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where (,) denotes the modulus of continuity of function with the step . An even more general result of four authors can be formulated as the following Jackson theorem. Theorem 3 : For every natural number n {\displaystyle n} , if f {\displaystyle f} is 2 π {\displaystyle 2\pi } -periodic continuous function, there exists a trigonometric ...
Using the fact that (,) =, the generalized Marcum Q-function can alternatively be defined as a finite integral as (,) = (+) ().However, it is preferable to have an integral representation of the Marcum Q-function such that (i) the limits of the integral are independent of the arguments of the function, (ii) and that the limits are finite, (iii) and that the integrand is a Gaussian function ...
The operator takes a locally integrable function f : R d → C and returns another function Mf. For any point x ∈ R d, the function Mf returns the maximum of a set of reals, namely the set of average values of f for all the balls B(x, r) of any radius r at x. Formally,
Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.
Then S N is identically equal to zero, hence E[S N] = 0, but E[X 1] = 1 / 2 and E[N] = 1 / 2 and therefore Wald's equation does not hold. Indeed, the assumptions , , and are satisfied, however, the equation in assumption holds for all n ∈ except for n = 1.
The Heaviside step function is an often-used step function. A constant function is a trivial example of a step function. Then there is only one interval, =. The sign function sgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
The Atkinson index is defined as: (, …,) = {(=) / (=) / = (,...,) = +where is individual income (i = 1, 2, ..., N) and is the mean income.. In other words, the Atkinson index is the complement to 1 of the ratio of the Hölder generalized mean of exponent 1−ε to the arithmetic mean of the incomes (where as usual the generalized mean of exponent 0 is interpreted as the geometric mean).
If α is the zero function and u is non-negative, then Grönwall's inequality implies that u is the zero function. The integrability of u with respect to μ is essential for the result. For a counterexample , let μ denote Lebesgue measure on the unit interval [0, 1] , define u (0) = 0 and u ( t ) = 1/ t for t ∈ (0, 1] , and let α be the ...