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Expectations in vector space $L^0$

Denote $\mathcal L^0$ as the set of all rvs (i.e. including simple rvs etc..) $\mathcal L^0:=\mathcal L^0(\Omega,\mathcal{F})$, where $\mathcal L^0$ is a vector space. We define a rv in $\mathcal L^0$ as $$ E(X): = \sup\{\hat{E}(Y):Y\le X \}$$ where $Y$ are contained in the set of all simple random variables. Hence this set is monotonic […]

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Limit Behaviour of Equivalent Processes

Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be two real-valued stochastic processes on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that their finite-dimensional distributions coincide. Assume furthermore that $$ \limsup\limits_{t\to \infty} \; X_t \; = \; +\infty \quad \text{ and } \quad \liminf\limits_{t\to \infty} \; X_t \; = \; -\infty$$ almost surely. I expect that in this […]

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Filtration at infinity and sigma-algebra of events strictly prior to infinity.

I’m a worker and I’m studying Stochasstic processes from the book Stochastic Calculus and Application by Cohen and Elliott. In particular, I am reading the Chapter 4 on Discrete Time Stochastic Processs, that is, processes of the form $X=\{X_n \}_{n \in\mathbb{T}}$ where $\mathbb{T}$ could be: $$ \mathbb{T}=\mathbb{Z}^+ \;\;\; \text{or}\;\;\;\mathbb{T}= \overline{\mathbb{Z}}^+= \mathbb{Z}^+\cup\{\infty\} $$ Before starting the […]

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Filtration at infinity and sigma-algebra of events strictly prior to infinity.

I’m a worker and I’m studying Stochasstic processes from the book Stochastic Calculus and Application by Cohen and Elliott. In particular, I am reading the Chapter 4 on Discrete Time Stochastic Processs, that is, processes of the form $X=\{X_n \}_{n \in\mathbb{T}}$ where $\mathbb{T}$ could be: $$ \mathbb{T}=\mathbb{Z}^+ \;\;\; \text{or}\;\;\;\mathbb{T}= \overline{\mathbb{Z}}^+= \mathbb{Z}^+\cup\{\infty\} $$ Before starting the […]

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Find a sequence of (probability) measures that are absolutely continuous to the measure $\mathbb{P}$

Let $\{\mathbb{P}_i\}$, for $ i\in \mathbb{N}$ be a sequence of probability measures on $(\Omega,\mathcal{F})$, then find a probability measure $\mathbb{P}$ where each sequence $\mathbb{P}_i$ is absolutely continuous to $\mathbb{P}$. I know that probability measure $\mathbb{P_i}$ is absolutely continuous with respect to another probability measure $\mathbb{P}$ if $\mathbb{P}(A)=0$ implies $\mathbb{P}_i(A)=0$ for any subset $\mathcal{F}$ of $\Omega$. […]

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Equilibrium triangle in the set $ \{ \mathbb{E}(1_{A}|\mathcal{G}) : \mathcal{G} \subset \mathcal{F} \}$?

Let us fix a probability space $\ (\Omega, \mathcal{F},\mathbb{P}) \ $ and any $\ A\in \mathcal{F}. \ $ I have proved (it is not hard), that the set $$\hat{\mathcal{S}}=\{ \mathbb{E}(\mathbb{1}_{A}|\mathcal{G}):\mathcal{G}\subset \mathcal{F}\}$$ is contained in a sphere $$\mathcal{S}=\Bigg\{Z\in L_{2}(\Omega) \ \ : \ \ \Bigg|\Bigg|Z-\frac{(\mathbb{1}_{A}+\mathbb{E}1_{A})}{2}\Bigg|\Bigg|^2= \frac{\mathbb{P}(A)(1-\mathbb{P}(A))}{4}\Bigg\},$$ where the norm is $\ ||X||^2:= \mathbb{E}X^{2}.$ I am trying […]

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Can a random probability measure be interpreted as random variable with values in the set of probability measures?

Suppose $(\Omega,\mathcal{F},P)$ is a complete probability space and denote by $\mathcal{B}(\mathbb R)$ the Borel $\sigma$-algebra on $\mathbb R$. Recall that a random probability measure is a function $k\colon \Omega\times \mathcal{B}(\mathbb R)\to [0,1]$ such that $E\mapsto k(\omega,E)$ is a probability measure of all $\omega\in\Omega$, $\omega\mapsto k(\omega,E)$ is $\mathcal{F}$-measurable for all $E\in\mathcal{B}(\mathbb R)$. My question is can […]

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Ergodicity of induced system

Suppose $(X,\mathcal{F},\mu,T)$ is an ergodic measure preserving dynamical system. Let $Y\subset X$ be such that $\mu(Y)>0$ and suppose there is an integrable function $R:Y\to \mathbb{N}$ such that $T^{R(y)}(y)\in Y$. Then we can define a function $F:Y\to Y$ by $F=T^R$ and consider the induced system $(Y,\mathcal{F}\cap Y, \mu|_Y,F)$. Can we say that $F$ is ergodic? When […]

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Exp. distributed r.v. as function on sample space

Let $(\Omega,\mathcal{F},P)$ denote the probability space upon which the exponentially distributed random variable $X (\omega)$ with $\omega\in\Omega$ is defined. Furthermore, let $E$ denote the space into which $X$ maps. Is it true that $\Omega = \left(0,\infty\right)$? And $E=\Omega$? And $X (\omega)=\omega$?

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$\sigma$-Algebra generated by a simple characteristic function

Suppose that $X:\Omega \rightarrow \mathbb{R}$. Let $A \in \mathcal{B}$ be a Borel set and $B=X^{-1}(A)$. Suppose that $X = \sum_i a_i\chi_{A_i}$ is a simple random variable defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Show that $A_i \in \sigma(X)$.