Quasi-arithmetic Mean - Properties

Properties

  • Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks.
M_f(x_1,\dots,x_{n\cdot k}) = M_f(M_f(x_1,\dots,x_{k}), M_f(x_{k+1},\dots,x_{2\cdot k}), \dots, M_f(x_{(n-1)\cdot k + 1},\dots,x_{n\cdot k}))
  • Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.
With it holds
  • The quasi-arithmetic mean is invariant with respect to offsets and scaling of :
.
  • If is monotonic, then is monotonic.
  • Any quasi-arithmetic mean of two variables has the mediality property and the self-distributivity property . Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél–Dhombres, Chapter 17.
  • Any quasi-arithmetic mean of two variables has the balancing property . An interesting problem is whether this condition (together with fixed-point, symmetry, monotonicity and continuity properties) implies that the mean is quasi-arthmetic. Georg Aumann showed in the 1930s that the answer is no in general, but that if one additionally assumes to be an analytic function then the answer is positive.

Read more about this topic:  Quasi-arithmetic Mean

Famous quotes containing the word properties:

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)