Let X n , n yen 1, be independent r v s such that X n curren M n a s with M n o(s n ) where Set and snow that Hint From the assumption M n o(s n ) it follows that so that Mn Eacute sn, n n0( n( Eacute )), Eacute 0, Write And since the first term tends to 0, as n , work with the second term only To ...

From M n os n it follows that for 0 there exists n 0 n such that M n s n n n 0 For n n 0 ...

Let X n , n ¥ 1, be independent r.v.s such that |X n | ¤ M

Question:

Let X_n, n ‰¥ 1, be independent r.v.s such that |X_n| ‰¤ M_na.s. with M_n= o(s_n) where

Set

and snow that

Hint: From the assumption

M_n = o(s_n)

it follows that

so that

Mn < É›sn, n > n0(=n(É›)), É› > 0,

Write

Let Xn, n ‰¥ 1, be independent r.v.s such that

And since the first term tends to 0, as n †’ ˆž, work with the second term only. To this end, set Ynj = (Xj - É›Xj)/Ï„_n where Ï„_n² = s_n² €“ s²_n0, and show that the r.v.s Y_nj, j = n₀ + 1€¦. n, satisfy the Liapounov condition (for Î´ = 1) (see Theorem 3).