Math 2.0

Probability theory and stochastic processes Proposition

Characteristic function of a sum of independent random variables

Builds on Characteristic functionExpectation of a product of independent random variablesIndependence of random variables

Statement

Let X1,…,XnX_1, \dots, X_n be independent real random variables, and let S=X1+⋯+XnS = X_1 + \dots + X_n. Then for each t∈ℝt \in \mathbb{R}, the characteristic functions satisfy φS(t)=φX1(t) φX2(t)⋯φXn(t).\varphi_S(t) = \varphi_{X_1}(t) \, \varphi_{X_2}(t) \cdots \varphi_{X_n}(t).

Remarks

Proof: eitS=eitX1⋯eitXne^{itS} = e^{itX_1} \cdots e^{itX_n}. The factors are independent and bounded, so the product rule for expectations applies to their real and imaginary parts. For i.i.d. XiX_i, φS(t)=φX1(t)n\varphi_S(t) = \varphi_{X_1}(t)^n. The sample mean has φS/n(t)=φX1(t/n)n\varphi_{S/n}(t) = \varphi_{X_1}(t/n)^n. The converse is false: φX+Y=φXφY\varphi_{X+Y} = \varphi_X \varphi_Y does not imply that XX and YY are independent.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

  1. Complex numbers object
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Real numbers object
  10. Euler's formula statement
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Power set definition
  28. Universal set definition
  29. Order on the reals definition
  30. Function definition
  31. Complement definition
  32. Interval definition
  33. Injective function definition
  34. Preimage of a set definition
  35. Sigma-algebra definition
  36. Countable set definition
  37. Event definition
  38. Borel sigma-algebra definition
  39. Probability axioms concept
  40. Probability measure definition
  41. Probability distribution definition
  42. Probability space definition
  43. Conditional probability definition
  44. Random variable definition
  45. Cumulative distribution function definition
  46. Discrete random variable definition
  47. Law of a random variable definition
  48. Independence of two events definition
  49. Random variable with a density definition
  50. Joint probability distribution definition
  51. Probability mass function definition
  52. Mutual independence of events definition
  53. Expected value definition
  54. Independence of two random variables definition
  55. Probability density function definition
  56. Marginal distribution definition
  57. Independent sigma-algebras definition
  58. Law of the unconscious statistician statement
  59. Characteristic function definition
  60. Independence of random variables definition
  61. Independence means the joint law is the product of the marginals statement
  62. Expectation of a product of independent random variables statement
  63. Characteristic function of a sum of independent random variables statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory), Basic manipulations of distributions
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Tag02JGYG0T7M
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Kindstatement, proposition
WritingW0 stub
MSC60E10
StatusThis copy of the site has no build data. Run m2 status 02JGYG0T7M.
UsesCharacteristic function, Expectation of a product of independent random variables, Independence of random variables

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