EXPLORING MODERN MATHEMATICS
CONTENTS SU?CIENT CONDITIONS FOR LORENZ ORDERING WITH COMMON ?NITE SUPPORT ERGODICITY AND STEADY STATE ANALYSIS FOR INTERFERENCE QUEUEING NETWORKS HOW STRONG CAN THE PARRONDO E?ECT BE? II BINARY RESPONSE MODELS COMPARISON USING THE ?-CHERNO?DIVERGENCE MEASURE AND EXPONENTIAL INTEGRAL FUNCTIONS NONLINEAR PARABOLIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS AND HARDY-LERAY TYPE INEQUALITIES BANACH SPACE VALUED WEAK SECOND ORDER STOCHASTIC PROCESSES ˆ ˆ EXPLICIT TRANSIENT PROBABILITIES OF VARIOUS MARKOV MODELS ON THE USE OF MARKOVIAN STICK-BREAKING PRIORS EULERIAN POLYNOMIALS AND QUASI-BIRTH-DEATH PROCESSES WITH TIME-VARYING-PERIODIC RATES RANDOM MEASURE ALGEBRAS FROM ADDITIVE TO SECOND-ORDER PROCESSES THE EXPONENTIAL-DUAL MATRIX METHOD: APPLICATIONS TO MARKOV CHAIN ANALYSIS TWO MOMENT CLOSURE TECHNIQUES FOR AN INTERACTING SPECIES MODEL SU?CIENT CONDITIONS FOR LORENZ ORDERING WITH COMMON ?NITE SUPPORT ABSTRACT ARNOLD AND GOKHALE (2017) PROVIDED A CHARACTERIZATION OF THE LORENZ INEQUALITY ORDER BETWEEN DISTRIBUTIONS WITH COMMON ?NITE SUPPORT IN THE MORE GENERAL LORENZ ORDER CONTEXT, A VARIETY OF PARTIAL ORDERS ARE OFTEN USED TO VERIFY THE EXISTENCE OF LORENZ ORDERING IN THIS PAPER WE INVESTIGATE WHETHER PARALLEL RESULTS CAN BE IDENTI?ED WITHIN THE COMMON ?NITE SUPPORT CONTEXT 1 INTRODUCTION THE LORENZ ORDER IS CLASSICALLY DE?NED ON THE CLASS OF ALL NON-NEGATIVE RANDOM VARIABLES WITH POSITIVE ?NITE EXPECTATIONS IN THIS PAPER OUR FOCUS IS ON THE LORENZ ORDER RESTRICTED TO A CLASS OF RANDOM VARIABLES WITH COMMON ?NITE SUPPORT IN PARTICULAR WE ARE CONCERNED WITH THE EXTENT TO WHICH CERTAIN SU?CIENT CONDITIONS FOR LORENZ ORDER IN THE GENERAL CONTEXT CONTINUE TO BE USEFUL IN THE RESTRICTED ?NITE SUPPORT SETTING WHILE PROGRESSIVE OR ROBIN HOOD TRANSFERS CAN BE USED TO CHAR- ACTERIZE LORENZ ORDER IN THE GENERAL SETTING, ARNOLD AND GOKHALE (2017) IDENTI?ED ANALOGOUS OPERATIONS THAT REDUCE INEQUALITY IN COMMON ?NITE SUPPORT CASES THESE SO-CALLED ROBIN HOOD EXCHANGES PLAY A PARALLEL ROLE IN THE ?NITE SUPPORT SETTING TO THAT PLAYED BY ROBIN HOOD TRANSFERS IN THE MORE GENERAL SETTING WE BEGIN OUR INVESTIGATION WITH A REVIEW OF NEEDED CONCEPTS DEALING WITH MAJORIZATION AND THE USUAL LORENZ ORDER 2 THE USUAL LORENZ ORDER AND THE ROLE OF ROBIN HOOD MORE DETAILED DISCUSSION OF THE TOPICS IN THIS SECTION MAY BE FOUND IN CHAPTER 17 OF MARSHALL, OLKIN AND ARNOLD (2011) CONSIDER A POPULATION OF N INDIVIDUALS, THE ITH MEMBER OF WHICH HAS INCOME OF XI INCOME UNITS A KEY DESIRABLE FEATURE OF SUITABLE INEQUALITY MEASURES OF SUCH POPULATIONS IS ONE ASSOCIATED WITH WHAT MAY BE CALLED A ROBIN HOOD TRANSFER ROBIN HOOD WAS KNOWN FOR TAKING MONEY FROM THE RICH AND GIVING IT TO THE POOR IT IS QUITE GENERALLY ACCEPTED THAT SUCH AN OPERATION WILL REDUCE INEQUALITY, AND ANY REASONABLE INEQUALITY MEASURE SHOULD BE REDUCED BY SUCH AN OPERATION IF ONE ACCEPTS THIS VIEW THEN ONE IS LED TO THE USE OF THE MAJORIZATION PARTIAL ORDER, DE?NED BY HARDY, LITTLEWOOD AND POLYA (1934), XI:N), K = 0, , N JOIN THESE N + 1 POINTS BY LINE SEGMENTS TO OBTAIN A CURVE CONNECTING THE ORIGIN WITH THE POINT (1, 1) THIS IS THE LORENZ CURVE CORRESPONDING TO THE VECTOR X, TO BE DENOTED BY LX(U) UNLESS ALL THE XI’S ARE EQUAL, THE LORENZ CURVE WILL BE CONVEX AND WILL LIE UNDER THE STRAIGHT LINE JOINING (0, 0) TO (1, 1) THE ASSOCIATED LORENZ ORDER RELATING VECTORS X AND Y, OF POSSIBLY DI?ERENT DIMENSIONS AND POSSIBLY DI?ERENT TOTALS, IS DENOTED BY X ?L Y AND IS DE?NED AS FOLLOWS DEFINITION 2 3 LORENZ ORDERING X ?L Y IF LX(U) ?LY(U) FOR ALL U ?[0, 1] LORENZ CURVES CAN CROSS ONCE OR SEVERAL TIMES SO THAT THE LORENZ ORDER IS ONLY A PARTIAL ORDER OBSERVE THAT IF X AND Y ARE OF THE SAME DIMENSION AND IF N I=1 XI = N I=1 YI THEN X ?L Y IF AND ONLY IF X ?Y THUS THE LORENZ ORDER APPEARS AS A NATURAL EXTENSION OF THE MAJORIZATION PARTIAL ORDER, ALLOWING US TO COMPARE POPULATIONS WITH DI?ERENT SIZES WITH DI?ERENT WEALTH TOTALS EXPLORING MODERN MATHEMATICS, BOOK X (Y)DY X (Y) = SUP{X : FX(X) ?Y}, 0 ?Y < 1, = SUP{X : FX(X) < 1}, Y = 1, IS THE RIGHT CONTINUOUS INVERSE DISTRIBUTION FUNCTION (OR QUANTILE FUNCTION) OF THE RANDOM VARIABLE X OBSERVE THAT IF X IS THE RANDOM VARIABLE ASSOCIATED WITH THE VECTOR X AS IN (2 2), THEN THE CORRESPONDING GASTWIRTH LORENZ CURVE LX(U) AND THE CURVE SUGGESTED BY LORENZ, LX(U), ARE IDENTICAL THE LORENZ ORDER CAN THEN BE EXTENDED TO ALLOW COMPARISON OF RANDOM VARIABLES AS FOLLOWS DEFINITION 2 5 FOR X, Y ?L+, WITH CORRESPONDING LORENZ CURVES LX AND LY , X IS LESS THAN Y IN THE LORENZ ORDER, WRITTEN AS X ?L Y IF LX(U) ?LY (U) FOR ALL U ?[0, 1] IT IS POSSIBLE TO PROVE A NATURAL EXTENSIONS OF THEOREMS 1, APPLYING TO THE LORENZ ORDER RATHER THAN BEING RESTRICTED TO MAJORIZATION THEOREM 2 6 FOR X, Y ?L+, X ?L Y IF AND ONLY IF E(G(X/E(X))) ? E(G(Y/E(Y ))) FOR EVERY CONTINUOUS CONVEX FUNCTION G SUCH THAT THE EXPECTATIONS EXIST THERE IS ANOTHER AVAILABLE CHARACTERIZATION OF THE LORENZ ORDER IN TERMS OF “ANGLE” FUNCTIONS (ORIGINALLY STATED BY HARDY, LITTLEWOOD AND POLYA IN THE CONTEXT OF MAJORIZATION) THUS THEOREM 2 7 SUPPOSE THAT X, Y ?L+ WITH E(X) = E(Y ) THEN X ?L Y IF AND ONLY IF E[(X ?C)+] ?E[(Y ?C)+] FOR EVERY C > 0 THIS RESULT WILL BE PARTICULARLY USEFUL WHEN WE TURN TO CONSIDER THE COMMON ?NITE SUPPORT CASE 3 OTHER PARTIAL ORDERS DE?NED ON L+ LET X, Y ?L WITH CORRESPONDING DISTRIBUTION FUNCTIONS FX AND FY STAR- SHAPED ORDERING OR, MORE BRIE?Y, STAR ORDERING IS DE?NED AS FOLLOWS Y (U) ARE AVAILAB1E IN CONVENIENT TRACTAB1E FORMS) 4 WHEN X AND Y HAVE COMMON ?NITE SUPPORT FOR A ?XED POSITIVE INTEGER N AND A ?XED SET OF N DISTINCT NUMBERS 0 < X1 < X2 < < XN?1 < XN, CONSIDER THE CLASS L(N) X OF ALL RANDOM VARIABLES WITH SUPPORT {X1, X2, , XN} A RANDOM VARIABLE X(P) IN THIS CLASS CAN BE ASSOCIATED WITH A PROBABILITY VECTOR P = (P1, P2, , PN) WHERE PI = P(X(P) = XI), I = 1, 2, , N HOW MUST TWO PROBABILITY VECTORS P AND Q BE RELATED IN ORDER THAT X(P) ?L X(Q) THE ?RST RESULT IN THIS DIRECTION IS PERHAPS SURPRISING THEOREM 4 1 FOR X(P), X(Q) ?L(N) X , IF X(P) ?L X(Q) THEN E(X(P)) = E(X(Q)) PROOF SUPPOSE THAT E(X(P)) > E(X(Q)) CONSIDER THE LOCAL BEHAVIOR OF THE CORRESPONDING LORENZ CURVES IN A NEIGHBORHOOD OF 0 AND IN A NEIGHBORHOOD OF 1 FIRST, IN A NEIGHBORHOOD OF 0, WE HAVE LX(P) < LX(Q), WHILE IN A NEIGHBORHOOD OF 1, WE HAVE LX(P) > LX(Q) BUT THIS IMPLIES THAT THE LORENZ CURVES MUST CROSS AND CANNOT BE NESTED ? I(P(1) I ?= P(2) I ) A ?NITE NUMBER OF SUCH EXCHANGES WILL BRING US TO P ARNOLD AND GOKHALE (2017) IDENTI?ED THE KEY ROLE OF SUCH EXCHANGES IN THE COMMON ?NITE SUPPORT SETTING, PARALLELING THE ROLE OF ROBIN HOOD TRANSFERS IN MA- JORIZATION SCENARIOS THUS WE HAVE X(Q)(U) IS NOT A NON-INCREASING FUNCTION OF U IN THIS CASE ALSO FOR U > U? THIS CONDITION IS READILY CHECKED BY CONSIDERING THE PARTIAL SUMS OF THE COORDINATES OF THE VECTORS P AND Q IT TURNS OUT THAT THE DENSITY CROSSING ORDERING, SU?CIENT FOR LORENZ ORDERING, IS THE MOST EASY TO CHECK ONCE MORE ASSUMING EQUAL MEANS, I E , E(X(P)) = E(X(Q)), TO ENSURE THAT X(P) ?L X(Q), IT IS SU?CIENT TO IDENTIFY TWO INTEGERS 1 < J1 < J2 < N SUCH THAT PJ ?QJ FOR J < J1 AND FOR J > J2, WHILE PJ ?QJ FOR J1 ?J ?J2 7 DISCUSSION EITHER SIGN CHANGE ORDERING OR DENSITY CROSSING ORDERING APPEAR TO BE THE TOOLS OF CHOICE TO VERIFY LORENZ ORDERING IN THE COMMON ?NITE SUPPORT CASE THE ROLE OF ROBIN HOOD EXCHANGES IS AN AID TO UNDERSTANDING LORENZ ORDERING IN THAT CONTEXT, BUT IDENTIFYING A SUITABLE EXCHANGE SEQUENCE IS NOT A CONVENIENT WAY TO CON?RM LORENZ DOMINANCE IT IS NOTEWORTHY THAT THE ACTUAL VALUES X1 < X2 < < XN INVOLVED IN THE DISCUSSION ARE CRUCIAL TO DETERMINE WHETHER E(X(P)) = E(X(Q)), BUT BEYOND THAT, DO NOT PLAY A ROLE IN DETERMINING LORENZ DOMINANCE REFERENCES [1] BARRY C ARNOLD AND D V GOKHALE, LORENZ ORDER WITH COMMON ?NITE SUPPORT, METRON