Intereting Posts

What does the function f: x ↦ y mean?
Matrices (Hermitian and Unitary)
How to generate $3 \times 3$ integer matrices with integer eigenvalues?
Integration of $\dfrac{x}{\sinh x}~dx$ from $-\infty$ to $\infty$
Interesting piece of math for high school students?
The action of SU(2) on the Riemann sphere
Can open set in $\mathbb{R}^n$ be written as the union of countable disjoint closed sets?
Weak limit and strong limit
Prove that any function can be represented as a sum of two injections
How to find $\int\frac{\sin x}{x}dx$
Probability a rotation has a small distance to a vector
Proving an entire, complex function with below bounded modulus is constant.
Partitions of $n$ into distinct odd and even parts proof
Image under an entire function.
Revisted$_2$: Are doubling and squaring well-defined on $\mathbb{R}/\mathbb{Z}$?

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below:

$\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$

Can it be proven that the result is also a characteristic function? If so, I am guessing this result could extend to any number of combinations $\alpha_i$ as long as $\sum_i\alpha_i=1$

- Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables
- Characteristic function of a square of normally distributed RV
- Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution
- Continuous probability distribution with no first moment but the characteristic function is differentiable
- Is $t\mapsto \left|\cos (t)\right|$ a characteristic function?
- Step Function and Simple Functions

Secondly, if $\phi$ is again a characteristic function, then $\mathfrak{R}e\phi(t)=\frac12(\phi(t)+\phi(-t))$ is also a characteristic function. I don’t even know how to begin attempting this proof as I am not sure what the $\mathfrak{R}$ represents.

Lastly, regarding the symmetry of characteristic functions,

$\phi$ is symmetric about zero iff it is real-valued iff the corresponding distribution is symmetric about zero.

Once again, my lack of familiarity with the complex plane leaves me in the dark here. Why can a complex-valued function not be symmetric about zero?

- convolution of characteristic functions
- measurability of a function -equivalent conditions
- How can I find a subset of a set with “half the size” of the original?
- Do random variables form a comma category?
- Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
- $f \in L^1 ((0,1))$, decreasing on $(0,1)$ implies $x f(x)\rightarrow 0$ as $x \rightarrow 0$
- Prove that a set $A$ is $\mu^\star$ measurable is and only if $\mu^\star (A) = l(X) - \mu^\star(A^{c})$
- Integrable function whose Fourier transform is not integrable
- Show that a function almost everywhere continuous is measurable
- Independence of a function and integral of a function

To prove that these are characteristic functions, using random variables yields simpler, more intuitive, proofs.

In the first case, assume that $\phi_1(t)=\mathrm E(\mathrm e^{itX_1})$ and $\phi_2(t)=\mathrm E(\mathrm e^{itX_2})$ for some random variables $X_1$ and $X_2$ defined on the same probability space and introduce a Bernoulli random variable $A$ such that $\mathrm P(A=1)=\alpha$ and $\mathrm P(A=0)=1-\alpha$, independent of $X_1$ and $X_2$. Then:

The function $\alpha\phi_1+(1-\alpha)\phi_2$ is the characteristic function of the random variable $AX_1+(1-A)X_2$.

The extension to more than two random variables is direct. Assume that $\phi_k(t)=\mathrm E(\mathrm e^{itX_k})$ for every $k$, for some random variables $X_k$ defined on the same probability space and introduce an integer valued random variable $A$ such that $\mathrm P(A=k)=\alpha_k$ for every $k$, independent of $(X_k)_k$. Then:

The function $\sum\limits_k\alpha_k\phi_k$ is the characteristic function of the random variable $X_A=\sum\limits_kX_k\mathbf 1_{A=k}$.

In the second case, assume that $\phi(t)=\mathrm E(\mathrm e^{itX})$ for some random variable $X$ and introduce a Bernoulli random variable $A$ such that $\mathrm P(A=1)=\mathrm P(A=-1)=\frac12$, independent of $X$. Then:

The function $t\mapsto\frac12(\phi(t)+\phi(-t))$ is the characteristic function of the random variable $AX$.

By Bochner’s theorem, a function $\phi : \mathbb{R} \to \mathbb{C}$ is the characteristic function of a probability measure if and only if

- $\phi$ is positive definite,
- $\phi(0) = 1$, and
- $\phi$ is continuous at the origin.

Since these properties are conserved under convex combination, your second statement is true whenever $\alpha_i$ are non-negative.

The symbol $\Re(z)$ means the real part of a complex number $z$. If $\phi$ is a characteristic function, then $\phi(-t) = \bar{\phi}(t)$, thus we have

$$\Re \phi(t) = \frac{\phi(t)+\bar{\phi}(t)}{2} = \frac{\phi(t)+\phi(-t)}{2}. $$

Now let $\phi$ be the characteristic function of a probability measure $\mu$. Then clearly the mapping $t \mapsto \phi(-t)$ is the characteristic function of the measure $\tilde{\mu} : E \mapsto \mu(-E)$. Thus in view of the first answer, $\Re \phi(t)$ is also a characteristic function.

If your *symmetry about zero* means $\phi(-t) = \phi(t)$, then the first assertion follows by our second answer. Now since $\phi(t) = \phi(-t)$, the corresponding measure must coincide, that is, we must have $\mu(E) = \tilde{\mu}(E) = \mu(-E)$ for any Borel measurable $E \subset \mathbb{R}$. Thus $\mu$ is symmetric.

- Why, historically, do we multiply matrices as we do?
- Shortest proof for 'hairy ball' theorem
- Ring isomorphism (polynomials in one variable)
- How many different ways can the signs be chosen so that $\pm 1\pm 2\pm 3 … \pm (n-1) \pm n = n+1$?
- Is the algebraic subextension of a finitely generated field extension finitely generated?
- Nonexistence of the limit $\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y^2}{x^3+y^3}$
- Prove or disprove: if $x$ and $y$ are representable as the sum of three squares, then so is $xy$.
- $\mathbb{A}^{2}$ not isomorphic to affine space minus the origin
- Spectrum of a Self-Adjoint Operator is Real
- Inductive Proof that $k!<k^k$, for $k\geq 2$.
- Equation of a plane containing a point and perpendicular to a line
- Proving ${n \choose p} \equiv \Bigl \ (\text{mod} \ p)$
- Existence of square root
- No. of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$
- Why not just define equivalence relations on objects and morphisms for equivalent categories?