Use this tag only if your question is 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 ...

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2answers
50 views

Is the probability of the union of events nondecreasing in the probability of the events?

Can it be shown that the probability $P(A_1 \cup \dots \cup A_n)$ is nondecreasing in the probability of any event $A_i$? This fact seems intuitive to me, independent of the fact whether $A_1, \dots, ...
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0answers
31 views

Application of dominated convergence theorem to conditional expectation

Suppose we have the following sequence of random variables such that \begin{align} Z_a&=E^2[V^2|aV+W]\\ \lim_{a \to 0} Z_a&= E^2[V^2] \quad (a.s.) \end{align} where $V$ and $W$ are ...
2
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1answer
30 views

Symmetric Difference Approximation of a Measurable Set [duplicate]

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{A}$ be an algebra of subsets of $\Omega$ such that $\sigma(\mathcal{A})=\mathcal{F}$. Prove that for all $B\in \mathcal{F}$ and for ...
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2answers
26 views

derive integration by parts for a stochastic integral

The question is to show the following identity: $\int_{0}^{T}tdW(t) = TW(T)-\int_{0}^{T}W(t)dt$ This can be done quite easily with ito's however the question explicitly says to show the identity ...
2
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0answers
23 views

Moving Average of an Ergodic Markov processes.

Let $ \{X(t); t\geq 0\} $ be an ergodic Markov process, and let G be a positive integrable function with $\int_0^\infty G(x)dx=1$. Does $$F(t)\doteq G*X(t) = \int_0^t G(t-s)X(s)ds $$ converge in ...
1
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1answer
13 views

Convergence rate of mean and standard deviation.

I have a random variable simulator with Normal distribution $(\mu,\sigma^2)$. I repeatedly conduction simulation. Each time, the simulation gives $N$ numbers $x_1,x_2,\ldots,x_N$. I use the $N$ ...
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0answers
42 views

How do I write this probability argument correct in the measure-theoretic sense?

We define the characteristic function of a random variable X, by $\phi_X(u)=E[e^{iuX}]$. My book has written an argument for the characteristic function random variable.That is, we assume N is ...
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0answers
13 views

On Laplace transforms - Applications in Probability Theory

I'm trying to find good bibliography on Laplace transforms for Applications in Probability Theory. I can't understand deeply the importance of this tool; nor I was taught very much on the subject. ...
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0answers
28 views

Find a Markov chain transition kernel

Let $f_{X}$ be a density we would like to sample from. For some reasons, $f_{X}$ may be analytically intractable or expensive to evaluate. A solution consists in considering a density $(x,y) \in X ...
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0answers
28 views

How can we prove that the derivative of a generalized Hilbert space valued Brownian motion is a Gaussian white noise?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\lambda$ be the Lebesgue measure on $[0,\infty)$ $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of ...
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0answers
14 views

How is the measure of states defined in Markov Chain [closed]

Given a Markov chain, is the sample space defined as "states"? how is the measure of the states defined? Does the measure depend on time?
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0answers
40 views

Tail estimate for $L^1$ functions.

Suppose $f\in{L^1(\mu)}$ for some probability measure $\mu$. Pick $\epsilon>0$ and let $A_n=\{x:|f(x)|>\epsilon{n}\}$. I want to show that $$\mu(A_1)+\mu(A_2)+\dots<\infty$$ My first ...
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0answers
14 views

Lower bound of KL-divergence between unknown and known distribution

Suppose I have two multivariate continuous random variables $X$ and $\hat{X}$ with underlying probability distributions $P_{X}$ and $P_{\hat{X}}$, where $P_{\hat{X}}$ is a Gaussian approximation of ...
2
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0answers
34 views

Martingale under conditional prob. measure (definition)

Suppose we are given a probability space $(\Omega, \mathcal{F},P)$ s.t. r.v.s $X$ and $(Y_i)_{i=1}^\infty$ are $\mathcal{F}$-measurable. The relevant filtration is given by ...
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2answers
48 views

How to find the joint PDF?

Ley X,Y two random variables with density $f(x,y)=8xy \ \text{if} \ 0<x<y<1$ Find the joint distribution F(x,y). I find that $F(x,y)=2x^2y^2-x^4\ \text{ if} \ 0<x<y<1$ but I ...
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0answers
16 views

Locally square integrable (local) martingales

I'm reading Protter and sometimes he says "locally square integrable martingale", and sometimes he says "locally square integrable local martingale", and I wonder if these two are the same. Protter's ...
1
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1answer
46 views

Random walk on $\mathbb Z/m\mathbb Z$ converges to uniform distribution

Let $(X_t)_t$ be the standard continuous time random walk on $\mathbb Z/m\mathbb Z$ with $X(0)=[0]$ almost surely, then I want to show that for $t \rightarrow \infty$, $\lim_{t \rightarrow \infty} ...
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1answer
24 views

Proving the impossibility of a particular binary sequence

Let $\Omega = \{0,1\}^{\mathbb{N}}$. My question is as follows. Can there exist an $\omega \in \Omega$ such that $$\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \omega_{k+a} = \frac{1}{2} \qquad ...
1
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1answer
22 views

Are these properties for independence false?

I have tried to prove that if: If $X$ is independent of $Y$, and $Y=Y_1+Y_2$, and $Y_1$ and $Y_2$ are independent. Then $X$ and $Y_1$ is independent. But I am not able to prove this, but I ...
2
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0answers
42 views

$X_{n}$ independent and there is $a_{n}\to0$ s.t $\lim\limits _{m\to\infty}a_{m}\sum_{n=1}^{m}X_{n}$ is finite w.p 1. Then the limit is constant.

I'm trying to prove the following claim: Suppose $X_{1},X_{2},...$ are independent and there exists a sequence $\left\{ a_{n}\right\} _{n\geq1}\subseteq\mathbb{R}$ s.t $\lim\limits ...
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0answers
31 views

heuristic for expected number of visits random walk

What is the heuristic argument that explains why, on $\mathbb{Z}^d$, $d \geq 3$, the expected number of visits of a random walk starting from the origin at $x$ is of order $$ O(|x|^{2-d})? $$
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0answers
50 views

Prove that the limit in probability of normally distributed random variables is normally distributed, too [duplicate]

Let $X_n\sim\mathcal N_{\mu_n,\sigma_n^2}$ for some $(\mu_n,\sigma_n^2)\in\mathbb R\times(0,\infty)$ and $X$ be a real-valued random variable with $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
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0answers
34 views

Convex closure of the support of a Levy Process

Given a Levy Process $X_{t}$ on a filtered Probability Space $(\Omega,\mathcal{F},\mathcal{F}_{t},P)$ with distribution-function $F_{t}$ for $X_{t}$. We look now for the cumulant transform $\phi_{1}$ ...
0
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1answer
28 views

The set of points in $\Omega$ which belong to exactly $k$ events is an event

This exercise if taken from Probability: An Introduction by Grimmett and Welsh. In what follows, $\Omega$ is a set and $\mathcal{F}$ is an event space of subsets of $\Omega$ (that is, $\mathcal{F}$ is ...
2
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0answers
33 views

Property derived form Monotone Convergence Theorem

Assume $\mathbb{E}|X_1| < \infty$ and $X_n \uparrow X$ a.s., then Monotone Convergence Theorem either provide $\mathbb{E}X_n \uparrow \mathbb{E} X < \infty$ or else $\mathbb{E}X \uparrow \infty$ ...
3
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1answer
24 views

$\sigma$-field generated by the continuity sets of a measure

Let $\mu$ be a probability measure on the Borel subsets of a topological space $X$ (a compact metric space if necessary). A Borel set $B$ is a $\mu$-continuity set if $\mu(\partial B)=0$, where ...
0
votes
1answer
13 views

marginalising probability

Given that $\sum_b p(a|b)p(b) = p(a)$ and its extension $\sum_b p(a|b,c)p(b,c)=p(a,c)=p(a|c)p(c)$ can I say anything about the following $\sum_b p(a|b,c)p(a,b)$? Whilst I clearly can't express ...
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3answers
302 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
2
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0answers
69 views

Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.

Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of IID random-variables s.t $\mathbb{E}\left[X_{i}\right]=0$ and $\left|X_{i}\right|\leq K$ . Let $S_{n}=\sum_{i=1}^{n}X_{i}$ , I want to show ...
2
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1answer
30 views

Find the expected number of edges in the graph.

It is given that we have a graph with n nodes labelled $\left\{1,2,...,n \right\} $. For each pair of nodes $\left(i\neq j\right)$ , A fair coin is tossed to decide if there should be an edge ...
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0answers
40 views

Distribution of Product of Matrices holding Multivariate Random Normal Variable Observations

We have matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ that hold observations from multivariate normal distributions. Each matrix represents $\boldsymbol{n}$ observations of $\boldsymbol{m}$ variables ...
0
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1answer
24 views

Show $P\{\xi_1+\dots+\xi_n=1\}=(\sum_{i=1}^n \lambda_i)\Delta + O(\Delta^2)$ (Shiryaev's Probability page 44)

I am trying to do the following exercise: Let $\xi_i,\dots, \xi_n$ be independent Bernoulli random variables such that $$ P\{\xi_i=0\}=1-\lambda_i \Delta\\ P\{\xi_i=1\}=\lambda_i \Delta $$ where ...
2
votes
1answer
57 views

If $X_1, X_2, …$ are independent and $E(X_n) =0$ and $E(X^2_n)$ is bounded and exists, how to show that the average converges almost surely to $0$?

I am trying to show that if $X_1, X_2, ...$ are independent and $E(X_n) =0$ and $E(X^2_n)\leq M < \infty$, then $\frac{X_1+X_2+ \ldots +X_n}{n} \to 0$ almost surely. I am able to show this in the ...
1
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0answers
46 views

Log concavity of $\int_{x \in \mathcal{C}_1} t^n \, e^{-\sum_i x_i t} \; dx$

Consider a real function $f(x) > 0$ with $n$ strictly positive arguments $x_i>0$ with the following properties: i) $f(x)$ is homogeneous of degree one in $x$ ii) Strictly increasing in all ...
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0answers
22 views

stationary distribution of outputs in Markov chain

consider a hidden Markov model with two states, with following transition/observation matrices: $T = \left( \begin{array}{cc} 0.9 & 0.1 \\ 0.1 & 0.9 \end{array} \right), O = \left( ...
1
vote
2answers
27 views

Finding the probability space of the given experiment.

Specify the probability space completely for the following experiment: tossing a fair coin till we see the first heads. Here is what I have done so far: The sample space is simply $T^n H$ where $n$ ...
1
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0answers
23 views

Does bounded and closed equal compact for sets of Borel probability measures?

Equip the space of Borel probability measures on a fixed closed subset X of the s-dimensional Euclidean space with the topology induced by weak convergence of probability measures. In this setting, ...
0
votes
1answer
29 views

Probability that $X$ is even for a Poisson?

I know that this question has been asked several times, and to my knowledge I have read through all the versions. However, I can't find a complete proof, and I am having trouble finishing the ...
0
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2answers
53 views

Show that a random variable $T_x$ is uniformly distributed given that $T$ is uniformly distributed?

We have a lifetime $T$, which is uniformly distributed over $(0,b)$. We then introduce a new r.v., $T_x=T-x$, which is defined on $0<x<b$. We want to show that given $T>x$, the variable ...
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0answers
42 views

Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?

All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties ...
1
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1answer
21 views

A one-sided continuous stochastic process is product measurable. Does the same hold true for almost surely continuous processes?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(X_t)_{t\ge 0}$ be a real-valued almost surely continuous stochastic process on $(\Omega,\mathcal A,\operatorname P)$. Let ...
1
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1answer
27 views

Expectation of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $\mathbb R$ and $$\langle ...
2
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0answers
28 views

Any rule of thumb that says any reasonable function I can write down is measurable?

Question arose in the context of probability: If $Y$ is $F-$measurable and $h$ is some function, $h(Y)$ is $F-$measurable if $h$ is a measurable function (Doob-Dynkin). For example, let $f(x,y)$ be ...
1
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1answer
25 views

Is $\phi B(\omega,\;\cdot\;)$ Lebesgue integrable over $[0,\infty)$ for a real-valued Brownian motion $B$ and $\phi\in C_c^\infty(\mathbb R)$?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $\lambda$ be the Lebesgue measure on $\mathbb R$. Is ...
0
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1answer
21 views

Why is this a Dynkin system: $\{A \in \mathcal{A} \colon E[X1_A]=E[Y1_A]\}$

Let $X,Y$ be two random variables on $(\Omega, \mathcal{A},P)$. Why is the set $\mathcal{D}:=\{A \in \mathcal{A} \colon E[X 1_A] = E[Y 1_A]\}$ a Dynkin system? Suppose $A \in \mathcal{D}$. Then ...
-1
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0answers
16 views

Martingale Convergence Theorem for non-negative supermartingales: Why is limit non-negative?

Consider a supermartingale $(X_n)$ that is non-negative. The martingale convergence theorem states that $X_n \rightarrow X$ P-a.s. with $X \geq 0$ and $E[X] \leq E[X_0]$. Why can we conclude that $X ...
1
vote
1answer
65 views

Simple CDF Computation for Products of Random Variable

Let $X(k)$ be i.i.d random variable governed by uniform distribution $[-1,1]$ for $k=0,1,2,...N$. I would like to compute the following CDF $$ P\left( {\prod\limits_{k = 0}^{N - 1} {(1 + X(} k)) ...
2
votes
1answer
23 views

What is the probability that there are $k$ people between $A$ and $B$ sitting around a circle?

I have $n$ people seated around a circular table and $2$ of the people are $A$ and $B$. What is the probability that there are $k$ people between $A$ and $B$? I have tried noting that the total ...
0
votes
0answers
33 views

3d symmetric random walk passes infinitely through any particular line

I'm trying to solve problem 27 from Chapter XIV An Introduction to Probability Theory Volume I by William Feller, ...
2
votes
1answer
35 views

Calculate the probability, that a man repair 20 machines in 8 hours. It is correct my work?

The problem statement said: The servicing of a machine requires two separate steps, with the time needed for the 1st step being an exponential random variable with mean 10 minutes and the ...