Secondary Measure - Sequence of Secondary Measures

Sequence of Secondary Measures

The secondary measure μ associated with a probability density function ρ has its moment of order 0 given by the formula

where c₁ and c₂ indicating the respective moments of order 1 and 2 of ρ.

To be able to iterate the process then, one 'normalizes' μ while defining ρ₁ = μ/d₀ which becomes in its turn a density of probability called naturally the normalised secondary measure associated with ρ.

We can then create from ρ₁ a secondary normalised measure ρ₂, then defining ρ₃ from ρ₂ and so on. We can therefore see a sequence of successive secondary measures, created from ρ₀ = ρ, is such that ρ_n+1 that is the secondary normalised measure deduced from ρ_n

It is possible to clarify the density ρ_n by using the orthogonal polynomials P_n for ρ, the secondary polynomials Q_n and the reducer associated φ. That gives the formula

The coefficient is easily obtained starting from the leading coefficients of the polynomials P_n−1 and P_n. We can also clarify the reducer φ_n associated with ρ_n, as well as the orthogonal polynomials corresponding to ρ_n.

A very beautiful result relates the evolution of these densities when the index tends towards the infinite and the support of the measure is the standard interval .

Let

be the classic recurrence relation in three terms. If

then the sequence {ρ_n} converges completely towards the Chebyshev density of the second form

.

These conditions about limits are checked by a very broad class of traditional densities.