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• A Measureable Function is less than or equal to another iff all its integrals are less than or equal to the others

  A Measureable Function is less than or equal to another iff all its integrals are less than or equal to the others

  ...Recall that , so we must have that $\mathbb{1}{A} (f - g) = 0$ . But $\forall x \in A$, $\mathbb{1}_{A} (f(x) - g(x)() > 0$, therefore $\mu(A) = 0$. Thus $f \leq g$ . $\blacksquare$ Other Outlinks ......

  11/7/2022

• A Sequence of L1 Random Variables is L1 convergent iff they converge in probability and are Uniformly Integrable

  A Sequence of L1 Random Variables is L1 convergent iff they converge in probability and are Uniformly Integrable

  - see Thm 6.6.1 in pg 191 ...

  11/7/2022

• Characteristic Function

  Characteristic Function

  ...\begin{align*} &\mathbb{E}(e^{tX} 1_{X \geq 0}) \leq \mathbb{E}(e^{tX})\\ &\mathbb{E}(e^{-tX} 1_{X < 0}) \leq \mathbb{E}(e^{-tX})\\ &\mathbb{E}(e^{tX} 1{X \geq 0}) + \mathbb{E}(e^{-tX} 1{X < 0}) = \mathbb{E}(e^{t|X|}) \leq m{X}(t) + m{X}(-t) \end{align*} Other Outlinks Encounters - Ch 9......

  11/7/2022

• Convergence in L1 implies convergence of Integrals

  Convergence in L1 implies convergence of Integrals

  Statement If $f{n} \overset{L^{1}}{\to} f$, then $\int\limits f{n} \to \int\limits f$. This follows because $\int\limits |f{n} - f| \geq \left| \int\limits f{n} -f \right| = \left| \int\limits f_{n} - \int\limits f \right| \geq 0$...

  11/7/2022

• Extended L1 Functions

  Extended L1 Functions

  ...and $\int\limits_{X} |f| d \mu = \infty$, then if $\int\limits{X} f^{+} d \mu = \infty$, we define $\int\limits{X} f d\mu = \infty$ if $\int\limits{X} f^{-} d \mu = \infty$, we define $\int\limits{X} f d \mu = -\infty$ Other Outlinks ......

  11/7/2022

• L1 is a Complete Metric Space

  L1 is a Complete Metric Space

  ...

  11/7/2022

• Martingale Difference Sequence

  Martingale Difference Sequence

  ...to $(\mathcal{B}{j})$ with $\mathbb{R}$. Then, it is a if $d_{j} \in L^{1}$ $\forall j \in \mathbb{N}$ $\mathbb{E}(d{j+1} | \mathcal{B}{j}) = 0$ $\forall j \in \mathbb{N}$ Properties (1) in other words: Other Outlinks ......

  11/7/2022

• Sum of Independent Random Variables is in L1 if and only if Random Variables are in L1

  Sum of Independent Random Variables is in L1 if and only if Random Variables are in L1

  ...F_{X}(dx)\\ &=\int\limits\mathbb{R} |x+y-y| F{X}(dx)\\ &\leq \int\limits\mathbb{R} |x+y| F{X}(dx) + \int\limits\mathbb{R} |y| F{X}(dx)\\ &=\int\limits\mathbb{R} |x+y| F{X}(dx) + |y|\\ &< \infty \end{align*}$$ and $X \in L^{1}(\Omega)$. Swap $X$ and $Y$ to get the same result for $Y$. $\checkmark$ $\blacksquare$ Other Outlinks ......

  11/7/2022