The probability it is 1 or greater is .1577, not too far below the .2 from the Chebyshev inequality. A Poisson distribution is divisible, so it is the sum of n IID random variables, in particular, n variables with a Poisson distributions of mean (3-2sqrt(2))/n.
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Question 2: If we know that P( |x_i| > a) = 0 for some real number a, is there a sharper bound on P( y_N > 2 s) using a?
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Yes. That puts an upper bound on the higher moments, and these are used in effective versions of the Central Limit Theorem. I don't know them offhand.