Probability theory and stochastic processes Proof

Proof of the sum of independent normal random variables by characteristic functions

Builds on Normal distribution Characteristic function Characteristic function of the normal distribution Characteristic function of a sum of independent random variables The characteristic function determines the distribution Independence of two random variables Product of exponentials

Proof

We compute the characteristic function of $X + Y$ and recognise it.

For a real random variable $Z$ , let $\varphi_Z(t) = E[e^{itZ}]$ be its characteristic function, for $t \in \R$ .
By the characteristic function of the normal distribution, $\varphi_X(t) = \exp\left(i \mu_1 t - \tfrac12 \sigma_1^2 t^2\right)$ and $\varphi_Y(t) = \exp\left(i \mu_2 t - \tfrac12 \sigma_2^2 t^2\right)$ .
$X$ and $Y$ are independent. By the characteristic function of a sum of independent random variables, $\varphi_{X+Y}(t) = \varphi_X(t) \, \varphi_Y(t)$ for each $t$ .
By the product of exponentials, $\varphi_{X+Y}(t) = \exp\left(i (\mu_1 + \mu_2) t - \tfrac12 (\sigma_1^2 + \sigma_2^2) t^2\right)$ .
Let $\mu = \mu_1 + \mu_2$ and $\sigma^2 = \sigma_1^2 + \sigma_2^2$ . Then $\sigma^2 > 0$ , so the normal distribution $\mathcal{N}(\mu, \sigma^2)$ is defined. Let $W \sim \mathcal{N}(\mu, \sigma^2)$ .
By the characteristic function of the normal distribution, $\varphi_W(t) = \exp\left(i \mu t - \tfrac12 \sigma^2 t^2\right)$ . By step 4, $\varphi_W(t) = \varphi_{X+Y}(t)$ for all $t \in \R$ .
By the theorem that the characteristic function determines the distribution, $X + Y$ and $W$ have the same law.
Hence $X + Y \sim \mathcal{N}(\mu_1 + \mu_2, \, \sigma_1^2 + \sigma_2^2)$ .

Idea

For independent variables, characteristic functions multiply. The characteristic function of a normal law is the exponential of a quadratic in $t$ . So in a product the means add and the variances add. Since a characteristic function fixes the law, the sum is normal.

Reading path

To understand this page from the ground up, read these 421 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of integration by parts, Proof of the Cauchy–Hadamard theorem, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the continuity of probability measures, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the integral of the exponential function, Proof of the probability of the complement, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the reverse triangle inequality, Proof that the exponential series has infinite radius of convergence, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof that i squared is minus one, Proof of the expectation of a product of independent random variables by simple functions, Proof of the linearity of differentiation, Proof of the angle addition formulas by differentiation, Proof of the mean value theorem for integrals, Proof of the monotonicity of probability, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of the Cauchy mean value theorem from Rolle's theorem, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the formula for the characteristic function of the normal distribution, Proof of the factorial formula for binomial coefficients by induction, Proof of integration by substitution, Proof of the product of exponentials by the Cauchy product, Proof of the second fundamental theorem of calculus, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof of Taylor's theorem with the Lagrange remainder by the Cauchy mean value theorem, Proof that differentiable functions are continuous, Proof that one-sided limits of a monotone function exist, Proof of the epsilon-delta form of the limit of a function, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof of the power series of the exponential function, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the formula for the moment-generating function of the normal distribution, Proof that a function with zero derivative on an open interval is constant, Proof of the monotonicity of the integral, Proof of the comparison test for improper integrals, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof of Euler's formula by power series, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the product formula for the characteristic function of a sum of independent random variables, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that sums and products of continuous functions are continuous, Proof that independence means the joint law is the product of the marginals, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

Binomial coefficient definition
Cartesian product concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Dynkin system concept
Element definition
Extended real number line object
Factorial definition
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Limit point of a set definition
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Mathematical induction concept
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Measure concept
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Peano axioms concept
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Probability measure concept
Product sigma-algebra concept
Proposition concept
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Real numbers object
Sequence definition
Set bounded above definition
Sigma-algebra concept
Sigma-finite measure concept
Simple function concept
Proof of Pascal's rule from the recursive definition proof
Complex exponential definition
Power series in the complex plane definition
Real and imaginary parts definition
Proof that i squared is minus one proof
Set definition
Ring definition
Ring of integers statement
Limit of a function definition
Monotone convergence theorem statement
Local minimum definition
Product sigma-algebra definition
Archimedean ordered field definition
Dynkin's π-λ theorem statement
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Mesh of a partition definition
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Riemann sum definition
Product measure definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
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Commutative ring definition
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Limit at infinity definition
One-sided limit definition
Union definition
Addition of natural numbers definition
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Proof of the Archimedean property in an Archimedean ordered field proof
Sample space definition
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Integral of a non-negative function definition
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Bounded sequence definition
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Geometric sequence definition
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Series definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Proof of the equality of complex numbers proof
Field definition
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Antiderivative definition
Differentiable function definition
One-sided derivative definition
Riemann integral definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Subset definition
Order on the natural numbers definition
The rationals form an ordered field statement
Mathematical proof concept
Propositional logic concept
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Limit superior definition
Upper and lower Darboux sums definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
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Partial sum definition
Power series definition
Equality of complex numbers statement
Factorial formula for binomial coefficients statement
Proof that a composition of continuous functions is continuous proof
Binary relation definition
Derivative function definition
Definite integral definition
Universal set definition
Improper integral definition
Order on the reals definition
Power set definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Tautology definition
Upper and lower Darboux integrals definition
Uniform convergence definition
Sum of a finite geometric series statement
Convergent series definition
Radius of convergence definition
Sine and cosine definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
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Proof that the complex numbers form a field proof
Complement definition
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Strong induction statement
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Logical equivalence definition
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Binomial theorem statement
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Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Sigma-algebra definition
Injective function definition
Indicator function definition
Preimage of a set definition
The complex numbers form a field statement
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Smooth function definition
Monotone function definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Countable set definition
Event definition
Dynkin system definition
Well-ordering principle statement
Borel sigma-algebra definition
Double negation law statement
Total order definition
Composition of continuous functions is continuous statement
Uniform continuity definition
Probability axioms concept
Taylor series definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Maclaurin series definition
Probability measure definition
Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Dirac measure definition
Probability distribution definition
Lipschitz functions are uniformly continuous statement
Probability space definition
The reals form an ordered field statement
Absolute value definition
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Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the existence of the infimum by negation proof
Cumulative distribution function definition
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Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Independence of two events definition
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Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
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Joint probability distribution definition
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Absolute value is multiplicative statement
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Probability density function definition
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Expected value definition
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Proof that one-sided limits of a monotone function exist proof
Proof of the intermediate value theorem by the supremum proof
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Marginal distribution definition
Proof of the reverse triangle inequality proof
Density of the normal distribution definition
Law of the unconscious statistician statement
Modulus of a complex number definition
Proof of the squeeze theorem for sequences proof
Proof that differentiable functions are continuous proof
Proof that limits preserve non-strict inequalities proof
Proof of the limit of a sum of sequences proof
Proof of the uniqueness of limits proof
Linearity of expectation statement
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Intermediate value theorem statement
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
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Proof that Cauchy sequences are bounded proof
Proof that subsequences of a convergent sequence converge to the same limit proof
The sequence 1/n tends to zero example
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Reverse triangle inequality statement
Normal distribution definition
Proof that a complex number times its conjugate is the square of its modulus proof
Moment definition
Differentiable functions are continuous statement
Proof of the probability of the complement proof
Independence of random variables definition
Proof that independence means the joint law is the product of the marginals proof
Linearity of the integral statement
Proof of the monotonicity of the integral proof
Floor function definition
Limits preserve non-strict inequalities statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Proof of the ratio formula for the radius of convergence proof
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are Cauchy statement
Subsequences of a convergent sequence converge to the same limit statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Independence means the joint law is the product of the marginals statement
A complex number times its conjugate statement
Moment-generating function definition
Probability of the complement statement
Proof of the limit of a product of sequences proof
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
One-sided limits of a monotone function exist statement
There is a natural number n with 1/n below any positive epsilon statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Proof that the modulus is multiplicative proof
Proof of the monotonicity of probability proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Limit of a product of sequences statement
Geometric series example
Decreasing sequence bounded below converges statement
Proof of Riemann's criterion for integrability proof
A series of nonnegative terms converges iff its partial sums are bounded statement
Modulus is multiplicative statement
Monotonicity of probability statement
Proof of Euler's formula by power series proof
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Proof of the expectation of a product of independent random variables by simple functions proof
Sequential characterization of limits statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Proof of the root test proof
Proof of the ratio test proof
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Euler's formula statement
Proof of the continuity of probability measures proof
Expectation of a product of independent random variables statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Comparison test statement
Cauchy product of absolutely convergent series statement
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Continuity of probability measures statement
Characteristic function definition
Chain rule statement
Product of continuous functions is continuous statement
Product rule statement
Additivity of the integral over intervals statement
Proof of the product of exponentials by the Cauchy product proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
The characteristic function determines the distribution statement
Product of exponentials statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the power rule by induction proof
Proof of the constant multiple rule proof
Proof of the first fundamental theorem of calculus proof
Boundedness theorem statement
Proof of the comparison test for improper integrals proof
Cauchy–Hadamard theorem statement
Ratio formula for the radius of convergence statement
Cauchy criterion for sequences statement
Power rule statement
Constant multiple rule statement
Comparison test for improper integrals statement
Proof of the product formula for the characteristic function of a sum of independent random variables proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Characteristic function of a sum of independent random variables statement
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that continuous functions are Riemann integrable proof
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Absolutely convergent series converge statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the mean value theorem for integrals proof
Proof of the Weierstrass M-test proof
Proof of the Cauchy mean value theorem from Rolle's theorem proof
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Weierstrass M-test statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Cauchy mean value theorem statement
Proof that a function with zero derivative on an open interval is constant proof
Proof that the integral of a continuous function is an antiderivative proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof of the second fundamental theorem of calculus proof
A function with zero derivative is constant statement
The integral of a continuous function is an antiderivative statement
Proof of Taylor's theorem with the Lagrange remainder by the Cauchy mean value theorem proof
Uniform convergence of derivatives allows term-by-term differentiation statement
Second fundamental theorem of calculus statement
Proof that the exponential series has infinite radius of convergence proof
Taylor's theorem with the Lagrange remainder statement
Proof of integration by parts proof
Proof of integration by substitution proof
Proof of term-by-term differentiation proof
The exponential series has infinite radius of convergence statement
Integration by substitution statement
Integration by parts statement
Term-by-term differentiation statement
Exponential function definition
Proof of the exponential of a sum by the Cauchy product proof
Proof that power series are differentiable inside the radius of convergence proof
Exponential of a sum statement
Power series are differentiable inside the radius of convergence statement
Proof of the formula for the moment-generating function of the normal distribution proof
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
Moment-generating function of the normal distribution statement
The exponential function is its own derivative statement
Derivative of cosine statement
Derivative of sine statement
Proof of the integral of the exponential function proof
Proof of the power series of the exponential function proof
Proof of the angle addition formulas by differentiation proof
Integral of the exponential function statement
Power series of the exponential function statement
Angle addition formulas statement
Integral of exp(-x) over the half line example
Proof of the formula for the characteristic function of the normal distribution proof
Characteristic function of the normal distribution statement
Proof of the sum of independent normal random variables by characteristic functions proof

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Sum of independent normal random variables

Tag	`G7Q5GT80HP`
URI	`m2://G7Q5GT80HP`
Kind	proof
Writing	W0 stub
MSC	60E05
Proves	Sum of independent normal random variables
Uses	Normal distribution, Characteristic function, Characteristic function of the normal distribution, Characteristic function of a sum of independent random variables, The characteristic function determines the distribution, Independence of two random variables, Product of exponentials
Changed	4 November 2026

Proof of the sum of independent normal random variables by characteristic functions

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