Properties of moments of random variables

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1 Properties of moments of rom variables Jean-Marie Dufour Université de Montréal First version: May 1995 Revised: January 23 This version: January 14, 23 Compiled: January 14, 23, 1:5pm This work was supported by the Canada Research Chair Program (Chair in Econometrics, Université de Montréal), the Alexer-von-Humboldt Foundation (Germany), the Canadian Network of Centres of Excellence [program on Mathematics of Information Technology Complex Systems (MITACS)], the Canada Council for the Arts (Killam Fellowship), the Natural Sciences Engineering Research Council of Canada, the Social Sciences Humanities Research Council of Canada, the Fonds FCAR (Government of Québec). Canada Research Chair Holder (Econometrics). Centre interuniversitaire de recherche en analyse des organisations (CIRANO), Centre interuniversitaire de recherche en économie quantitative (CIREQ), Département de sciences économiques, Université de Montréal. Mailing address: Département de sciences économiques, Université de Montréal, C.P succursale Centre-ville, Montréal, Québec, Canada H3C 3J7. TEL: ; FAX: ; jean.marie.dufour@umontreal.ca. Web page:

2 Contents List of Definitions, Propositions Theorems ii 1. Existence of moments 1 2. Moment inequalities 1 3. Markov-type inequalities 2 4. Moments behavior of tail areas 4 5. Moments of sums of rom variables 6 6. Proofs 9 List of Definitions, Propositions Theorems 1.1 Proposition : Existence of absolute ordinary moments Proposition : Monotonicity of L r Proposition : c r -inequality Proposition : Mean form of c r -inequality Proposition : Closure of L r Proposition : Hölder inequality Proposition : Cauchy-Schwarz inequality Proposition : Minkowski inequality Proposition : Moment monotonicity Theorem : Liapunov theorem Proposition : Lower bounds on the moments of a sum Proposition : Jensen inequality Proposition : Markov inequalities Proposition : Chebyshev inequalities Proposition : Refined Markov inequalities Proposition : Moment existence tail area decay Proposition : Distribution decomposition of r-moments Proposition : Distribution decomposition of the first absolute moment Corollary : Moment-tail area inequalities Proposition : Mean-tail area inequalities Proposition : Bounds on the absolute moments of a sum of rom variables 6 i

3 5.2 Proposition : Minkowski inequality for n variables Proposition : Bounds on the absolute moments of a sum of rom variables under conditional symmetry Proposition : Bounds on the absolute moments of a sum of rom variables under martingale condition Proposition : Bounds on the absolute moments of a sum of rom variables under two-sided martingale condition Proposition : Bounds on the absolute moments of a sum of independent rom variables Proof of Theorem Proof of Proposition ii

4 Let X Y be real rom variables, let r s be real positive constants (r >, s > ). The distribution functions of X Y are denoted F X (x) =P[X x] F Y (x) =P[Y x]. 1. Existence of moments 1.1 EXISTENCE OF ABSOLUTE AND ORDINARY MOMENTS. E( X ) always exists in the extended real numbers R R { } { } E( X ) [, ]; i.e., either E( X ) is a non-negative real number or E( X ) =. 1.2 E(X) exists is finite E( X ) <. 1.3 E( X ) < E(X) E( X ) <. 1.4 If <r s, then E( X s ) < E( X r ) <. (1.1) 1.5 MONOTONICITY OF L r. L s L r for <s r. 1.6 E( X r ) < E(X k ) exists is finite for all integers k such that <k r. 2. Moment inequalities 2.1 c r -INEQUALITY. E( X + Y r ) c r [E( X r )+E( Y r )] (2.1) where c r =1, if <r 1, =2 r 1, if r>1. (2.2) 2.2 MEAN FORM OF c r -INEQUALITY. E( 1 2 (X + Y ) r ) ( 1 2) r [E( X r )+E( Y r )], if <r 1, 1 2 [E( X r )+E( Y r )], if r>1. (2.3) 2.3 CLOSURE OF L r. Let a b be real numbers. Then X L r Y L r ax + by L r. (2.4) 1

5 2.4 HÖLDER INEQUALITY. Ifr>1 1 r + 1 s =1,then 2.5 CAUCHY-SCHWARZ INEQUALITY. 2.6 MINKOWSKI INEQUALITY. If r 1, then E( XY ) [E( X r )] 1/r [E( Y s )] 1/s. (2.5) E( XY ) [E(X 2 )] 1/2 [E(Y 2 )] 1/2. (2.6) E( X + Y r ) 1/r [E( X r )] 1/r +[E( Y r )] 1/r. (2.7) 2.7 MOMENT MONOTONICITY. [E( X r )] 1/r is a non-decreasing function of r, i.e. <r s [E( X r )] 1/r [E( X s )] 1/s. (2.8) 2.8 Theorem LIAPUNOV THEOREM. log[e( X r )] is a convex function of r, i.e. for any λ [, 1], log[e( X λr+(1 λ)s )] λ log[e( X r )] + (1 λ)log[e( X s )]. (2.9) 2.9 LOWER BOUNDS ON THE MOMENTS OF A SUM. IfE( X r ) <, E( Y r ) < E(Y X) =, then E( X + Y r ) E( X r ), for r 1. (2.1) 2.1 JENSEN INEQUALITY. If g(x) is a convex function on R E( X ) <, then, for any constant c R, g(c) E[g(X EX + c)] (2.11), in particular, 3. Markov-type inequalities g(ex) E[g(X)]. (2.12) 3.1 MARKOV INEQUALITIES. Let g : R R be a function such that g(x) is a real rom variable, E( g(x) ) < P[ g(x) M] =1 (3.1) 2

6 where M [, ]. If g(x) is a non-decreasing function on R, then, for all a R, E [g (X)] g (a) M P[X a] E [g (X)] g (a). (3.2) If g(x) is a non-decreasing function on [, ) g(x) =g( x) for any x, then, for all a, E [g (X)] g (a) E [g (X)] P[ X a] (3.3) M g (a) where / CHEBYSHEV INEQUALITIES. If P[ X M] =1,whereM [, ], then, for all a, E ( X r ) a r P[ X a] E ( X r ). (3.4) M r a r 3.3 Theorem REFINED MARKOV INEQUALITIES. Let g : R R be a function such that g(x) is a real rom variable, E( g(x) ) < g(x) M U for x A U, (3.5) g(x) M L for x A L, (3.6) where M U, M L, A U A L. Let also C U (g, a) = g(x) df X (x), C L (g, a) = g(x) df X (x). (3.7) [a, ) (,a] (a) If g(x) is nondecreasing on [A U, ), then, for a A U, C U (g, a) P[X a] C U(g, a). (3.8) M U g (a) (b) If g(x) is nonincreasing on (, A L ], then, for a A L, C L (g, a) P[X a] C L(g, a). (3.9) M L g (a) (c) If g(x) is nondecreasing on [A U, ) nonincreasing on (, A L ], then, for a 3

7 max{ A U, A L }, P[ X a] C U(g, a) g (a) + C L(g, a) g ( a) C U(g, a)+c L (g, a) min{g (a),g( a)}, (3.1) P[ X a] C U(g, a) + C L(g, a) M U M L C U(g, a)+c L (g, a). (3.11) max{m U,M L } 4. Moments behavior of tail areas 4.1 Proposition MOMENT EXISTENCE AND TAIL AREA DECAY. Let r >. If E( X r ) <, then In particular, if E( X ) <, then lim x {xr P [X x]} = lim x { x r P [X x]} = lim {x r P [ X x]} =. (4.1) x lim {x P [X x]} = lim { x P [X x]} x x = lim {x P [ X x]} =. (4.2) x 4.2 Theorem DISTRIBUTION DECOMPOSITION OF r-moments. Forany r>, x r df X (x) =r x r 1 [1 F X (x)]dx, (4.3) E( X r ) = r = r x r 1 P( X x)dx x r 1 [1 F X (x)+f X ( x)]dx, (4.4) E( X r ) < x r 1 P( X x) is integrable on (, + ) 4

8 x r 1 [1 F X (x)+f X ( x)] is integrable on (, + ) x r 1 [1 F X (x)]dx < x r 1 F X (x)dx <. (4.5) 4.3 Proposition DISTRIBUTION DECOMPOSITION OF THE FIRST ABSOLUTE MOMENT. xdf X (x) = E X = [1 F X (x)]dx, (4.6) P( X x)dx, (4.7) E( X ) < P( X x) is integrable on (, + ) [1 F X (x)+f X ( x)] is integrable on (, + ) [1 F X (x)]dx < F X (x)dx <. (4.8) 4.4 Proposition MOMENT-TAIL AREA INEQUALITIES. Let g(x) be a nonnegative strictly increasing function on [, ) let g 1 (x) be the inverse function of g. Then, P [ X g 1 (n) ] E[g(X)] n=1 P [ X >g 1 (n) ]. (4.9) n= In particular, for any r>, P( X n 1/r ) E( X r ) P( X >n 1/r ) n=1 1+ n= P(X >n 1/r ). (4.1) 4.5 Corollary MEAN-TAIL AREA INEQUALITIES. IfX is a positive rom variable, n=1 P(X n) E(X) 1+ n=1 P(X >n). (4.11) n=1 5

9 5. Moments of sums of rom variables In this section, we consider a sequence X 1,...,X n of rom variables, study the moments of the corresponding sum average: S n = X i, Xn = S n /n. (5.1) 5.1 Proposition BOUNDS ON THE ABSOLUTE MOMENTS OF A SUM OF RANDOM VARI- ABLES. E( S n r ) n E( X i r ), if <r 1, n r 1 n (5.2) E( X i r ), if r>1, E( X n r ) ( ) 1 r E( X n i r ), if <r 1, E( X i r ), if r>1. 1 n 5.2 Proposition MINKOWSKI INEQUALITY FOR n VARIABLES. Ifr 1, then (5.3) [E( S n r )] 1/r [E( X i r )] 1/r (5.4) [E( X n r )] 1/r 1 n [E( X i r )] 1/r { 1 n 1/r. E( X i )} r (5.5) 5.3 Proposition BOUNDS ON THE ABSOLUTE MOMENTS OF A SUM OF RANDOM VARI- ABLES UNDER CONDITIONAL SYMMETRY. If the distribution of X k+1 given S i is symmetric about zero for k =1,..., n 1, E( X i r ) <,,...,n,then E( S n r ) E( X i r ) for 1 r 2, (5.6) 6

10 E( X n r ) with equality holding when r =2. ( ) r 1 E( X i r ) for 1 r 2, (5.7) n 5.4 Proposition BOUNDS ON THE ABSOLUTE MOMENTS OF A SUM OF RANDOM VARI- ABLES UNDER MARTINGALE CONDITION. If E( X i r ) <,,..., n,then Furthermore, for r =2, E(X k+1 S k )= a.s., k =1,...,n 1, (5.8) E( S n r ) 2 E( X n r ) 2 E( X i r ), for 1 r 2, (5.9) ( ) r 1 E( X i r ), for 1 r 2. (5.1) n E(S n 2 )= E(Xi 2 ). (5.11) 5.5 Proposition BOUNDS ON THE ABSOLUTE MOMENTS OF A SUM OF RANDOM VARI- ABLES UNDER TWO-SIDED MARTINGALE CONDITION. Let S m(k) = m+1, i k X i, 1 k m +1 n. (5.12) If E(X k S m(k) )= a.s., for 1 k m +1 n, (5.13) E( X i r ) <,,..., n,then E( S n r ) ( 2 1 ) E( X i r ), for 1 r 2, (5.14) n E( X n r ) ( ) r ( ) E( X i r ), for 1 r 2. (5.15) n n 7

11 5.6 Proposition BOUNDS ON THE ABSOLUTE MOMENTS OF A SUM OF INDEPEN- DENT RANDOM VARIABLES. Let the rom variables X 1,...,X n be independent with E(X i )= E( X i r ) <,,..., n, let If D(r) < 1 1 r 2, then D(r) =[13.52/(2.6π) r ] Γ (r)sin(rπ/2). (5.16) E( S n r ) [1 D(r)] 1 E( X i r ), (5.17) E( X n r ) ( ) r 1 [1 D(r)] 1 E( X i r ), for 1 r 2. (5.18) n 8

12 6. Proofs 1.1 to 3.2. See Loève (1977, Volume I, Sections , pp ). For Jensen inequality, see also Chow Teicher (1988, Section 4.3, pp ). Hannan (1985), Lehmann Shaffer (1988), Piegorsch Casella (1988) Khuri Casella (22) discussed conditions for the existence of the moments of 1/X See von Bahr Esseen (1965, Lemma 3). PROOF OF THEOREM 3.3 (a) For x a A U, we have g (x) g (a) g (x) M U, hence C U (g, a) = g (x) df X (x) g (a) df X (x) =g (a) P [X a] [a, ) [a, ) g (x) df X (x) M U P [X a], [a, ) from which we get the inequality C U (g, a) P [X a] C U (g, a). M U g (a) (b) For x a A L, we have g (x) g (a) g (x) M L, hence C L (g, a) = g (x) df X (x) g (a) df X (x) =g (a) P [X a] [,a) [,a) g (x) df X (x) M L P [X a] [,a) from which we get the inequality C L (g, a) P [X a] C L (g, a). M L g (a) (c) For a max ( A U, A L ), we have a A U a A L, hence P [ X a] = P [X a]+p [X a] C U (g, a) + C L (g, a) g (a) g(a) 9

13 C U (g, a)+c L (g, a) min {g ( a),g( a)} P [ X a] C U (g, a) + C L (g, a) M U M L C U (g, a)+c L (g, a). max (M U,M L ) 4.1 to 4.3. See Feller (1966, Section V.6, Lemma 1), Chung (1974, Exercises 17-18), Serfling (198, Section 1.14, pp ) Chow Teicher (1988, Section 4.3, pp ). For other inequalities involving absolute moments, the reader may consult Beesack (1984) See Chow Teicher (1988, Section 4.1, Corollary 3, p. 9) The inequality (4.11) is given by Chung (1974, Theorem 3.2.1) Serfling (198, Section 1.3, p. 12) See von Bahr Esseen (1965), Chung (1974, p. 48) Chow Teicher (1988, p. 18) See Chung (1974, p. 48). PROOF OF PROPOSITION 5.2 The first inequality follows by recursion on applying the Minkowski inequality for two variables. The first part of the second inequality is obtained by multiplying both sides of the first one by (1/n). The second part follows on observing that the function x 1/r is concave in x for x> when r> See von Bahr Esseen (1965, Theorem 1) See von Bahr Esseen (1965, Theorem 2) See von Bahr Esseen (1965, Theorem 3) See von Bahr Esseen (1965, Theorem 4). 1

14 References BEESACK, P. R. (1984): Inequalities for Absolute Moments of a Distribution: From Laplace to von Mises, Journal of Mathematical Analysis Applications, 98, CHOW, Y.S., AND H. TEICHER (1988): Probability Theory. Independence, Interchangeability, Martingales. Second Edition. Springer-Verlag, New York. CHUNG, K. L. (1974): A Course in Probability Theory. Academic Press, New York, second edn. FELLER, W. (1966): An Introduction to Probability Theory its Applications, Volume II. John Wiley & Sons, New York. HANNAN, E. J. (1985): Letter to the Editor: "Sufficient Necessary Conditions for Finite Negative Moments", The American Statistician, 39, 326. KHURI, A., AND G. CASELLA (22): The Existence of the First Negative Moment Revisited, The American Statistician, 56, LEHMANN, E.L., AND J. P. SHAFFER (1988): Inverted Distributions, The American Statistician, 42, LOÈVE, M. (1977): Probability Theory, Volumes I II. Springer-Verlag, New York, 4th edn. PIEGORSCH, W. W., AND G. CASELLA (1988): The Existence of the First Negative Moment, The American Statistician, 39, SERFLING, R. J. (198): Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York. VON BAHR, B., AND C.-G. ESSEEN (1965): Inequalities for the rth Absolute Moment of a Sum of Rom Variables, 1 r 2, The Annals of Mathematical Statistics, 36,