Convolution Operation for Summing Independent Random Variables
The core principle asserts that the sum of independent random variables converges in distribution to a specific form determined by their combined means and variances via linear convolution. Formally, this mechanism relies on the property where the probability density function (PDF) or characteristic function of the aggregate variable is derived from the product of individual transforms under independence assumptions within measure-theoretic probability theory. This concept serves as a fundamental theorem in stochastic processes and mathematical statistics regarding the structural evolution of cumulative distributions through additive combinations.
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The core principle asserts that the sum of independent random variables converges in distribution to a specific form determined by their combined means and variances via linear convolution. Formally, this mechanism relies on the property where the probability density function (PDF) or characteristic function of the aggregate variable is derived from the product of individual transforms under independence assumptions within measure-theoretic probability theory. This concept serves as a fundamental theorem in stochastic processes and mathematical statistics regarding the structural evolution of cumulative distributions through additive combinations.
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