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Kolmogorov inequalities for the partial sum of independent Bernoulli random variables
ARTICLE IN PRESS
Statistics & Probability Letters 77 (2007) 1117–1122 www.elsevier.com/locate/stapro
Kolmogorov inequalities for the partial sum of independent Bernoulli random variables Petroula M. Mavrikiou Department of Business Administration, Frederick Institute of Technology, P.O. Box 24729, 1303 Nicosia, Cyprus Received 19 April 2006; received in revised form 21 December 2006; accepted 21 February 2007 Available online 12 March 2007
Abstract In this paper two Kolmogorov inequalities are presented for the sample average of independent (but not necessarily identically distributed) Bernoulli random variables. r 2007 Elsevier B.V. All rights reserved. Keywords: Kolmogorov inequalities; Bernoulli random variables
1. Introduction For a sequence of independent and identically distributed Bernoulli random variables X 1 ; X 2 ; . . . ; with EðX 1 Þ ¼ p, Kolmogorov (1963) provided the following inequality: 2 P sup jX¯ k pj4 p2e2n ð1Þ , kXn
where k 1X X¯ k ¼ X i; k i¼1
40.
Improvements, extensions and related results can be found in Hoeffding (1963), Young et al. (1987, 1988), Turner et al. (1992), Christofides (1991, 1994), Kambo and Kotz (1966) and Banjevic (1985). In this paper, two Kolmogorov inequalities are provided for the case of independent but not necessarily identically distributed Bernoulli random variables.
Tel.: +357 22 345159x132; fax: +357 22 438234.
E-mail address: bus.mp@fit.ac.cy. 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.02.001
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1118
2. Preliminaries The following results will be used: P Lemma 2.1. Let p1 ; . . . ; pn be positive real numbers and p¯ ¼ ð1=nÞ ni¼1 pi . Then for t40 n Y ðpi et þ 1 pi Þpð¯pet þ 1 p¯ Þn . i¼1
Proof. From the arithmetic–geometric mean inequality, we have " #n n n X Y 1 t t ðp e þ 1 pi Þ ðpi e þ 1 pi Þp n i i¼1 i¼1 ¼ ð¯pet þ 1 p¯ Þn :
&
The following result is due to Christofides (1994). Lemma 2.2. Let o12 and
p¯ þ 1 p¯ . gð¯p; Þ ¼ ð1 p¯ Þ ln þ ð¯p þ Þ ln p¯ 1 p¯
Then for p¯ þ o12 or 12 þ 13 p¯pp1 gð¯p; ÞX 12 lnð1 42 Þ. Lemma 2.3. Let x ¼ 2ð¯p þ Þ 1 and y ¼ 1 2¯p. Then, 1 X
1 ½x2r þ ð2r 1Þy2r þ 2rxy2r1 ¼ gð¯p; Þ, 2rð2r 1Þ r¼1
where gð¯p; Þ is the function defined in Lemma 2.2. Proof. We have p¯ þ ¼ ðx þ 1Þ=2 and p¯ ¼ ð1 yÞ=2. Then, gð¯p; Þ from Lemma 2.2 is 1x 1x 1þx 1þx ln þ ln . gð¯p; Þ ¼ 2 1þy 2 1y Using the Taylor series expansions lnð1 þ xÞ ¼
1 X xr ð1Þr1 r r¼1
and
lnð1 xÞ ¼
1 X xr r¼1
r
we have that " # # " 1 1 1 1 r r r X X ð1 xÞ xr X 1þx X y r1 y r1 x þ ð1Þ ð1Þ gð¯p; Þ ¼ þ 2 2 r r r r r¼1 r¼1 r¼1 r¼1 and after algebraic manipulations we arrive at gð¯p; Þ ¼
1 X
1 ½x2r þ ð2r 1Þy2r þ 2rxy2r1 : 2rð2r 1Þ r¼1
Lemma 2.4. Let n be a positive integer and x41. Then 22n2 ðx2n1 þ 1Þ ðx þ 1Þ2n1 40.
&
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1119
Proof. Let F ðxÞ ¼ 22n2 ðx2n1 þ 1Þ ðx þ 1Þ2n1 . Then F 0 ðxÞ ¼ 22n2 ð2n 1Þx2n2 ð2n 1Þðx þ 1Þ2n2 ¼ ð2n 1Þ½22n2 x2n2 ðx þ 1Þ2n2 ¼ ð2n 1Þ½ð2xÞ2n2 ðx þ 1Þ2n2 40
since x41.
Thus, F ðxÞ is an increasing function and F ðxÞ4F ð1Þ ¼ 0.
&
Lemma 2.5. Let yX1 and n ¼ 1; 2; . . . : Then HðyÞ ¼ y2n þ 2n 1 þ 2ny
4 ðy þ 1Þ2n X0. 22n
Proof. 4 H 0 ðyÞ ¼ 2ny2n1 þ 2n 2n 2nðy þ 1Þ2n1 2 4 22n 2n1 22n y ðy þ 1Þ2n1 ¼ 2n 2n þ 4 4 2 ¼ 232n n½22n2 y2n1 þ 22n2 ðy þ 1Þ2n1 ¼ 232n n½22n2 ðy2n1 þ 1Þ ðy þ 1Þ2n1 . By Lemma 2.4 H 0 ðyÞ40 implying that H is increasing and therefore HðyÞXHð1Þ ¼ 4n4X0.
&
Lemma 2.6. Let x; y be as in Lemma 2.3 and r ¼ 1; 2; . . . . Then x2r þ ð2r 1Þy2r þ 2rxy2r1 X4
x þ y2r 2
.
Proof. By Lemma 2.5 for cX1 c2r þ 2r 1 þ 2rc
4 ðc þ 1Þ2r X0. 22r
Take c ¼ x=y. Then 2r x2r x 4 x þ1 , þ 2r 1 þ 2r X 2r y 2 y y2r
x2r þ ð2r 1Þy2r þ 2rxy2r1 X4
x þ y2r 2
.
3. Main results
Theorem 3.1. Let Y 1 ; Y 2 ; . . . ; Y n be a sequence of independent Bernoulli random variables with EðY i Þ ¼ pi , i ¼ 1; . . . ; n and o12. Then, for p¯ þ o12 or p¯ X12 þ 13 , PðY¯ p¯ 4Þpð1 42 Þn=2 , P P where Y¯ ¼ ð1=nÞ ni¼1 Y i and p¯ ¼ ð1=nÞ ni¼1 pi .
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1120
Proof. Let s40. Then, PðY¯ p¯ 4Þ ¼ P½sðY¯ p¯ Þ40 ¯
pE½esðY ¯pÞ ¯
¼ esð¯pþÞ E½eðsY Þ
Pn ¼ esð¯pþÞ Eðesð1=nÞ i¼1 Y i Þ n Y Eðeðs=nÞY i Þ ¼ esð¯pþÞ i¼1 n Y ¼ esð¯pþÞ ðpi es=n þ 1 pi Þ i¼1
pesð¯pþÞ ð¯pes=n þ 1 p¯ Þn ¼ ef ðsÞ ,
ð1Þ
where f ðsÞ ¼ sð¯p þ Þ n lnð¯pes=n þ 1 p¯ Þ and the last inequality follows from Lemma 2.1. The function f is maximized at smax ¼ n ln½ð1 p¯ Þð¯p þ Þ=¯pð1 p¯ Þ and ð¯p þ Þð1 p¯ Þ 1 p¯ max f ðs Þ ¼ nð¯p þ Þ ln n ln p¯ ð1 p¯ Þ 1 p¯ p¯ þ 1 p¯ ¼ n ð1 p¯ Þ ln þ ð¯p þ Þ ln p¯ 1 p¯ ¼ ngð¯p; Þ. Thus, by Lemma 2.2 n f ðsmax Þ ¼ ngð¯p; ÞX lnð1 42 Þ 2 and therefore PðY¯ p¯ 4Þpð1 42 Þn=2 . Notice that the restrictions on p¯ and arise from Lemma 2.2 which is due to Christofides (1994). & The following theorem gives an exponential bound under different conditions on p¯ and . Theorem 3.2. Let Y 1 ; Y 2 ; . . . ; Y n be a sequence of independent Bernoulli random variables, with EðY i Þ ¼ pi , i ¼ 1; . . . ; n. Then for p¯ þ 412 or p¯ o12 and 8o1, 4 2 =4
2
PðY¯ p¯ 4Þpenð2 þð1=3Þ e Þ , P P where Y¯ ¼ ð1=nÞ ni¼1 Y i and p¯ ¼ ð1=nÞ ni¼1 pi . Proof. From the proof of Theorem 3.1 PðY¯ p¯ 4Þpengð¯p;Þ , where p¯ þ 1 p¯ gð¯p; Þ ¼ ð1 p¯ Þ ln . þ ð¯p þ Þ ln p¯ 1 p¯
By Lemma 2.3 gð¯p; Þ ¼
1 X
1 ½x2r þ ð2r 1Þy2r þ 2rxy2r1 , 2rð2r 1Þ r¼1
where x and y are positive values, imposing the restrictions on p¯ and .
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1121
Now, since x þ y ¼ 2 and using Lemma 2.6 we have 2r 1 X 1 2 4 gð¯p; ÞX 2rð2r 1Þ 2 r¼1 ¼
1 X
22r rð2r 1Þ r¼1
¼ 22 þ 4
1 X 2ðÞ2r4 rð2r 1Þ r¼2
X22 þ 24
1 X
ð2 Þr2 ðr 2Þ! r¼2 6:4 r2
since rð2r 1Þp6:4r2 ðr 2Þ! for rX2. The last inequality is clearly true for r ¼ 2, and can be established (e.g., by induction) for r42. Then, 1 2 r2 X 1 2 41 gð¯p; ÞX2 þ 2 6 r¼2 4 ðr 2Þ! 1 2 ¼ 22 þ 4 e =4 . 3 Thus 2
4
PðY¯ p¯ 4Þpenð2 þð1=3Þ :e and the proof is complete.
2 =4
Þ
&
Theorem 3.3. Let Y 1 ; Y 2 ; . . . ; be a sequence of independent Bernoulli random variables with EðY i Þ ¼ pi . Then, (1) 8o12 and for p¯ k þ o12 or p¯ k X12 þ 13 , P supðY¯ k p¯ k Þ4 pð1 42 Þn=2 ;
n ¼ 0; 1; 2; . . .
kXn
while (2) 8o1 and for p¯ k þ 412 or p¯ k o12, 2 4 2 =4 P supðY¯ k p¯ k Þ4 penð2 þð1=3Þ e Þ ;
n ¼ 0; 1; 2; . . . ,
kXn
P P where Y¯ k ¼ ð1=kÞ ki¼1 Y i and p¯ k ¼ ð1=kÞ ki¼1 pi . Proof. Using Lemma 1 of Turner et al. (1995) we can arrive at inequality (1) of Theorem 3.1 having the quantity P½supkXn ðY¯ k p¯ k Þ4 as our left-hand side. Then, one can follow the same steps of Theorems 3.1 and 3.2 to reach the right-hand side, and the proof is complete. & Remarks. (1) Theorem 3.3 provides sharper bounds than that of the main result of Turner et al. (1995), under of course restrictions on p¯ and . (2) Theorem 3.1 is an extension of Corollary 3.2 of Christofides (1994).
References Banjevic, D., 1985. On a Kolmogorov inequality. Theory Probab. Appl. 29, 391–394. Christofides, T.C., 1991. Probability inequalities with exponential bounds for U-statistics. Statist. Probab. Lett. 12, 257–261.
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P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
Christofides, T.C., 1994. A Kolmogorov inequality for U-statistics based on Bernoulli kernels. Statist. Probab. Lett. 21, 357–362. Hoeffding, W., 1963. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30. Kambo, N.S., Kotz, S., 1966. On exponential bounds for binomial probabilities. Ann. Inst. Statist. Math. 18, 277–287. Kolmogorov, A.N., 1963. On tables of random numbers. Sankhya¯ Ser. A 25, 369–376. Turner, D.W., Young, D.M., Seaman, J.W., 1992. Improved Kolmogorov inequalities for the binomial distribution. Statist. Probab. Lett. 13, 223–227. Turner, D.W., Young, D.M., Seaman, J.W., 1995. A Kolmogorov inequality for the sum of independent Bernoulli random variables with unequal means. Statist. Probab. Lett. 23, 243–245. Young, D.M., Seaman, J.W., Marco, V.R., 1987. A note on a Kolmogorov inequality. Statist. Probab. Lett. 5, 217–218. Young, D.M., Turner, D.W., Seaman, J.W., 1988. A note on a Kolmogorov inequality for the Binomial distribution. Theory Probab. Appl. 33, 747–749.
Statistics & Probability Letters 77 (2007) 1117–1122 www.elsevier.com/locate/stapro
Kolmogorov inequalities for the partial sum of independent Bernoulli random variables Petroula M. Mavrikiou Department of Business Administration, Frederick Institute of Technology, P.O. Box 24729, 1303 Nicosia, Cyprus Received 19 April 2006; received in revised form 21 December 2006; accepted 21 February 2007 Available online 12 March 2007
Abstract In this paper two Kolmogorov inequalities are presented for the sample average of independent (but not necessarily identically distributed) Bernoulli random variables. r 2007 Elsevier B.V. All rights reserved. Keywords: Kolmogorov inequalities; Bernoulli random variables
1. Introduction For a sequence of independent and identically distributed Bernoulli random variables X 1 ; X 2 ; . . . ; with EðX 1 Þ ¼ p, Kolmogorov (1963) provided the following inequality: 2 P sup jX¯ k pj4 p2e2n ð1Þ , kXn
where k 1X X¯ k ¼ X i; k i¼1
40.
Improvements, extensions and related results can be found in Hoeffding (1963), Young et al. (1987, 1988), Turner et al. (1992), Christofides (1991, 1994), Kambo and Kotz (1966) and Banjevic (1985). In this paper, two Kolmogorov inequalities are provided for the case of independent but not necessarily identically distributed Bernoulli random variables.
Tel.: +357 22 345159x132; fax: +357 22 438234.
E-mail address: bus.mp@fit.ac.cy. 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.02.001
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1118
2. Preliminaries The following results will be used: P Lemma 2.1. Let p1 ; . . . ; pn be positive real numbers and p¯ ¼ ð1=nÞ ni¼1 pi . Then for t40 n Y ðpi et þ 1 pi Þpð¯pet þ 1 p¯ Þn . i¼1
Proof. From the arithmetic–geometric mean inequality, we have " #n n n X Y 1 t t ðp e þ 1 pi Þ ðpi e þ 1 pi Þp n i i¼1 i¼1 ¼ ð¯pet þ 1 p¯ Þn :
&
The following result is due to Christofides (1994). Lemma 2.2. Let o12 and
p¯ þ 1 p¯ . gð¯p; Þ ¼ ð1 p¯ Þ ln þ ð¯p þ Þ ln p¯ 1 p¯
Then for p¯ þ o12 or 12 þ 13 p¯pp1 gð¯p; ÞX 12 lnð1 42 Þ. Lemma 2.3. Let x ¼ 2ð¯p þ Þ 1 and y ¼ 1 2¯p. Then, 1 X
1 ½x2r þ ð2r 1Þy2r þ 2rxy2r1 ¼ gð¯p; Þ, 2rð2r 1Þ r¼1
where gð¯p; Þ is the function defined in Lemma 2.2. Proof. We have p¯ þ ¼ ðx þ 1Þ=2 and p¯ ¼ ð1 yÞ=2. Then, gð¯p; Þ from Lemma 2.2 is 1x 1x 1þx 1þx ln þ ln . gð¯p; Þ ¼ 2 1þy 2 1y Using the Taylor series expansions lnð1 þ xÞ ¼
1 X xr ð1Þr1 r r¼1
and
lnð1 xÞ ¼
1 X xr r¼1
r
we have that " # # " 1 1 1 1 r r r X X ð1 xÞ xr X 1þx X y r1 y r1 x þ ð1Þ ð1Þ gð¯p; Þ ¼ þ 2 2 r r r r r¼1 r¼1 r¼1 r¼1 and after algebraic manipulations we arrive at gð¯p; Þ ¼
1 X
1 ½x2r þ ð2r 1Þy2r þ 2rxy2r1 : 2rð2r 1Þ r¼1
Lemma 2.4. Let n be a positive integer and x41. Then 22n2 ðx2n1 þ 1Þ ðx þ 1Þ2n1 40.
&
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1119
Proof. Let F ðxÞ ¼ 22n2 ðx2n1 þ 1Þ ðx þ 1Þ2n1 . Then F 0 ðxÞ ¼ 22n2 ð2n 1Þx2n2 ð2n 1Þðx þ 1Þ2n2 ¼ ð2n 1Þ½22n2 x2n2 ðx þ 1Þ2n2 ¼ ð2n 1Þ½ð2xÞ2n2 ðx þ 1Þ2n2 40
since x41.
Thus, F ðxÞ is an increasing function and F ðxÞ4F ð1Þ ¼ 0.
&
Lemma 2.5. Let yX1 and n ¼ 1; 2; . . . : Then HðyÞ ¼ y2n þ 2n 1 þ 2ny
4 ðy þ 1Þ2n X0. 22n
Proof. 4 H 0 ðyÞ ¼ 2ny2n1 þ 2n 2n 2nðy þ 1Þ2n1 2 4 22n 2n1 22n y ðy þ 1Þ2n1 ¼ 2n 2n þ 4 4 2 ¼ 232n n½22n2 y2n1 þ 22n2 ðy þ 1Þ2n1 ¼ 232n n½22n2 ðy2n1 þ 1Þ ðy þ 1Þ2n1 . By Lemma 2.4 H 0 ðyÞ40 implying that H is increasing and therefore HðyÞXHð1Þ ¼ 4n4X0.
&
Lemma 2.6. Let x; y be as in Lemma 2.3 and r ¼ 1; 2; . . . . Then x2r þ ð2r 1Þy2r þ 2rxy2r1 X4
x þ y2r 2
.
Proof. By Lemma 2.5 for cX1 c2r þ 2r 1 þ 2rc
4 ðc þ 1Þ2r X0. 22r
Take c ¼ x=y. Then 2r x2r x 4 x þ1 , þ 2r 1 þ 2r X 2r y 2 y y2r
x2r þ ð2r 1Þy2r þ 2rxy2r1 X4
x þ y2r 2
.
3. Main results
Theorem 3.1. Let Y 1 ; Y 2 ; . . . ; Y n be a sequence of independent Bernoulli random variables with EðY i Þ ¼ pi , i ¼ 1; . . . ; n and o12. Then, for p¯ þ o12 or p¯ X12 þ 13 , PðY¯ p¯ 4Þpð1 42 Þn=2 , P P where Y¯ ¼ ð1=nÞ ni¼1 Y i and p¯ ¼ ð1=nÞ ni¼1 pi .
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1120
Proof. Let s40. Then, PðY¯ p¯ 4Þ ¼ P½sðY¯ p¯ Þ40 ¯
pE½esðY ¯pÞ ¯
¼ esð¯pþÞ E½eðsY Þ
Pn ¼ esð¯pþÞ Eðesð1=nÞ i¼1 Y i Þ n Y Eðeðs=nÞY i Þ ¼ esð¯pþÞ i¼1 n Y ¼ esð¯pþÞ ðpi es=n þ 1 pi Þ i¼1
pesð¯pþÞ ð¯pes=n þ 1 p¯ Þn ¼ ef ðsÞ ,
ð1Þ
where f ðsÞ ¼ sð¯p þ Þ n lnð¯pes=n þ 1 p¯ Þ and the last inequality follows from Lemma 2.1. The function f is maximized at smax ¼ n ln½ð1 p¯ Þð¯p þ Þ=¯pð1 p¯ Þ and ð¯p þ Þð1 p¯ Þ 1 p¯ max f ðs Þ ¼ nð¯p þ Þ ln n ln p¯ ð1 p¯ Þ 1 p¯ p¯ þ 1 p¯ ¼ n ð1 p¯ Þ ln þ ð¯p þ Þ ln p¯ 1 p¯ ¼ ngð¯p; Þ. Thus, by Lemma 2.2 n f ðsmax Þ ¼ ngð¯p; ÞX lnð1 42 Þ 2 and therefore PðY¯ p¯ 4Þpð1 42 Þn=2 . Notice that the restrictions on p¯ and arise from Lemma 2.2 which is due to Christofides (1994). & The following theorem gives an exponential bound under different conditions on p¯ and . Theorem 3.2. Let Y 1 ; Y 2 ; . . . ; Y n be a sequence of independent Bernoulli random variables, with EðY i Þ ¼ pi , i ¼ 1; . . . ; n. Then for p¯ þ 412 or p¯ o12 and 8o1, 4 2 =4
2
PðY¯ p¯ 4Þpenð2 þð1=3Þ e Þ , P P where Y¯ ¼ ð1=nÞ ni¼1 Y i and p¯ ¼ ð1=nÞ ni¼1 pi . Proof. From the proof of Theorem 3.1 PðY¯ p¯ 4Þpengð¯p;Þ , where p¯ þ 1 p¯ gð¯p; Þ ¼ ð1 p¯ Þ ln . þ ð¯p þ Þ ln p¯ 1 p¯
By Lemma 2.3 gð¯p; Þ ¼
1 X
1 ½x2r þ ð2r 1Þy2r þ 2rxy2r1 , 2rð2r 1Þ r¼1
where x and y are positive values, imposing the restrictions on p¯ and .
ARTICLE IN PRESS P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
1121
Now, since x þ y ¼ 2 and using Lemma 2.6 we have 2r 1 X 1 2 4 gð¯p; ÞX 2rð2r 1Þ 2 r¼1 ¼
1 X
22r rð2r 1Þ r¼1
¼ 22 þ 4
1 X 2ðÞ2r4 rð2r 1Þ r¼2
X22 þ 24
1 X
ð2 Þr2 ðr 2Þ! r¼2 6:4 r2
since rð2r 1Þp6:4r2 ðr 2Þ! for rX2. The last inequality is clearly true for r ¼ 2, and can be established (e.g., by induction) for r42. Then, 1 2 r2 X 1 2 41 gð¯p; ÞX2 þ 2 6 r¼2 4 ðr 2Þ! 1 2 ¼ 22 þ 4 e =4 . 3 Thus 2
4
PðY¯ p¯ 4Þpenð2 þð1=3Þ :e and the proof is complete.
2 =4
Þ
&
Theorem 3.3. Let Y 1 ; Y 2 ; . . . ; be a sequence of independent Bernoulli random variables with EðY i Þ ¼ pi . Then, (1) 8o12 and for p¯ k þ o12 or p¯ k X12 þ 13 , P supðY¯ k p¯ k Þ4 pð1 42 Þn=2 ;
n ¼ 0; 1; 2; . . .
kXn
while (2) 8o1 and for p¯ k þ 412 or p¯ k o12, 2 4 2 =4 P supðY¯ k p¯ k Þ4 penð2 þð1=3Þ e Þ ;
n ¼ 0; 1; 2; . . . ,
kXn
P P where Y¯ k ¼ ð1=kÞ ki¼1 Y i and p¯ k ¼ ð1=kÞ ki¼1 pi . Proof. Using Lemma 1 of Turner et al. (1995) we can arrive at inequality (1) of Theorem 3.1 having the quantity P½supkXn ðY¯ k p¯ k Þ4 as our left-hand side. Then, one can follow the same steps of Theorems 3.1 and 3.2 to reach the right-hand side, and the proof is complete. & Remarks. (1) Theorem 3.3 provides sharper bounds than that of the main result of Turner et al. (1995), under of course restrictions on p¯ and . (2) Theorem 3.1 is an extension of Corollary 3.2 of Christofides (1994).
References Banjevic, D., 1985. On a Kolmogorov inequality. Theory Probab. Appl. 29, 391–394. Christofides, T.C., 1991. Probability inequalities with exponential bounds for U-statistics. Statist. Probab. Lett. 12, 257–261.
ARTICLE IN PRESS 1122
P.M. Mavrikiou / Statistics & Probability Letters 77 (2007) 1117–1122
Christofides, T.C., 1994. A Kolmogorov inequality for U-statistics based on Bernoulli kernels. Statist. Probab. Lett. 21, 357–362. Hoeffding, W., 1963. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30. Kambo, N.S., Kotz, S., 1966. On exponential bounds for binomial probabilities. Ann. Inst. Statist. Math. 18, 277–287. Kolmogorov, A.N., 1963. On tables of random numbers. Sankhya¯ Ser. A 25, 369–376. Turner, D.W., Young, D.M., Seaman, J.W., 1992. Improved Kolmogorov inequalities for the binomial distribution. Statist. Probab. Lett. 13, 223–227. Turner, D.W., Young, D.M., Seaman, J.W., 1995. A Kolmogorov inequality for the sum of independent Bernoulli random variables with unequal means. Statist. Probab. Lett. 23, 243–245. Young, D.M., Seaman, J.W., Marco, V.R., 1987. A note on a Kolmogorov inequality. Statist. Probab. Lett. 5, 217–218. Young, D.M., Turner, D.W., Seaman, J.W., 1988. A note on a Kolmogorov inequality for the Binomial distribution. Theory Probab. Appl. 33, 747–749.