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That each probability distribution has only one characteristic funciton is obvious; the non-trivial part is in the other direction: different distributions cannot share one characteristic f |
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If ''X'' is a [[vector space|vector]]-valued random variable, one takes the argument ''t'' to be a vector and ''tX'' to be a [[dot product]].
A characteristic function exists for any random variable. More than that, there is a bijection between cumulative probability functions and characteristic functions. In other words, two probability distributtions never share the same characteristic function.▼
▲More than that, there is a bijection between cumulative probability functions and characteristic functions.
Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function ''F'':
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In general this is an [[improper integral]]; the function being integrated may be only conditionally integrable rather than [[Lebesgue integral|Lebesgue-integrable]], i.e. the integral of its absolute value may be infinite.
Characteristic functions are used in the most frequently seen proof of the [[central limit theorem]].
Characteristic functions can also be used to find [[moment (mathematics)|moments]] of random variable. Provided that ''n''-th moment exists, characteristic function can be differentiated ''n'' times and
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