MATEMATIQKI VESNIK
originalni nauqni rad
research paper
62, 4 (2010), 251–258
December 2010
A NOTE ON THE EXPONENTIAL CONVERGENCE
RATE FOR PRODUCTS OF SUMS
Yu Miao and Bin Qian
Abstract. In this paper, we establish an exponential convergence theorem for products of
sums of independent identically distributed positive random variables.
1. Introduction
Let (Xn )n≥1 be a sequence of independent identically
(i.i.d.) posPdistributed
n
X
and
the
product
itive random variables
and
define
the
partial
sums
S
=
i
n
i=1
Pn
of sums Tn = k=1 Sk for n ≥ 1. In the past decade, there have been many studies
about the asymptotic properties for the products of partial sums Tn .
The study for the product of partial sums was initiated by Arnold and Villaseñr
[1] who considered the limiting properties of the sums of records. In their paper,
the authors obtained the following version of the central limit theorem (CLT) for
a sequence of i.i.d. exponential r.v.’s (Xn )n≥1 with the mean equal to one:
Pn
k=1 log Sk − n log n + n L
√
→ N, as n → ∞,
2n
where N is a standard normal r.v. Here we think that it is interesting to recall that
the products of i.i.d. positive, square integrable random variables are asymptotically log-normal. This fact is an immediate consequence of the classical CLT. In
[11], Rempala and Wesolowski have noted that this limit behavior of the product
of partial sums has a universal character and holds for any sequence of square integrable, positive i.i.d. random variables. Namely, they have proved the following
result.
Theorem RW. Let (Xn )n≥1 be a sequence of i.i.d. positive square integrable
random variables with EX1 = µ, V arX1 = σ 2 > 0 and the coefficient of variation
2010 AMS Subject Classification: 60F10, 60G05.
Keywords and phrases: The exponential convergence; moderate deviation principle; products
of sums; positive random variables.
251
252
γ = σ/µ. Then
Yu Miao, Bin Qian
µ Qn
k=1
Sk
n! µn
¶1/(γ √n)
L
→e
√
2N
.
Recently, Gonchigdanzan and Rempala [3] obtained the first almost sure central limit theorem (ASCLT) for the product of the partial sums of i.i.d. positive
random variables as follows.
Theorem GR. Let (Xn )n≥1 be a sequence of i.i.d. positive square integrable
random variables with EX1 = µ > 0, V arX1 = σ 2 . Denote by γ = σ/µ the
coefficient of variation. Then for any real x,
µµ Qn
¶
¶1/(γ √n)
N
1 X1
k=1 Sk
lim
I
≤
x
= F (x), a.s.
N →∞ log N
n
n!µn
n=1
where F (·) is the distribution function of the r.v. e
√
2N
.
For the further discussions of the CLT, the author refers to [5, 10]. In [4], Huang
and Zhang obtained the invariance principle of the product of sums of random
variables. It is perhaps worth to notice that by the strong law of large numbers
and the property of the geometric mean it follows directly that
µ Qn
¶1/n
a.s.
k=1 Sk
→ µ
(1.1)
n!
if only existence of the first moment is assumed. Very recently the first author [6,
7] obtained CLT and ASCLT for the product of some general partial sums.
The studies on the products of partial sums are usually concentrated on the
classic limiting theory, such as, CLT, ASCLT, LIL. The main purpose of this short
note is to establish a exponential convergence theorem for the product of sums of
i.i.d. positive random variables.
2. Main results
2.1. A moderate deviation principle for the weighted sums
In this subsection, we establish a moderate deviation principle for the weighted
sums which will play a key role in proving our main result.
Lemma 2.1. Let (Yn )n≥1 be a sequence of i.i.d. positive random variables with
EY1 = 0 and E(Y12 ) = 1. Assume that the sequence of positive numbers (bn ) is the
moderate deviation scale satisfying
bn → ∞,
bn log n
√
→ 0, as n → ∞.
n
If we suppose that the following exponential integrability condition holds: there
exists a positive number δ such that
E exp (δ|Y1 |) < ∞,
(2.1)
Exponential convergence rate for products of sums
253
then for any r > 0,
¯ n
¯
µ
¶
¯
1
1 ¯¯X
r2
¯
√
log
P
b
Y
≥
r
=− ,
i,n i ¯
¯
2
n→∞ bn
2
bn 2n i=1
lim
Pn
where bi,n = k=i k1 , 1 ≤ i ≤ n.
Proof. For any λ ∈ R, considering the following logarithmic moment generating
function
µ
¶
n
λ X
√
Λn (λ) := log E exp
bi,n Yi
bn 2n i=1
by the Gärtner-Ellis theorem [2, 12], we need to calculate the following limit,
Λ(λ) := lim
n→∞
1
Λn (b2n λ).
b2n
From the independence, Taylor formula and the condition (2.1), for all n large
enough, we obtain
µ
¶
n
1
1
λbn X
2
√
λ)
=
b
Y
Λ
(b
log
E
exp
i,n i
n n
b2n
b2n
2n i=1
µ
¶
n
λbn
1 X
log E exp √ bi,n Yi
= 2
bn i=1
2n
µ 2 2 ¶¶
µ
n
bn bi,n
1 X
λbn
λ2 b2n 2
2
= 2
log 1 + √ bi,n EYi +
bi,n EYi + o
.
bn i=1
4n
n
2n
Since EYi = 0, EYi2 = 1, and the following fact
n
X
b2i,n
i=1
n
n
n k−1
X
X
X
X1
k−1
1
= b1,n + 2
= 2n − b1,n = 2n −
,
= b1,n + 2
k
k
i
i=1
i=1
k=2
k=2
we have
µ
¶
n
n
X
1
λ2 X 2
λ2
1
λ2
2
lim 2 Λn (bn λ) = lim
bi,n = lim
2n −
=
.
n→∞ bn
n→∞ 4n
n→∞ 4n
i
2
i=1
i=1
By the Gärtner-Ellis theorem, the desired result can be obtained.
2.2. Moderate deviation principle for the product of sums
Theorem 2.2. Let (Xn )n≥1 be a sequence of i.i.d. positive random variables.
Denote µ = E(X1 ) > 0, the coefficient of variation γ = σ/µ, where σ 2 = V ar(X1 ),
and Sk = X1 + · · · + Xk , k = 1, 2, . . . In addition, assume that the sequence of
positive numbers (bn ) is the moderate deviation scale satisfying
bn → ∞,
bn log n
√
→ 0,
n
as n → ∞.
254
Yu Miao, Bin Qian
If there exists a sequence of positive real numbers (αn ) such that
√
αn
αn
n log n
√ → ∞,
αn → ∞,
→ 0,
→0
n
αn bn
bn n
(2.2)
and for all t > 0
¯ αn
¯
µ
¶
Sk ¯¯
1
1 ¯¯X
√
lim
log P
log
≥ t = −∞,
n→∞ b2
k ¯
bn n ¯ i=1
n
(2.3)
then we have for any r ≥ 1,
1
lim
log P
n→∞ b2
n
µµ Qn
k=1 Sk
n!µn
¶
1√
γbn 2n
¶
≥r
=−
(log r)2
;
2
(2.4)
=−
(log r)2
.
2
(2.5)
and for any 0 < r < 1,
1
lim
log P
n→∞ b2
n
µµ Qn
k=1 Sk
n!µn
¶
1√
γbn 2n
¶
≤r
Proof. Without loss of generality, let µ = 1, σ 2 = 1, then γ = 1. Let Ck =
Sk /k, k = 1, 2 · · · . For any r > 0, 0 < ε < 1/2, it follows that
µ
¶
n
1 X
√
P
log(Ck ) ≥ r
bn 2n k=1
µ
¶
n
1 X
√
=P
log(Ck ) ≥ r, max |Ck − 1| ≥ ε
αn ≤k≤n
bn 2n k=1
µ
¶
n
1 X
√
+P
log(Ck ) ≥ r, max |Ck − 1| < ε
αn ≤k≤n
bn 2n k=1
=: An + Bn .
(2.6)
By the comparison
µ
P max
inequality in [8, Corollary 4], for any ε > 0, it is obvious that
¯
¯
¯ k
¯
¶
µ
¶
¯ Sk
¯
¯X
¯
¯
¯
¯
¯
− 1¯ ≥ ε ≤ P max ¯ (Xi − 1)¯ ≥ αn ε
αn ≤k≤n¯ k
αn ≤k≤n
i=1
¯
µ¯X
¶
¯ n
¯
¯
¯
≤ cP ¯ (Xi − 1)¯ ≥ αn ε/c
µ
= cP
i=1
¯ n
¯
¶
¯
αn ε
1 ¯¯X
√ ¯ (Xi − 1)¯¯ ≥ √
,
bn n i=1
bn nc
(2.7)
where c > 0 is a constant. From the assumption bnα√n n → ∞ (n → ∞) and the
classic moderate deviation principle (cf. [2, 12]), it follows that for any ε > 0,
¯
¯
¶
µ
¯
¯ Sk
1
− 1¯¯ ≥ ε = −∞.
(2.8)
lim 2 log P max ¯¯
n→∞ bn
αn ≤k≤n k
255
Exponential convergence rate for products of sums
From this we know that the term An in (2.6) is negligible in the sense of the moderate deviation principle. To estimate the term Bn , we will expand the logarithm:
x2
log(1 + x) = x − (1+θx)
2 , where θ ∈ (0, 1) depends on x ∈ (−1/2, 1/2). Let En
denote the event {maxαn ≤k≤n |Ck − 1| < ε}, thus
³ 1
´
n
P
√
Bn = P
log(Ck ) ≥ r, maxαn ≤k≤n |Ck − 1| < ε
bn 2n k=1
³ 1 ³P
´
´
αn
n
n
P
P
(Ck − 1)2
√
=P
log(Ck ) +
(Ck − 1) −
≥
r,
E
n
2
bn 2n k=1
k=αn +1
k=αn +1 (1 + θk (Ck − 1))
h P
³ 1 ³P
i
´
´
αn
n
n
P
(Ck − 1)2
√
log(Ck ) +
(Ck − 1) −
=P
I
≥
r
E
n
2
bn 2n k=1
k=αn +1
k=αn +1 (1 + θk (Ck − 1))
³ ´
´
´
³ 1 ³P
αn
n
P
√
log Ck +
(Ck − 1) ≥ r, Enc
−P
bn 2n k=1
k=αn +1
=: Dn − Fn .
By the same reason as for the term An , we know that the term Fn is also negligible
in the sense of the moderate
¯Pαn deviation
¡ ¢¯ principle. Furthermore, by the condition
1
¯
(2.3), the term b √
log
Ck ¯ is negligible with respect to the moderate
k=1
n 2n
¯Pαn
¯
1
¯
(Ck − 1)¯ can
deviation principle. Similarly as for (2.7), we know that √
bn
2n
k=1
be neglected in the sense of the moderate deviation principle, so, from Lemma 2.1,
we have
µ
¶
n
X
1
1
r2
√
lim 2 log P
(Ck − 1) ≥ r = − .
n→∞ bn
2
bn 2n
k=αn +1
Next if we can prove the claim: for any ε > 0,
µ
· X
¸
¶
n
1
(Ck − 1)2
1
√
≥
I
ε
= −∞,
lim 2 log P
En
n→∞ bn
(1 + θk (Ck − 1))2
bn n
(2.9)
k=αn +1
then the desired results can be obtained. Note that for |x| < 1/2 and any θk ∈ (0, 1),
2
it follows that (1+θxk x)2 ≤ 4x2 . Therefore we have
µ
· X
¸
¶
µ
¶
n
n
X
1
(Ck − 1)2
1
2
√
√
P
I
≥
4ε
≤
P
(C
−1)
≥
ε
.
En
k
(1 + θk (Ck − 1))2
bn n
bn n
k=αn +1
k=αn +1
By Theorem 15 and Lemma 5 in [9, Chapter III], for all n sufficiently large, it
follows that
¯ s
µ
¶
µ¯ X
¶
n
n
X
X
¯1 k
¯
1
εbn
2
√
P
(Ck − 1) ≥ ε ≤
P ¯¯
Xi − 1¯¯ ≥ √
k i=1
bn n
n
k=αn +1
k=αn +1
µ
¶
µ
¶
½
¾
n
−1
X
bn
kbn
αn bn
−c √
n
≤2
exp −c √
exp −c √
≤ 1−e
n
n
k=αn
½
½
¾
¾
αn bn
4
αn bn
8n
exp −c √
≤
≤
,
¡ b ¢2 exp −c √
bn
n
cbn
n
n
− 1 c√
c√
n
2
n
256
Yu Miao, Bin Qian
where c is a positive constant. Therefore we have
¶
¶
µ
µ
n
X
αn
1
1
1
4n
2
√
√
(C
−
1)
≥
ε
≤
−c
+
→ −∞.
log
P
log
k
b2n
cbn
bn n
bn n b2n
k=αn +1
Thus the claim (2.9) holds.
b2
Remark 2.3. If the sequence (bn ) satisfies lognn → ∞, then there exists
affirmatively a sequence (αn ) with the properties (2.2).
Remark 2.4. By the Jensen’s inequality, we have
µ
¶
αn
αn
k
Sk
Sk
Sk
1X
1 X
1 X
log
≤ log
, log
≥
log Xi , a.e.
αn
k
αn
k
k
k i=1
k=1
k=1
Hence, in order to make the condition (2.3) hold, it is sufficient to show the following
relations: for any t > 0,
µ
¶
µ
¶
αn
αn
√
√
bn nt
bn nt
Sk
1
1 X
1
1 X
αn
αn
log
P
≥
e
=
log
P
b
X
≥
e
→ −∞,
i,αn i
b2n
αn i=1 k
b2n
αn i=1
and
µ X
¶
αn
k
√
1X
1
log
P
−
(log
X
)
≥
tb
n
i
n
b2n
k i=1
k=1
µ X
¶
αn
√
1
= 2 log P −
bi,αn (log Xi ) ≥ tbn n → −∞,
bn
i=1
Pαn 1
where bi,αn = k=i
k.
Example 2.5. (Bounded random variables) Let (Xn ) be a sequence of i.i.d.
n
bounded random variables with a < X1 < b, where 0 < a < b < ∞. If log
b2n → 0
then the assumption (2.3) holds.
√
Noting E log X1 ≤ log EX1 = 0, for any t > 0, let tn = tbn n/(4|E log X1 |),
Ptn
Ptn 2
then by the Hoeffding’s inequality and the facts i=1 bi,tn = tn , i=1
bi,tn ∼ 2tn ,
there exists a constant c > 0 such that
µ
¶
µ X
√ ¶
tn
tn
k
1 X
Sk
1X
tbn n
√
P
log
≤ −t ≤ P −
(log Xi ) ≥
k
k i=1
2
bn n i=1
k=1
µ X
√ ¶
tn
√
tbn n
=P −
bi,tn (log Xi − E log Xi ) ≥
≤ e−cbn n (2.10)
4
i=1
and, by the inequality e−x ≤ 1 − x + 12 x2 , x > 0, for all n large enough
¶
µ
µ
√ ¶
αn
αn
X
X
tbn n
1
Sk
Sk
√
≤ −t ≤
≥
P
log
P − log
k
k
2αn
bn n i=t +1
n
k=tn +1
Exponential convergence rate for products of sums
257
µ
¶
√
tbn n
Sk
≥ 1 − e− 2αn
P 1−
k
k=tn +1
µ
√ ¶
αn
b2 n
X
−ctn n2
Sk
tbn n
αn
P 1−
≤
≥
≤ αn e
.
k
4αn
(2.11)
k=tn +1
=
αn
X
√
Hence if we take αn = bn n(log n)1/2 , then
¶
µ
tn
1
Sk
1 X
√
lim
log
≤ −t = −∞.
log P
n→∞ b2
k
bn n i=1
n
(2.12)
Moreover, by the inequality 1 + x ≤ ex , x ≥ 0, then
µ
¶
αn
√
bn nt
1
1 X
α
log P
bi,αn Xi ≥ e n
b2n
αn i=1
µ
√ ¶
αn
1
1 X
bn nt
→ −∞
≤ 2 log P
bi,αn (Xi − 1) ≥
bn
αn i=1
αn
by the Hoeffiding’s inequality again. So from the above discussions, the condition
(2.3) holds.
Example 2.6. (Exponential random variable) Let (Xn ) be a sequence of i.i.d.
n
exponential random variables with density function f (x) = e−x , x > 0. If log
b2n → 0,
by using the exponential inequalities in [9, Chapter III], it is not difficult to get
the inequalities (2.10)-(2.12), which yields the condition (2.3). So we omit these
proofs.
Acknowledgements. The authors are very grateful to the anonymous referees for their valuable comments which improve the presentation of this work.
REFERENCES
[1] B.C. Arnold, J.A. Villaseñr, The asymptotic distribution of sums of records, Extremes 1
(1998), 351–363.
[2] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, 2nd Ed., Springer,
New York 1998.
[3] K. Gonchigdanzan, G.A. Rempala, A note on the almost sure limit theorem for the product
of partial sums, Appl. Math. Lett. 19 (2006), 191–196.
[4] W. Huang, L.X. Zhang, A note on the invariance principle of the product of sums of random
variables, Electron. Comm. Probab. 12 (2007), 51–56.
[5] X.W. Lu, Y.C. Qi, A note on asymptotic distribution of products of sums, Statist. Probab.
Lett. 68 (2004), 407–413.
[6] Y. Miao, Central limit theorem and almost sure central limit theorem for the product of some
partial sums, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), 289–294.
[7] Y. Miao, An extension of almost sure central limit theory for the product of partial sums, J.
Dyn. Syst. Geom. Theor. 7 (2009), 49–60.
[8] S.J. Montgomery-Smith, Comparison of sums of independent identically distributed random
vectors, Probab. Math. Statist. 14 (1993), 281–285.
258
Yu Miao, Bin Qian
[9] V.V. Petrov, Sums of Independent Random Variables, Springer, New York, 1975.
[10] Y.C. Qi, Limit distributions for products of sums, Statist. Probab. Lett. 62 (2003), 93–100.
[11] G.A. Rempala, J. Wesolowski, Asymptotics for products of sums and U -statistics, Electron.
Comm. Probab. 7 (2002), 47–54.
[12] L.M. Wu, An introduction to large deviations, In: Several Topics in Stochastic Analysis (J.A.
Yan, S. Peng, S. Fang, L. Wu, Eds.), pp. 225–336, Academic Press of China, Beijing, 1997
(In Chinese)
(received 14.07.2009; in revised form 15.04.2010)
Yu Miao, College of Mathematics and Information Science, Henan Normal University, 453007
Henan, China.
E-mail: yumiao728@yahoo.com.cn
Bin Qian, Department of Mathematics, Changshu Institute of Technology, Changshu, 215500
Jiangsu, China.
E-mail: binqiancn@yahoo.com.cn