In this section, we will present some complete convergence for a sequence of acceptable random variables by using the probability inequality.
Theorem 3.1 Let be a sequence of acceptable random variables with and for each . Denote , . For fixed , suppose that there exists a positive number H such that (2.11) holds true. If for any ,
(3.1)
and
(3.2)
where is a sequence of positive numbers, then completely as .
Proof By Corollary 2.2, we have for any ,
which implies that
This completes the proof of the theorem. □
It is easily seen that (3.2) holds if . So, we have the following corollary.
Corollary 3.1 Let be a sequence of acceptable random variables with and for each . Denote , . Suppose that conditions (2.11) and (3.1) hold with . Then completely as .
Theorem 3.2 Let be a sequence of acceptable random variables with and for each . Denote and
If for any ,
(3.3)
then completely as .
Proof By Markov’s inequality and Definition 1.2, for any and ,
Taking in the inequality above, we can get that
It follows from the inequality above and (2.5) that
which completes the proof of the theorem. □
Hanson and Wright (1971) obtained a bound on tail probabilities for quadratic forms in independent random variables using the following condition: for all and all , there exist positive constants M and γ such that
(3.4)
Wright [11] proved that the bound established by Hanson and Wright [12] for independent symmetric random variables also holds when the random variables are not symmetric but condition (3.4) is valid. We will study the complete convergence for a sequence of acceptable random variables under condition (3.4). The main result is as follows.
Theorem 3.3 Let be a sequence of acceptable random variables satisfying condition (3.4) for all and all , where M and γ are positive constants. Suppose that there exists a positive constant C not depending on n such that
(3.5)
Then for all , completely as .
Proof By Markov’s inequality and assumption (3.5), we have that for any ,
In the following, we will estimate . By (3.4), we can see that
Hence,
This completes the proof of the theorem. □