The law of the iterated logarithm (LIL) is a mathematical theorem that relates to the behavior of the maximum values of a sequence of independent and identically distributed random variables. It characterizes the fluctuations and growth rates of these maximum values as the number of variables in the sequence increases.
According to the LIL, for a sequence of random variables X₁, X₂, X₃, ... that are independent, identically distributed, with finite mean μ and variance σ², the maximum of the standardized partial sums of the variables will grow at most logarithmically, and this growth is asymptotically optimal.
Formally, the LIL states that as n tends to infinity, the probability lim sup (maximum) of the standardized partial sum of the first n variables, multiplied by (√(2n log log n))/(σ√log n), equals 1 with probability 1/2, and equals -1 with probability 1/2.
In simpler terms, the law of the iterated logarithm explains that as the number of variables in a sequence increases towards infinity, the maximum value of the sum of these variables will grow at most logarithmically. This logarithmic growth is essentially the fastest rate at which the maximum can increase.
The law of the iterated logarithm is a fundamental result in probability theory and has important applications in fields such as statistics, finance, and computer science. It helps in understanding the behavior of extreme values and their probabilities in random processes.
