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In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if x {\displaystyle x} is a function on Euclidean space R and n {\displaystyle n} is a natural number, then the convolution power is defined by
where * denotes the convolution operation of functions on R and δ0 is the Dirac delta distribution. This definition makes sense if x is an integrable function , a rapidly decreasing distribution or is a finite Borel measure.
If x is the distribution function of a random variable on the real line, then the n 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 and L with mean zero and variance σ, then
where Φ is the cumulative standard normal distribution on the real line. Equivalently, x ∗ n / σ n {\displaystyle x^{*n}/\sigma {\sqrt {n}}} tends weakly to the standard normal distribution.