Question: Let the r.v.s Xj, j ¥ 1, be distributed as follows: Show that the Lindeberg condition (relation (12.24)) holds, if and only if α <

Let the r.v.s Xj, j ‰¥ 1, be distributed as follows:

1 1 P(X; = ± jª) = a > 1. P(X; =0) = 1 – 6j2(a-1)' %3D 3 j2(a-1) '

Show that the Lindeberg condition (relation (12.24)) holds, if and only if α < 3/2. Conclude that

Let the r.v.s Xj, j ‰¥ 1, be distributed as

Let the r.v.s Xj, j ‰¥ 1, be distributed as

For α < 3/2, show that

Let the r.v.s Xj, j ‰¥ 1, be distributed as

which is implied by n2a < ɛ2sn2 for large n, so that gn (ɛ) = 0. Next,

gn(É›) ‰¥ 1 - É›2/18 (l €“ 1/k)(2- 1/k) k2a/É›2sn2 k3-2a,

Where k = [(É›sn)l/α], and conclude that the expression on the right-hand side does not converge to 0 for α ‰¥ 3/2.

1 1 P(X; = j) = a > 1. P(X; =0) = 1 6j2(a-1)' %3D 3 j2(a-1) '

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