The Law of the Iterated Logarithm (LIL) is a result in probability theory that describes the asymptotic behavior of sums of independent and identically distributed (i.i.d.) random variables. It provides a precise way to understand the fluctuations of a normalized random walk. To put it more formally, consider a sequence of i.i.d.
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For independent identically distributed centered variables of variance one under standard moment assumptions, the law of the iterated logarithm describes the almost-sure envelope of their partial sums on the scale .