The jensen s measure is a riskadjusted performance measure that represents the average return on a portfolio or investment above or below that predicted by the capital asset. As the proof is very similar with the proof of theorem 2. The simplest form of jensens inequality is that if 4x is a convex function, and m is. Basic inequalities when working in the set of real numbers, we have a law of trichotomy. Jensen s inequality is an inequality involving convexity of a function. Consider the american derivative security with payoff g s k in period k. A story about pedro strong law of large numbers jensens inequality. A good explanation of jensens inequality should get the oh, duh reaction.
Lecture 32 strong law of large numbers and jensens. Jia studied the ndimensional \n1\ jensens inequality for gexpectations and got the result that the ndimensional \n1\ jensens inequality holds for gexpectations if and only if g is independent of y and linear with respect to z, in other words, the corresponding gexpectation must be linear. Measure theorylp spaces wikibooks, open books for an open. Remember that variance of every random variable x is a positive value, i. Whether is always positive or always negative depends upon the numerical value of. If we define gx x2, we can write the above inequality as egx.
Reverses of the jensentype inequalities for signed measures. In mathematics, jensens inequality, named after the danish mathematician johan jensen, relates the value of a convex function of an integral to the integral of the convex function. Jensen inequality if and for a convex function then proof for a simple r. Especially, we refine jensen s classical inequality. Jensens inequality theroem in hindi real analysis youtube. For a more detailed discussion of jensens inequality see the electronic appendix. On jensens type inequalities via generalized majorization. If x is a nondegenerate random variable taking values in an interval r, s, and if ux is a strictly convex function on r, s, then meanuxumeanx, providing that meanx and meanux exist and are finite. Convexity, inequalities, and norms 9 applying the same reasoning using the integral version of jensens inequality gives p q z x fpd 1p x fqd 1q for any l1 function f. Convergence in measure converge almost everywhere converges in measure converge in measure some subsequence converges almost everywhere dominated convergence theorem holds for convergence in measure. The value of this security is the same as the value of the simple european derivative security with.
Convexity, jensens inequality and benefits of noisy. Operations on measurable functions sums, products, composition realvalued measurable functions. Jensens inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum likelihood estimators. So under the assumption that g is continuous with respect to t, some papers, such as 18, 19, 25, 27, 28 have been devoted to jensens inequality for gexpectations, with the help of the theory of bsdes, they have obtained the necessary and sufficient conditions under which jensens inequality for gexpectations holds in general.
Let f be a function with a positive second derivative. The same inequality holds, almost surely, if the expectation in is replaced by the conditional expectation e. This is one of fundamental result of the measure theory, having enormous applications in probability theory, statistics and other branch of mathematics. Pdf inequalities corresponding to the classical jensens inequality. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the lefthand and the right. If its that obvious, itll be easy to extrapolate to other related circumstances. Jensen inequality basics this is a revision of material that appeared earlier in sep 37 and reprinted in pvi. It is also possible to obtain the jensen inequality for monetary utility functions with respect to certain concave functions theorem 2. The classical form of jensen s inequality involves several numbers and weights.
Jensens inequality convex functions and a proof for. In this paper we derive refinements of the jensen type inequalities in the case of real stieltjes measure, not necessarily positive, which are generalizations of jensen s inequality and its reverses for positive measures. Application for fdivergence measures including the kullbackleibler and je. Jensen s inequality if f is a real convex function, x1. A refinement of the integral form of jensens inequality. Specifically, no more than 1k 2 of the distributions values can be more than k standard deviations away from the mean or equivalently, at. Intuitively a function is convex if it is upward bending. There are similar inequalities for concave and strictly concave functions by reversing the inequality signs. Examples of jensen inequalities stanford university. A refinement of jensens inequality with applications for fdivergence measures s. Jensens inequality wikimili, the best wikipedia reader. Some integral inequalities are proved in probability spaces, which go back to some discrete variants of jensens inequality. For instance, x x k is the payoff function for an american call. The function f is called concave if the reverse inequality in 1 holds.
The function f is said to be strictly convex if the inequality in 1 strictly holds for each x 1, x 2 and t 20. In the probabilistic setting, the inequality can be further generalized to its full strength. If we use the generalized jensens inequality instead, we can get the weighted amgm inequality. The function plotted here, showing relationship of radius in cm horizontal axis to surface area in cm2 vertical axis is a convex function. Oxi 1, e is a set of finite positive measure,f l is the class of functions. Pdf some integral inequalities are proved in probability spaces, which go back to some discrete. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the lefthand and the righthand side of these inequalities and give. One of the most important inequality in mathematics and statistics is the jensen inequality. The inequality can be stated quite generally using either the language of measure theory or equivalently probability.
An inequality is said to be sharp, if it cannot be relaxed and still be valid in general. Given its generality, the inequality appears in many forms depending on the context. The refined jensens inequality is used to prove a minkowski inequality. Prepared for handbook of income distribution, edited by a b atkinson and f bourguignon. The classical integral form of jensens inequality states that. In practice we may see the dimensionless form, in which the ratio instead of the difference of the two terms is used a most important inequality in information theory and thermodynamics is the one based on,where is a small positive number tending to zero. Jensens inequality in measure theory mathematics stack exchange. This is one of fundamental result of the measure theory, having enormous applications in probability theory, statistics and other branch of. To start, note by convexity that for any a r that admits a supporting hyperplane at the barycenter of. Pdf inequalities corresponding to the classical jensens. Norms a norm is a function that measures the lengths of vectors in a vector space. The jensens measure, or jensens alpha, is a riskadjusted performance measure that represents the average return on a portfolio or investment, above or below that predicted by the capital asset pricing model capm, given the portfolios or investments beta and the average market return. Farenick and fei zhou 1 mar 2007 journal of mathematical analysis and applications, vol. The notion of convexity is very close to jensens inequality.
Uniform integrability in terms of uniform integrability test functions 18. The analysis of inequality is placed in the context of recent developments in economics and statistics. Lecture 32 strong law of large numbers and jensens inequality scott she eld mit 18. Jensens inequality is an inequality involving convexity of a function. Jensen s inequality relative to matrixvalued measures douglas r.
The classical form of jensens inequality involves several numbers and weights. In mathematics, jensen s inequality, named after the danish mathematician johan jensen, relates the value of a convex function of an integral to the integral of the convex function. I if the segment between any two points taken on its graph. There are a lot of refinements of the discrete jensens inequality, and this problem has been studied by many authors. It is also a natural problem to give analogous results for the classical jensens inequality. Measure theorylp spaces wikibooks, open books for an. In spite of this, few papers have been published dealing with this problem. On jensens inequality, holders inequality, and minkowskis. Convex functions jensens inequality holder and minkowski inequalities. Lecture notes measure and integration mathematics mit. We even establish this result for weak measurable, selfadjoint, abelian fields x it t.
The lecture notes were prepared in latex by ethan brown, a former student in the class. The properties that deal with multiplication and division state that for any real numbers, a, b and nonzero c. The lebesgue integral, chebyshevs inequality, and the. In this paper we derive refinements of the jensen type inequalities in the case of real stieltjes measure, not necessarily positive, which are generalizations of jensens inequality and its reverses for positive measures. In the probabilistic setting, the inequality can be further generalized to its full strength finite form.
Jensens inequality jensens inequality applies to convex functions. Pdf on jensens inequality, holders inequality, and. On jensens inequality for generalized choquet integral with an. Jensens inequality states that, for any convex function g, we have egx. Especially, we refine jensens classical inequality. The function gx x2 is an example of convex function. Strong law of large numbers and jensens inequality scott she eld mit 18. A refinement of jensens inequality with applications for f. On jensens inequality for generalized choquet integral with.
Sta 711 week 5 r l wolpert theorem 1 jensens inequality let. Refining jensens inequality shoshana abramovich, graham. Lp spaces, 1 leq p leq infty normed spaces, banach spaces. What is an intuitive explanation of jensens inequality. The purpose of this paper is to give some refinements of the classical jensens inequality. He used professor viaclovskys handwritten notes in producing them. From wikibooks, open books for an open world jensen inequality for monetary utility functions with respect to certain concave functions theorem 2. On jensens inequality for generalized choquet integral. Exponentiating both sides, we get the amgm inequality. Theorem a1 jensens inequality let be a positive measure on a algebra. The jensens measure is a riskadjusted performance measure that represents the average return on a portfolio or investment above or.
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