# Questions tagged [probability-theory]

For questions solely about the modern theoretical footing for probability, for example, probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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### Notation in Kuo's Introduction to Stochastic Integration

I am reading this book by Kuo (really like it so far) and I really don't understand the notation used in section 10.5. In particular, when he writes $(X_{t_2} \in dx_2)$. I can't find anywhere else... ...
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### Variance for random variable with known density

let $t,h,g\in \mathbb{R}$ and $$h(z|x)=\frac{1}{\sqrt{2\pi}}e^{-1/2(z-t-hx-gx^2)^2}$$ $$f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ and $g(x,z)=f(x)h(z|x)$. Find the variance of $Z$ I have found the ...
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### If $X$ and $Y$ are independent $\implies$ $E(X|Y)=E(X)$

Exercices : Let $X$ et $Y$ be a random variables integrable such that $XY$ also integrable . Prove that : if $X$ and $Y$ are independent $\implies$ $E(X|Y)=E(X)$ $\implies~~~E(XY)=E(X)E(Y)$ My ...
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### uniformly bounded Tail probabilities by the tail probability of another random variable

Consider the following setting. Let $1 < p < 2$, let $(X_i)_{i \in \mathbb{N}}$ be a sequence of non-negative (real-valued) random variables and let $(D_i)_{i \in \mathbb{N}}$ be a sequence of ...
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### Ascending ladder variables and epochs when the time is constraint

Let $\{S_k\}_{k\ge0}$ be a random walk with $S_0=0$. Set $T_0=0$ and define $$T_1=\inf\{k>0: S_k>0\}$$ and $$T_j=\inf\{k>T_{j-1}:S_k>S_{T_{j-1}}\}, j\ge2.$$ If no such $k$ exists we ...
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### What is the probability of finding 2 in set of primes?

Let $$S=\{p :p\in\mathbb{P}\}$$ Be the set of primes. So $2$ is also a member of it right? $$P[\text{ finding 2 in S}]=\frac{1}{\infty}=0 ??$$ Does that mean that probability of finding $2$ from the ...
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### Klenke's Proof of De Finetti's Theorem

There's a technical problem I ran into when working through Klenke's proof of De Finetti's theorem (Theorem 12.24 on Klenke's Probability Theory: A Comprehensive Course, pages 239-240). The notation ...
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### a symmetrical die is thrown, then a coin is thrown as many times as the die indicates .

Exercices : a symmetrical die is thrown, then a coin is thrown as many times as the die indicates points. let $X$ be the number of Tails obtained . Determine $E(X)$ My attempts : First it's clearly ...
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### Versions of stochastic processes have the same generated sigma algebra

Let $X_t$ and $Y_t$ be two stochastic processes such that they are versions of each other, viz, $\mathbb{P}(X_t = Y_t) = 1, \forall t$. Is it then so that the sigma albegra generated by $X$ is the ...
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### Is this Markov process Gaussian ? $Y_t= X_t - (1-t) X_0 - tX_1, \, t \in [0,1]$

Let $(X_t)$ be a real sample continuous stochastic process with density function $f_t$. Let $$Y_t= X_t - (1-t) X_0 - tX_1, \, t \in [0,1].$$ Suppose that $(Y_t)$ is Markov with regard to its own ...
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### Prove or disprove with $\lim_{n \rightarrow \infty}\ \varphi \ v_{n}$ the existence $V$ with $V_{n}\overset{d}{\rightarrow} V ,n\rightarrow \infty$

Prove or disprove with $\lim_{n \rightarrow \infty}\ \varphi \ v_{n}$ the existence of a random variable $V$ with $V_{n}\overset{d}{\rightarrow} V ,n\rightarrow \infty$ $V_n$ is a sequence of random ...
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### $\mathbb{P}(-1\leqslant X\leqslant\frac{1}{2})$ from $\varphi_{X}(t)=\frac{1}{7}\left(2+e^{-it}+e^{it}+3e^{2it}\right).$

Let $X$ be a random variable with characteristic function given by $$\varphi_{X}(t)=\frac{1}{7}\left(2+e^{-it}+e^{it}+3e^{2it}\right).$$ Determine $\mathbb{P}(-1\leqslant X\leqslant\frac{1}{2})$. ...
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### Question on interchanging of random variables with the same distribution inside expectation

Let $T\in\mathbb{N}$ and let $(S_t)_{0\leq t\leq T}$ be such that the increments $S_1-S_0,\dots,S_T-S_{T-1}$ are independent and identically distributed, and let $(\mathcal{F}_t)_{0\leq t\leq T}$ be ...
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### Durrett's Probability: Theorem 6.2.6

I am having some difficulty understanding the concept of measure preserving, invariance, and ergodic. Here is a proof from Theorem 6.2.6 in Durrett's Probability: Theory and Examples, 5e (p.338) (...
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### Are stochastic processes defined on product of probability spaces?

The following question has confused me lately. Suppose that you have a sequence of random variables $\{\xi_n(\omega)\}$ defined on some probability space $\left( \Omega,\mathcal{F},\mathbb{P} \right)$,...
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### Determine the weak convergence and applicable the weak limit

$(\mu_n)_{n \in \mathbb{N}}$ is the probability measure, $\lambda$ a Lebesgue-measure on $(\mathbb{R}, \cal{B}\mathbb{(R)})$ (\mu_n)_{n \in \mathbb{N}} = f_n \lambda, f_n(x) = \sqrt{\frac{n}{2 \pi}...
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### A statement about finite Markov chains

The following quantity $\tilde\pi$ is defined in the textbook Markov chains and mixing times by David A. Levin. Here $\tau_z^+ = \min \left\{t\geq 1| X_t = z\right\}$. Let $z \in \mathcal{X}$ be an ...
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How do we compute the transition probabilities in a continuous time Markov chain? Supposing $h$ is sufficiently small then how would I compute $p_{i,j}(h)$, I am aware of the relation to the generator ...