If the independent r.v.s Xj, j ≥ 2, are distributed as follows:
P(Xj = -ja) = P(Xj= ja) = 1/jβ P(Xj=0)
= 1 - 2/jβ (α, β > 0),
Show that the restriction β < 1 ensures that the Lindeberg condition (relation (12.24)) holds.
Show that the restriction β < 1 ensures that the set of js with j = 1, ..., n, and | ± ja| ≥ ɛsn, is empty for all ɛ > 0, for large n. For an arbitrary, but fixed β (< 1), it is to be understood that j ≥ j0, where j0 = 21/β, if 21/β is an integer, or j0 = [21/β] + 1 otherwise. This ensures that 1 - 2/jβ j$ is nonnegative.