Strong Convergence Properties and Strong Stability for Weighted Sums of AANA Random Variables

Definition 1 (cf. Wu [1]). A sequence  {X_n, n ≥ 1}  of random variables is said to be stochastically dominated by a random variable  X  if there exists a constant  C such that

for all  x ≥ 0  and  n ≥ 1.

Definition 2 (cf. Chow and Teicher [2]). A sequence  {Y_n, n ≥ 1}  of random variables is said to be strongly stable if there exist two constant sequences  {b_n, n ≥ 1}  and  {d_n, n ≥ 1}  with  0 < b_n↑∞  such that

Definition 3 (cf. Wu [1]). A function  l(x) > 0  (x > 0)  is said to be quasimonotonically increasing function if there exist  x₀ > 0  and constant  C > 0  with  ∀ t ≥ x ≥ x₀  such that  l(t) ≥ Cl(x). A function  l(x) > 0(x > 0)  is said to be quasimonotonically decreasing function if there exist  x₀ > 0  and constant  C > 0  with  ∀ t ≥ x ≥ x₀  such that  l(t) ≤ Cl(x).

Definition 4 (cf. Wu [1]). A real-valued function  l(x), positive and measurable on  (0, ∞), is said to be slowly varying if

for each  λ > 0.

Definition 5 (cf. Joag-Dev and Proschan [3]). A finite collection of random variables  X₁, X₂, …, X_n  is said to be negatively associated (NA, in short) if for every pair of disjoint subsets  A₁, A₂  of  {1,2, …, n},

whenever  f  and  g  are coordinatewise nondecreasing such that this covariance exists. An infinite sequence  {X_n, n ≥ 1}  is NA if every finite subcollection is NA.

Definition 6 (cf. Chandra and Ghosal [4]). A sequence  {X_n, n ≥ 1}  of random variables is called asymptotically almost negatively associated (AANA) if there exists a nonnegative sequence  q(n) → 0  as  n → ∞  such that

for all  n, k ≥ 1  and for all coordinatewise nondecreasing continuous functions  f  and  g  whenever the variances exist.

Lemma 7 (cf. Yuan and An [8]). Let  {X_n,  n ≥ 1}  be a sequence of AANA random variables with mixing coefficients  {q(n), n ≥ 1}, and let  f₁, f₂, …  be all nondecreasing (or nonincreasing) continuous functions; then  {f_n(X_n), n ≥ 1}  is still a sequence of AANA random variables with mixing coefficients  {q(n), n ≥ 1}.

Lemma 8 (cf. Wang et al. [9]). Let  1 < p ≤ 2  and  {X_n, n ≥ 1}  be a sequence of AANA random variables with mixing coefficients  {q(n),  n ≥ 1}. Assume that  EX_n = 0  for all  n ≥ 1  and$$; then there exists a positive constant  C_p  depending only on  p  such that

for all  n ≥ 1, where$$.

Corollary 9 (Khintchine-Kolmogorov-type convergence theorem). Let  {X_n,  n ≥ 1}  be a sequence of AANA random variables with mixing coefficients  {q(n),  n ≥ 1}  and$$. If

then$$converges almost surely.

Corollary 10 (three-series theorem for AANA random variables). Let  {X_n, n ≥ 1}  be a sequence of AANA random variables with mixing coefficients  {q(n), n ≥ 1}  and$$. Assume that for some  c > 0,

Then,$$converges almost surely.

Remark 11. Since NA implies AANA, Corollaries 9 and 10 extend corresponding results for NA random variables (see Matula [18]) to AANA random variables without adding any extra condition.

Lemma 12 (cf. Wu [19] or Shen [20]). Let  {X_n,  n ≥ 1}  be a sequence of random variables which is stochastically dominated by a random variable  X. For any  α > 0  and  b > 0, the following two statements hold:

where  C₁  and  C₂  are positive constants.

Lemma 13 (cf. Wu [1]). Let  h(x) > 0  be a slowly varying function; then for any  δ > 0,  x^δh(x)  is a quasimonotonically increasing function and  x^−δh(x)  is a quasimonotonically decreasing function.

Theorem 14. Let  {X_n, n ≥ 1}  be a sequence of AANA random variables with$$. Assume that  {g_n(x), n ≥ 1}  is a sequence of even functions defined on  R. For each  n ≥ 1,  g_n(x)  is a positive and nondecreasing function in  (0, ∞)  and satisfies one of the following conditions:

• (i)

  for some  0 < r ≤ 1,  x^r/g_n(x)  is a nondecreasing function in  (0, ∞);

• (ii)

  for some  1 < r ≤ 2,  x/g_n(x)  and  g_n(x)/x^r  are nonincreasing functions in  (0, ∞); furthermore, assume that  EX_n = 0  for each  n ≥ 1.

For any positive number sequence  {a_n,  n ≥ 1}  with  a_n↑∞ such that

then $$converges a.s., and

Proof. For each  n ≥ 1, denote

By Lemma 7, we can see that, for fixed  n ≥ 1,$$is still a sequence of AANA random variables. So by Corollary 10 in order to prove (11), we need only to prove the convergence of three series of (8), where  c = 1.

Firstly, we prove that $$under condition (i) or (ii).

For each  n ≥ 1, if  g_n(x)  satisfies condition (i), noting that  {g_n(x), n ≥ 1}  is a sequence of positive and nondecreasing even function in  (0, +∞). Combining Markov’s inequality with (10), it follows that

If  g_n(x)  satisfies condition (ii), it is easy to prove that (13) also holds.

Secondly, we will show$$.

If  g_n(x)  satisfies (i), when   | x | ≤ a_n, we have$$, which implies that

Note that  {g_n(x), n ≥ 1}  is a sequence of positive and nondecreasing functions in  (0, +∞), so  0 ≤ (g_n(x)/g_n(a_n)) ≤ 1  when   | x | ≤ a_n. Consequently, On the other hand, if  g_n(x)  satisfies condition (ii), then we can also get that Therefore, whether even function  g_n(x)  satisfies condition (i) or (ii), we can obtain Therefore, it follows from (10) that

Finally, we prove that$$.

If  g_n(x)  satisfies condition (i), when   | x | ≤ a_n, we have$$, for  0 < r ≤ 1. It follows that

If  g_n(x)  satisfies condition (ii), then by the fact that  EX_n = 0  and  x/g_n(x)  is a nonincreasing function in  (0, ∞), we get

Therefore, whether  g_n(x)  satisfies condition (i) or (ii), it also follows from (10) that The proof of Theorem 14 is completed by (13), (18), and (21).

Corollary 15. Let  {X_n,  n ≥ 1}  be a sequence of AANA random variables with$$, and let  {a_n, n ≥ 1}  be a sequence of positive numbers with  a_n↑∞. There exists some  0 < p ≤ 2  such that

If  1 < p ≤ 2, we further assume that  EX_n = 0. Then, (11) holds.

Proof. We take  g_n(x) = |x|^r,  x ∈ (−∞, ∞). If  0 < r ≤ 1, one can find that  x^r/g_n(x) ≡ 1  for  x ≥ 0. So condition (i) of Theorem 14 is satisfied. If  1 < r ≤ 2, we have that  x/g_n(x) = x^1−r  and  g_n(x)/x^r ≡ 1  for  x ≥ 0. Thus, condition (ii) of Theorem 14 is satisfied. Consequently, the desired result (11) follows from Theorem 14 immediately.

Remark 16. If taking  r = 1  in (i) and  r = 2  in (ii), Theorem 14 and Corollary 15 extend the corresponding ones for NA random variables (see Gan [21]) to AANA random variables.

Theorem 17. Let  1 < α < 2  and  {X_n,  n ≥ 1} be a sequence of AANA random variables with$$and identical distribution

where  L(x)  is a slowly varying function. Let  {a_n,  n ≥ 1}  and  {b_n,  n ≥ 1}  be sequences of positive constants satisfying  0 < b_n↑∞. Denote  c_n = b_n/a_n  for each  n ≥ 1. Assume that then

Proof. Since (23) and (24) imply that  c_k ≥ 1  for all sufficiently large  k. Without loss of generality, we assume  c_k ≥ 1  for all  k ≥ 1.

By Borel-Cantelli Lemma, it is easily seen that (24) implies that

Denote thus,  {Y_k, k ≥ 1}  is still AANA from Lemma 7. It is easy to check that In order to show that $$a.s., we only need to show that the first three terms above are  o(b_n)  or  o(b_n)  a.s.

By  C_r  inequality, Theorem  1b in [22, page 281] (or see Adler [23]) and (24), we can get

It follows from Corollary 9 and Kronecker’s lemma that By (24) again, which implies that By Kronecker’s lemma, it follows that By Theorem  1b in [22, page 281] (or see Adler [23]) and (24) again, we have which implies that By Kronecker’s Lemma, it follows that Hence, the desired result (25) follows from (26)–(36) immediately.

Remark 18. Theorem 17 generalizes and extends the corresponding one for NA random variables (see Wang et al. [24]) to AANA random variables.

Theorem 19. Let  1 < r < 2  and  {X_n, n ≥ 1}  a sequence of mean zero AANA random variables with$$, which is stochastically dominated by a random variable  X. Let  {a_n, n ≥ 1}  be a sequence of positive constants satisfying$$. Denote  c_n = A_n/a_n  for each  n ≥ 1. Assume that

then

Proof. Let  N(0) = 0  and denote

It follows by (37) that By the equality above and Borel-Cantelli lemma, we can get$$. Therefore, in order to prove (38), we only need to prove that

By  C_r  inequality, Lemma 12, and (37) again,

Hence, by the inequality above, Corollary 9 and Kronecker’s lemma, we have In order to prove (41), it suffices to prove that Notice that  EX_n = 0  for each  n ≥ 1, we have It follows from Lemma 12 and (37) that, By Kronecker’s lemma, we can get (44) immediately. The proof is complete.

Theorem 20. Let  {X_n, n ≥ 1}  be a sequence of AANA random variables with$$, which is stochastically dominated by a random variable  X. Let  {a_n, n ≥ 1}  and  {b_n, n ≥ 1}  be two sequences of positive numbers with  c_n = b_n/a_n  and  b_n↑∞. Denote  N(x) = Card{n : c_n ≤ x}, x > 0. If the following conditions are satisfied:

• (i)

  EN(|X|) < ∞;

• (ii)

  $$, for some  p ∈ [1,2],

then there exist  d_n ∈ R,  n = 1,2, …, such that

Proof. For each  i ≥ 1, denote

By Definition 1 and conditions (i), we can obtain By Borel-Cantelli lemma for any sequence  {d_n, n ≥ 1} ⊂ R, with probability 1, the sequences$$and$$converge on the same set and to the same limit. We will prove that$$., which implies (6) with$$. According to Lemma 7,$$is a sequence of AANA random variables with mean zero. By  C_r  inequality and Lemma 12, we have Notice that where the last inequality follows from the fact that for  t > 0   By (50), (51) and condition (i), (ii), we can get that Therefore, it follows from (53) and Corollary 15 that The proof is complete.

Corollary 21. Suppose that the conditions of Theorem 20 are satisfied and  EX_n = 0  for each  n ≥ 1. If$$, then$$.

Proof. According to the proof of Theorem 20, we need only to prove that

Notice that  EX_n = 0  for each  n ≥ 1; then By Kronecker’s lemma, we can get (55) immediately.

Theorem 22. Let  {X_n, n ≥ 1}  be a sequence of AANA random variables with mean zero and$$, which is stochastically dominated by a random variable  X. Let  {a_n, n ≥ 1}  and  {b_n, n ≥ 1}  be two sequences of positive numbers with  c_n = b_n/a_n  and  b_n↑∞. Denote  N(x) = Card{n : c_n ≤ x},  x > 0. If the following conditions are satisfied:

• (i)

  EN(|X|) < ∞;

• (ii)

  $$;

• (iii)

  $$, for some  p ∈ [1,2],

then

Proof. By (49), condition (i), and Borel-Cantelli lemma, it suffices to prove$$,  a.s.  So we need only to prove

We can get (60) from the proof of Corollary 21. In the following, we prove (59). Put  ε₀ = 0  and  ε_n = max _1≤j≤nc_j  for  n ≥ 1. According to Lemma 7,$$is a sequence of AANA random variables with mean zero. By  C_r  inequality and Lemma 12, It is easy to see that Therefore, follows from condition (i), (61) and (62). By Corollary 15 and (63), we can obtain (59) immediately. The proof is complete.

Theorem 23. Let  {X_n, n ≥ 1}  be a sequence of AANA random variables with$$, which is stochastically dominated by a random variable  X. Let  {a_n, n ≥ 1}  and  {b_n, n ≥ 1}  be two sequences of positive numbers with  c_n = b_n/a_n  and  b_n↑∞. Define  N(x) = Card{n : c_n ≤ x},$$,  x > 0. If

• (i)

  N(x) < ∞  for any  x > 0;

• (ii)

  $$;

• (iii)

  E(X²R(|X|)) < ∞,

then there exist  d_n ∈ R,  n = 1,2, …, such that

Proof. According to Lemma 7,$$and$$are sequences of AANA random variables.

Since  N(x)  is nondecreasing, then for any  x > 0  

which implies that  EN(|X|) ≤ 2E(X²R(|X|)) < ∞. Therefore, By Borel-Cantelli lemma for any sequence  {d_n, n ≥ 1} ⊂ R, with probability 1, the sequences$$and$$converge on the same set and to the same limit. We will prove that$$., which implies the theorem with$$. By  C_r  inequality and Lemma 12, Since  N(1) = Card{n : c_n ≤ 1} ≤ 2R(1) < ∞, following from (65) and condition (ii), then we have  I₁ < ∞. To prove  I₂ < ∞, we need to prove that  I₂₁ < ∞  and  I₂₂ < ∞: Since  N(x)  is nondecreasing and  R(x)  is nonincreasing, then By (67)–(71), we can get that Therefore, it follows from Corollary 9 and Kronecker’s lemma that Taking$$,  n ≥ 1, we can get (64). The proof is complete.

Remark 24. Since NA implies AANA, Theorem 20 extends corresponding result for NA random variable (see Wang et al. [24]) to AANA random variables without adding any extra condition.

Similar to the proof of Corollary 21, we can get the following corollary.

Corollary 25. Let the conditions of Theorem 23 be satisfied and  EX_n = 0  for each  n ≥ 1. If$$, then$$.

Corollary 26. Let  {X_n, n ≥ 1}  be a sequence of AANA random variables with$$, which is stochastically dominated by a random variable  X. Let  {a_n, n ≥ 1}  and  {b_n, n ≥ 1}  be two sequences of positive numbers with  c_n = b_n/a_n  and  b_n↑∞. Let  f(x) = x^rh(x), where  h(x) > 0  is a slowly varying function as  x → ∞,  1 < r < 2. Define  N(x) = Card{n : c_n ≤ x},  x > 0. If

• (i)

  N(n) = O(f(n))  for each  n ≥ 1;

• (ii)

  Ef(|X|) < ∞.

Then there exist  d_n ∈ R,  n = 1,2, …, such that

Proof. It is easy to verify that conditions (i)–(iii) of Theorem 23 hold under the conditions of Corollary 25. So Corollary 25 is true by Theorem 23.

Corollary 27. Suppose that the conditions of Corollary 26 are satisfied. If  EX_n = 0  for each  n ≥ 1, then$$.

Proof. According to Corollary 26, we need only to prove (60). By  EX_n = 0  for each  n ≥ 1  and Lemma 12, we have

Since  r > 1, we can take  δ > 0  such that  r − δ > 1. By Lemma 13 and differential mean value theorem, we can obtain It is easily seen that  x^r−1h(x)  is a quasimonotonically increasing function by Lemma 13. Hence, we have by (75) and (76) that By Kronecker’s lemma, we can get (60) immediately. The proof is complete.