nhaliday + series   14

Hoeffding’s Inequality
basic idea of standard pf: bound e^{tX} by line segment (convexity) then use Taylor expansion (in p = b/(b-a), the fraction of range to right of 0) of logarithm
pdf  lecture-notes  exposition  nibble  concentration-of-measure  estimate  proofs  ground-up  acm  probability  series  s:null
february 2017 by nhaliday
Existence of the moment generating function and variance - Cross Validated
This question provides a nice opportunity to collect some facts on moment-generating functions (mgf).

In the answer below, we do the following:
1. Show that if the mgf is finite for at least one (strictly) positive value and one negative value, then all positive moments of X are finite (including nonintegral moments).
2. Prove that the condition in the first item above is equivalent to the distribution of X having exponentially bounded tails. In other words, the tails of X fall off at least as fast as those of an exponential random variable Z (up to a constant).
3. Provide a quick note on the characterization of the distribution by its mgf provided it satisfies the condition in item 1.
4. Explore some examples and counterexamples to aid our intuition and, particularly, to show that we should not read undue importance into the lack of finiteness of the mgf.
q-n-a  overflow  math  stats  acm  probability  characterization  concept  moments  distribution  examples  counterexample  tails  rigidity  nibble  existence  s:null  convergence  series
january 2017 by nhaliday
pr.probability - When are probability distributions completely determined by their moments? - MathOverflow
Roughly speaking, if the sequence of moments doesn't grow too quickly, then the distribution is determined by its moments. One sufficient condition is that if the moment generating function of a random variable has positive radius of convergence, then that random variable is determined by its moments.
q-n-a  overflow  math  acm  probability  characterization  tidbits  moments  rigidity  nibble  existence  convergence  series
january 2017 by nhaliday

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