the following excerpt is from den hollander:
i just don't get the point. what is meant by cheapest realization? is it, loosely speaking, the realization occuring with highest probability because of the minus preceding the infimum? what, then, do the least unlikely of all the unlikely ways correspond to? is the least unlikely way the minimizer?
besides, is the theory of large deviations in the form of cramer's theorem, say, somehow related to the law of the iterated logarithm or to the theory of extreme values?
i'd appreciate some comments!
Yes to “realization occurring with highest probability”; yes to minus preceeding the infimum. They are all VERY unlikely to happen;
think of n fair coin tosses; it is quite unlikely to have exactly half of them to be heads, but it is the most likely."Think of n fair coin tosses. Then it is quite unlikely to have around 70% of heads, but it is the most likely scenario if one wants to get at least 70% of heads" (changed after comment below) You can read about Sanov’s theorem if you want a more concrete example of large deviations.