Lesson plan
Convolution power
In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if x is a function on Euclidean space R d and n is a natural number, then the convolution power is defined by : x^{*n} = \underbrace{x * x * x * \cdots * x * x}_n,\quad x^{*0}=\delta_0 where ∗ denotes the convolution operation of functions on R d and δ 0 is the Dirac delta distribution. This definition makes sense if x is an integrable function (in L 1 ), a rapidly decreasing distribution (in particular, a compactly supported distribution) or is a finite Borel measure. If x is the distribution function of a random variable on the real line, then the n th convolution power of x gives the distribution function of the sum of n independent random variables with identical distribution x. The central limit theorem states that if x is in L 1 and L 2 with mean zero and variance σ 2 , then : P\left(\frac{x^{*n}}{\sigma\sqrt{n}} where Φ is the cumulative standard normal distribution on the real line. Equivalently, x^{*n}/\sigma\sqrt{n} tends weakly to the standard normal distribution. In some cases, it is possible to define powers x *t for arbitrary real t > 0. If μ is a probability measure, then μ is infinitely divisible provided there exists, for each positive integer n, a probability measure μ 1/n such that : \mu_{1/n}^{* n} = \mu.…
Article summary
In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if x is a function on Euclidean space R d and n is a natural number, then the convolution power is defined by : x^{*n} = \underbrace{x * x * x * \cdots * x * x}_n,\quad x^{*0}=\delta_0 where ∗ denotes the convolution operation of functions on R d and δ 0 is the Dirac delta distribution. This definition makes sense if x is an integrable function (in L 1 ), a rapidly decreasing distribution (in particular, a compactly supported distribution) or is a finite Borel measure. If x is the distribution function of a random variable on the real line, then the n th convolution power of x gives the distribution function of the sum of n independent random variables with identical distribution x. The central limit theorem states that if x is in L 1 and L 2 with mean zero and variance σ 2 , then : P\left(\frac{x^{*n}}{\sigma\sqrt{n}} where Φ is the cumulative standard normal distribution on the real line. Equivalently, x^{*n}/\sigma\sqrt{n} tends weakly to the standard normal distribution. In some cases, it is possible to define powers x *t for arbitrary real t > 0. If μ is a probability measure, then μ is infinitely divisible provided there exists, for each positive integer n, a probability measure μ 1/n such that : \mu_{1/n}^{* n} = \mu.…
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Discussion questions
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