Law of The Iterated Logarithm - Discussion

Discussion

The law of iterated logarithms operates “in between” the law of large numbers and the central limit theorem. There are two versions of the law of large numbers — the weak and the strong — and they both claim that the sums Sn, scaled by n−1, converge to zero, respectively in probability and almost surely:

\frac{S_n}{n} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{n} \ \xrightarrow{a.s.} 0, \qquad \text{as}\ \ n\to\infty.

On the other hand, the central limit theorem states that the sums Sn scaled by the factor n−½ converge in distribution to a standard normal distribution. Combined with Kolmogorov's zero-one law, this implies that these quantities converge neither in probability nor almost surely:

\frac{S_n}{\sqrt n} \ \stackrel{p}{\nrightarrow}\ \forall, \qquad \frac{S_n}{\sqrt n} \ \stackrel{a.s.}{\nrightarrow}\ \forall, \qquad \text{as}\ \ n\to\infty.

The law of the iterated logarithm provides the scaling factor where the two limits become different:

\frac{S_n}{\sqrt{n\log\log n}} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{\sqrt{n\log\log n}} \ \stackrel{a.s.}{\nrightarrow}\ 0, \qquad \text{as}\ \ n\to\infty.

Thus, although the quantity is less than any predefined ε > 0 with probability approaching one, that quantity will nevertheless be dropping out of that interval infinitely often, and in fact will be visiting the neighborhoods of any point in the interval (0,√2) almost surely.

Read more about this topic:  Law Of The Iterated Logarithm

Famous quotes containing the word discussion:

    “We cannot set aside an hour for discussion with our children and hope that it will be a time of deep encounter. The special moments of intimacy are more likely to happen while baking a cake together, or playing hide and seek, or just sitting in the waiting room of the orthodontist.”
    —Neil Kurshan (20th century)

    “There exist few things more tedious than a discussion of general ideas inflicted by author or reader upon a work of fiction.”
    —Vladimir Nabokov (1899–1977)

    “Opinions are formed in a process of open discussion and public debate, and where no opportunity for the forming of opinions exists, there may be moods—moods of the masses and moods of individuals, the latter no less fickle and unreliable than the former—but no opinion.”
    —Hannah Arendt (1906–1975)