Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Qual martingale theory question

Suppose $Y,X_{1},...,X_{n},\ldots$ have the following properties: $Y$ has the exponential distribution. That is, $P(Y>t)=\exp(-t)$ for $t>0$; Conditionally on $Y, X_{1},\ldots,X_{n}$ are i.i.d. ...
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0answers
71 views

Qual Question concerning martingale

Suppose $X_n$ is a sequence of random variables that has the property that $\sup|X_n| \leq 1$ a.s. Then use Doob's decomposition to prove that $\sum_{n\geq 1} X_n$ converges a.s. iff the sum ...
2
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1answer
33 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
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3answers
48 views

Independence of $X$ and $2X$

Are these two random variables independent? Unfortunately, I don't know probability theory enough to answer this question. I know for a fact that if $X$ and $Y$ are independent random variables and ...
3
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1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
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0answers
24 views

Number of trials for two successes

Let Y denote the number of successes required for two successes in a series of Bernoulli trials with parameters p and q. I want to know whether $P[Y=n]=\binom{n}{2}p^2q^{n-2}$ or ...
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0answers
27 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
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0answers
19 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
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0answers
12 views

How to Prove this Multinomial Distribution Inequality

I have the following lemma, but there seems to be one (or two) mistakes in the proof found in this paper (lemma 3). The lemma states that for $Multinomial(n,p_1,\ldots,p_k)$ distributed ...
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1answer
54 views

Probability to pick an integer number $k\in\mathbb{Z}$ from the field $\mathbb{R}$

In the field of the real numbers $\mathbb{R}$, we build a subset $A\subset\mathbb{R}$ $A=\{k\}, k\in\mathbb{Z}, k=-N,-N+1,...,0,1,2,..,N$. If we pick infinite times a number $x$ from $\mathbb{R}$ with ...
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0answers
9 views

About the space $((R^n)^{[0,\infty)},\mathcal{B},\tilde{Q}^x)$ in Oksendal SDE book

I am reading the book Stochastic differential equations (6th ed.) by Oksendal. I am not sure about the meaning on P.146. (Below Theorem 8.3.1) It says that ``if we identify each $\omega \in \Omega$ ...
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1answer
66 views

Convergence in Probability (weak law of large numbers) [closed]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d. standard normal random variables. Prove that $$\frac{X_1X_2 + X_2X_3 + X_3X_4 + \cdots + X_{n−1}X_n}{n}$$ converges in probability to 0. I started this by ...
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0answers
15 views

Are functions of independent random variables related to each other by a constant independent

I have $6$ random variables $a,b,c,d,f,g$, each having independent Gaussian distribution. Now I define following three random variables \begin{equation} X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ...
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2answers
24 views

Reliability Probability problem

What is the Probability that at least one close path is formed from A to B where each switch has a Probability of close = p and each switch acts independent of the other Proposed Solution Let ...
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1answer
35 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
2
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0answers
40 views

Sequence of independent RV [duplicate]

Let $(X_n)$ be a sequence of independent RVs that converges in probability to some RV $X$. Show that $X$ is constant a.e. My attempt So from definition of convergence I have: $$\forall\epsilon>0 ...
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1answer
26 views

Q-matrix vs. P-matrix description of a Markov chain

Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$. The ...
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0answers
14 views

When is a characteristic function of a distribution real analytic?

I know that a characteristic function of a random vector $X$ is real analytic, if e.g. $X$ takes values in a bounded set of $\mathbb R^n$ or in other words if the density $p_X$ has compact support. ...
3
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2answers
46 views

recurrent events-Probability of even number of successes

Let E be the event of an even number of successes. $u_n$:Probability of E occurring at the nth trial not necessarily for the first time $f_n$:Probability of E occurring at the nth trial for the first ...
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0answers
30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
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1answer
22 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
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1answer
44 views

What is the limit of $nf(x+n)$ as $n\rightarrow \infty$? Here $f(x)$ is prob. density function.

I tried the cases when $f(x)$ are the densities of normal and student t distribution. In both cases, the limit is $0$. I guess this conclusion might hold in general. I tried the following. Let ...
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1answer
26 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
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1answer
40 views

Find $\liminf X_n$ where $X_n=1_{[n,n+1]}$?

My attempt: Suppose $\omega=n_0$. Then choose $N\geq n_0+1$.Threfore, $X_N(\omega)=0$. Therefore, $\inf_{k\geq N}X_k(\omega)=0$. Does it suffice to prove that $\liminf\limits_{n \rightarrow \infty} ...
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0answers
25 views

Mutual Information for Gaussian Process (and also Fano's Inequality)

According to this presentation: Bounding Gaussian Process Information Gain we have a closed-form expression for the information gain as follows: $$ I\left(\vec{y} \mid f\right) = \frac{1}{2} \log\det ...
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1answer
64 views

Exponential Distribution question

I'm having some trouble understanding the mechanics of how to solve with this distribution. The question: The number of years that a washing machine functions is a random variable whose hazard rate ...
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1answer
19 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
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1answer
19 views

Prokhorov-like convergence

Let $(X,d)$ be a metric space, and for any $A\subseteq X$ define $$ A^\delta:=\{y\in X:\exists x\in A \text{ such that }d(x,y)\leq \delta\}. $$ Under which conditions on $(X,d)$, $A \subseteq X$ and ...
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1answer
28 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
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1answer
39 views

Property of Wiener process sample path

What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$? We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
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35 views

Derive $f(Ax)=f(x)g(A)$: the property for scale invariance.

This is part of a proof for the following question: Show that if a probability density function $f(x)$ with $x>0$ is scale invariant then $f(Ax)=\frac{1}Af(x)$ where $A$ is a real constant. ...
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1answer
18 views

$f$ nonnegative map on $X$ implies $\exists g$ s.t. $S \rightarrow \sum_{\omega \in S} g(\omega) f(\omega)$ is probability measure on $2^X$

This is exercise 3.3 in Chapter B of Efe Ok Probability with Economic Applications (freely available online). The claim we are asked to prove is : Let X be a countably infinite set, and $f$ a ...
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1answer
12 views

$X$ countable and $\exists$ $f : X \rightarrow [0,1] \Rightarrow \big(X,2^X, \sum_{\omega \in S} f(\omega)\big)$ is a probability space.

This is exercise 3.1 in Chapter B of Efe Ok Probability with Economic Applications (freely available online). Part of the claim we are asked to prove is: Let X be a nonempty countable set. If ...
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1answer
16 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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0answers
35 views

What is the variance of an arbitrary “good” function of several independent normally distributed random variables

During my studies years ago I came over a formula that states something like if $x_i$ are independent normally distributed variables with variances $\sigma^2_i$ and $f(x_i)$ is differentiable (and ...
2
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1answer
54 views

Filling of a tank - recurrence relation

Suppose a tank has a maximum limit of 100 units. Each day 2,1 and 0 units are added to the water level with probability p,r and q. Any excess water would overflow and if it reaches the minimum level ...
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0answers
53 views

$X_n=n^21_{(0,1/n)}$. What is $\limsup\limits_{n \rightarrow \infty} X_n$

$X_n=n^21_{(0,1/n)}$. What is $\limsup\limits_{n \rightarrow \infty} X_n$? What about its $\liminf\limits_{n \rightarrow \infty} X_n$? My attempt: For each $n$ on $\{0, [1/n,1]\}$, we have ...
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0answers
38 views

Prove that E$(\liminf\limits_{n\rightarrow \infty} X_n )\leq \liminf\limits_{n\rightarrow \infty}E(X_n)$

I want to prove that if $X_n\geq 0$, then E$(\liminf\limits_{n\rightarrow\infty} X_n )\leq \liminf\limits_{n\rightarrow\infty}E(X_n)$. My attempt: $E(\liminf\limits_{n \rightarrow \infty} ...
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1answer
30 views

How to compute a marginal probability

Given a weighted graph, using the Kirchhoff's matrix tree theorem, how can I compute the marginal edge presence probability: $$P_\beta(ij)=Z_\beta^{-1}\sum_{\text{T spanning tree:$(i,j)\in ...
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2answers
70 views

Expected number of coin tosses needed until all coins show heads

We flip $n$ fair coins every iteration of the game. Every coin that shows heads is removed from the game and we use the remaining $n-k$ coins to play the game again (where $k$ is the number of heads ...
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1answer
25 views

convergence in measure of min $(f_n,g)$

I was reading a proof of a convergence in measure variant of fatou's lemma earlier and there was a seemingly easy part of it I just could not verify. Assume $(f_n)_{n \in \mathbb N}$ is a sequence of ...
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1answer
25 views

Almost Trivial $\sigma-$fields

I am trying to understand the proof of the following Lemma form the book A probability path by sidney Resnick. Lemma: Let $\mathcal{G}$ be an almost trivial $\sigma-\text{field}$ and let $X$ be a ...
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2answers
144 views

Are random selections from iid random variables independent?

Let us have identically independently distributed random variables $x_1, x_2, \dots, x_{10}$. Now let us pick indices $\alpha, \beta$ uniformly independently from $1,2,\dots,10$. Are variables ...
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1answer
46 views

Determining whether an uncountable set of integral equations yield a unique solution

I am interested in the set of numbers $\alpha>0$ for which there exists a function $g:\mathbb{R}\to[0,1]$ satisfying $$ \forall r\in \mathbb{R} \qquad f(r) = \int\limits_\mathbb{R}\! g(\alpha ...
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1answer
30 views

Every martingale is also a martingale with respect to its own filtration

I want to prove the following: Let $A_0, A_1, ..$ be a martingale with respect to the sequence $B_0, B_1, ..$. then $(A_i)_{i\geq0}$ is also a martingale with respect to itself. I have no idea how ...
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Almost surely convergence with stationary random vectors

I dont seem to be able to incorporate the stationarity condition into any of limit theorems I know. I cannot see how the Birkhoff almost everywhere ergodic theorem could be used as I cannot see how ...
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2answers
42 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
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2answers
24 views

What is the probability when i attempt twice?

// I have never got probabilities' lessons , I only know some basics Let's say the probability of me hitting the target ( let's consider hitting a bottle with a soccer ball ) in ONE ATTEMPT is ...
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0answers
41 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
0
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1answer
71 views

How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...