Intereting Posts

Denseness of the set $\{ m+n\alpha : m\in\mathbb{N},n\in\mathbb{Z}\}$ with $\alpha$ irrational
A strange little number – $6174$.
Measurability of a function defined on a product measure space, and related to a measurable function
Least-upper-bound property Rudin book
Classify sphere bundles over a sphere
Prove that $1.49<\sum_{k=1}^{99}\frac{1}{k^2}<1.99$
Prove that undirected connected graph w/ |V| >= 2, 2 nodes have same degree
Recursive Mapping
Sum of $k {n \choose k}$ is $n2^{n-1}$
$\sqrt{ab}=\sqrt{a}\sqrt{b}$ for complex number $a$ and $b.$
How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?
How do you orthogonally diagonalize the matrix?
Evaluation of $ \lim_{x\to 0}\left\lfloor \frac{x^2}{\sin x\cdot \tan x}\right\rfloor$
Factoring a hard polynomial
Known bounds for the number of groups of a given order.

In the following theorem, I have a problem about the second part. That is showing if $f$ is strictly convex then $X=EX$ with probability $1$. While I can see this must be true, I don’t know how to show it holds with probability $1$. I am specially interested in the case where $X$ is s continuous random variable in reals.

Thanks a lot in advance for explaining.

- Infinite series $\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$
- Schwarz Lemma of Complex Analysis
- What is the Domain of $f(x)=x^{\frac{1}{x}}$
- Type of singularity of $\log z$ at $z=0$
- proof Intermediate Value Theorem
- Prove for every $n,\;\;$ $\sum\limits_{k=1}^{\infty}\left\lfloor \frac{n}{2^{k}}+\frac{1}{2} \right\rfloor=n $

- If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that exactly k characters will be the same?
- Why does this expected value simplify as shown?
- Prove that $ f$ is uniformly continuous
- $\limsup $ and $\liminf$ of a sequence of subsets relative to a topology
- Fourier series of almost periodic functions and regularity
- Central Limit Theorem for exponential distribution
- Expected outcome for repeated dice rolls with dice fixing
- What is the domain of $x^x$ when $ x<0$
- Solutions of autonomous ODEs are monotonic
- Sine function dense in $$

Let $f : \mathbb{R} \to \mathbb{R}$ be convex. This means that at every point $a \in \mathbb{R}$, there is an affine linear function $l_a : \mathbb{R} \to \mathbb{R}$ which is dominated by $f$, i.e.

$$

l_a(x) \leq f(x)

$$

and $l_a(a) = f(a)$. When $f$ is differentiable, for example, then $l_a$ is the tangent to $f$ at $a$.

When $f$ is strictly convex, we have the additional condition

$$

l_a(x) = f(x) ~\Rightarrow ~ x = a

$$

Before we define $a$ in this particular problem (and sweeping integrability problems under the rug), notice that

$$

l_a(X) \leq f(X)

$$

holds, hence $E l_a(X) \leq E f(X)$. Moreover $E l_a(X) = l_a(E X)$ because of the linearity of $l_a$. Finally, we set $a = EX$, and have obtained

$$

f(EX) \leq E f(X)

$$

Suppose now that $f(EX) = E(fX)$, which can be written as $E l_a(X) = E f(X)$ with our choice $a = E X$.

With this setup, consider $E [f(X) – l_a(X)] = 0$. Inside the expectation we have a nonnegative random variable (because of convexity) and it has expectation zero. We conclude that $f(X) = l_a(X)$ almost everywhere (because we used the integral to do so! the integral doesn’t see measure zero sets.)

Now we use strict convexity: $f(X) = l_a(X) ~\Rightarrow~ X = a = EX$ almost surely, i.e. $X$ is a constant.

Addendum:

Claim: If $Y$ is a nonnegative-valued random variable and $E Y = 0$, then $Y = 0$ almost surely.

To see this, let $A_n = \{Y \geq 1/n\}$, i.e. the set where $Y$ is greater than $1/n$. Note that $\cup_n A_n = A := \{Y > 0\}$. Let’s show that $P A_n = 0$ for any $n$, where $P$ is the probability measure.

$$

\frac{1}{n} P A_n \leq E (Y I_{A_n}) \leq E Y = 0

$$

Now recall that $P \cup_n A_n \leq \sum_n P A_n$, which is often called the ‘countable subadditivity’ property. This implies that $P A = 0$, and the claim follows.

Here is an alternative proof (given several years later) that is a bit more general as it does not require existence of an affine bounding function (subgradients do not always exist for convex functions defined over restricted domains).

Fix $n$ as a positive integer, let $\mathcal{X} \subseteq \mathbb{R}^n$ be a convex set, and let $f:\mathcal{X}\rightarrow\mathbb{R}$ be a strictly convex function, meaning that

$$f(px + (1-p)y) < pf(x) + (1-p)f(y)$$

whenever $0<p<1$ and $x, y \in \mathcal{X}$, $x \neq y$.

Let $X$ be a random vector that takes values in $\mathcal{X}$ and that has a finite expectation $E[X]$. We know that $E[X] \in \mathcal{X}$ (this is a precursor to Jensen’s inequality). Suppose that $f(E[X]) = E[f(X)]$. We show that $X=E[X]$ with prob 1.

Define $m=E[X]$. Suppose $P[X>m] >0$ (we reach a contradiction).

Case 1: Suppose $P[X>m]=1$. Then $X-m$ is a positive random variable with prob 1 and so $E[X-m]>0$, meaning $m-m>0$, a contradiction.

Case 2: Suppose $0 < P[X>m] < 1$. Define $m_1 = E[X|X\leq m]$ and $m_2 = E[X|X>m]$. Note that $m_1 \leq m < m_2$ and

$$m_1P[X\leq m] + m_2 P[X>m] = m$$

Also

\begin{align}

f(m) &\overset{(a)}{=} E[f(X)] \\

&= E[f(X)|X\leq m]P[X\leq m] + E[f(X)|X>m]P[X>m] \\

&\overset{(b)}{\geq} f(E[X|X\leq m])P[X\leq m] + f(E[X|X>m])P[X>m] \\

&= f(m_1)P[X\leq m] + f(m_2)P[X>m] \\

&\overset{(c)}{>} f(m_1 P[X\leq m] + m_2 P[X>m])\\

&= f(m)

\end{align}

where (a) holds by the assumption $f(E[X]) = E[f(X)]$; (b) holds by Jensen’s inequality applied to the conditional expectations; (c) holds by strict convexity. Hence, $f(m)>f(m)$, a contradiction.

Cases 1 and 2 together imply that $P[X>m]=0$. Similarly it can be shown that $P[X<m]=0$. $\Box$

- Simple question about the definition of Brownian motion
- General term of this interesting sequence
- Does Euclidean geometry require a complete metric space?
- Can $\pi$ be rational in some base radix
- Value of $\sum\limits_n x^n$
- Which optimization class does the following problem falls into (LP, MIP, CP..) and which solver to use
- On the element orders of finite group
- What is the motivation for semidirect products?
- Rational roots of polynomials
- UPDATE: How to find the order of elliptic curve over finite field extension
- How to evaluate $\int_{0}^{1} \frac{\ln x}{x+1} dx$
- How can I find the square root using pen and paper?
- $n = 2^k + 1$ is a prime iff $3^{\frac{n-1}{2}} \equiv -1 \pmod n$
- Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
- Proving Galmarino's Test