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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57 views

Generalization of classic 3 roll die game to $n$ rolls

I am trying to generalize the following well-known 3 roll die problem: "We roll a single die no more than 3 times. We can stop immediately after the first roll, immediately after the second roll, or ...
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0answers
12 views

Fast Convergence of the Tails of a Random Variable [closed]

Suppose $X$ is a random variable with $\lim_{n\to\infty}\int_{\Omega}n\mathbb{I}_{\{|X_1|>n\}}(\omega)dP(\omega)=0$. What does this say about ...
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2answers
33 views

Ratio of Expected values of Boys to Girls

In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of ...
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0answers
26 views

Does a analytic joint distribution necessarily have continuous marginals?

Although the question actually popped up in a course about evolution equations, it seemed most natural to ask this in the context of joint distributions. Namely: Suppose $X$ and $Y$ are two ...
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1answer
40 views

Characteristic functions of random variables are non-negative definite

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X:\Omega\to\mathbb{R}$ a random variable. How to prove that the characteristic function $\varphi_X(t) = E[e^{itX}] ...
3
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1answer
42 views

Characterization of Normal RVs by uni variate version?

If $X$ is a symmetric $n$-dimensional random vector with mean $0$ then is it true that: \begin{align*} & X \text{ follows a multivariate normal law} \\ & \text{iff} \\ & \|X\| \text{is a ...
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1answer
37 views

Counterexample that a measurable function does not guarantee almost sure convergence

I try to find a counterexample that if $X_n\xrightarrow{\text{a.s.}}X$ then for a measurable function $g:\mathbb{R}\to \mathbb{R}$ this does not imply that $$g(X_n)\xrightarrow{\text{a.s.}}g(X)$$ ...
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0answers
30 views

Changing a measure? [closed]

If I have a probability measure $Q$, what does it mean to say that we "change the measure" to $Q' \colon = XQ$ where $X$ is some random variable? What's that supposed to mean?
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1answer
24 views

Characteristic function of Laplace distribution

I'm trying to derive the characteristic function for the Laplace distribution with density $$\frac{1}{2}\exp\{-|x|\}$$ My attempt: $$\frac{1}{2}\int_{\Omega}e^{itx-|x|}\mathrm{d}x$$ ...
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0answers
21 views

Convergence in distribution to a constant implies convergence in probability to that constant.

Suppose that $(X_1,X_2,...)$ is a sequence of random variables and that the distribution of $X_n$ converges to the distribution of the constant $c$ as $n\to\infty$. Then $X_n\to c$ as $n\to\infty$ in ...
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0answers
26 views

Integrating over random boundary

What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"?
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1answer
52 views

If X, Y, Z are independent random variables, then X + Y, Z are independent random variables. [duplicate]

I found the same question (X,Y,Z are mutually independent random variables. Is X and Y+Z independent? here), but the answer uses characteristic functions and fourier inversion theorem, but this is ...
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0answers
31 views

Prove that ($Y_n$,$F_n$) is martingale. What does ($Y_n$,$F_n$) mean in this problem?

$X_1$, .... , $X_n$ are independent variables, $P(X_i=1)=p$, $P(X_i=-1)=q$, where $0<p<1$. $F_n=\sigma(X_1,.....,X_n)$ and $Y_n=(\frac{q}{p})^{X_1+....+X_n}$. The task is to prove that ...
6
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0answers
55 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
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1answer
30 views

Does this set of infinite binary sequences have positive probability?

The AMM article "What is a random sequence?" argues (at the end of Sec. 2) that if, from the set of all binary sequences, we remove those (countably many) that have "computable regularities", then the ...
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0answers
26 views

The series $\sum_{i=1}^n \frac{Y_i}{n} _{n\ge 1}$ converges? Where $Y_i = min(1,X_i)$ and $X_i$ is a serie of random variables. [on hold]

I've been finding some problems to solve this kind of question when studying to my probability's exam. We should probably use the law of great numbers, but I can't see how. The complete exercice: Let ...
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0answers
27 views

Information in Filtrations

Is the “information” kept track of by filtrations the same as information-theoretic “information”? If not, is there some way the two concepts can be reconciled?
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1answer
27 views

Lévy-process property

I get a problem that comes up in the construction of the Lévy-Itõ decomposition. For a Lévy-process $X$ there is a independently scattered poisson random measure $N$, such that for each t, and for ...
2
votes
1answer
72 views

Notation i.i.d sample

I am learning measure theory and sometimes I am not sure if I am using the correct notations, especially with respect to distributions of random variables. In the following I try to formulate the ...
2
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0answers
33 views

Indicator Functions properties in Probability

This is probably quite a straightforward question but I just thought I should double check my working. Say for instance we have two fixed values $K_{1}$ and $K_{2}$ such that $K_{2} > K_{1}$ and ...
2
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1answer
19 views

Max Likelihood Examples, Stuck in Calculation [closed]

We get samples 2,4,8,16 be random instances that get from distribution with following PDF. maximum likelihood estimation of $ (\alpha, \sigma) $ is : $ \frac {2}{3 ln 2}, 2$. $ f_{\alpha, ...
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0answers
47 views

Lim Sup and Measurability of one Random Variable with respect to Another

Here, there is a common proposition in probability theory : Let $X,Y: (\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{R})$ where $\mathcal{R}$ are the Borel Sets for the Reals. Show that ...
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0answers
11 views

K-wise identical marginal distributions

Suppose I have two joint distributions described by the two sequences of random variables,$X_1, \ldots, X_n$; $Y_1, \ldots, Y_n$. Is there a name/theory/reference for when these two distributions ...
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0answers
36 views

The probability density function of the product of independent exponentially distributed RVs

When $X$ ~ exponential distribution(10) and $Y$ ~ exponential distribution(15), and they are independent, find the probability density function of $Z = XY$ I just took an exam for probability ...
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0answers
19 views

Ito's lemma in infinite dimensional spaces

I'm trying to use the ito's lemma in infinite dimensional spaces applicatte to $F(X)=\Vert AX\Vert^{2}$, where $A$ is a linear map. But i I have trouble calculating the integral $\int_{0}^{t}\langle ...
1
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1answer
17 views

Relative entropy between discrete and continuous random variables

Is this possible to define relative entropy between discrete and continuous random variables? Say $P$ is a discrete pmf and $Q$ is a continuous pdf, what is $D(P||Q)$?
2
votes
1answer
72 views

Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
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0answers
49 views

If $X_n\to0$ in probability then $X_n/r_n\to0$ in probability, for some $r_n\to0$

Suppose that a sequence of positive random variables $X_n$ is such that for all $\epsilon >0$, $P(X_n>\epsilon)\rightarrow 0$ as $n\rightarrow\infty$, that is $X_n$ converges in probability ...
4
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0answers
76 views

Suppose $E[X_1] <\infty$. Show that $lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s.

Let $X_1,X_2,X_3,...$ be i.i.d. with $P(X_1 >0)=1$. Define $S_n =\Sigma_{i=1}^{n} X_i$. (a) Suppose $\mathbb{E}[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s. I ...
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0answers
42 views

Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
0
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1answer
38 views

Maximum process of Brownian motion

Consider the linear standard Brownian motion $(B_t)_{t \ge 0}$. We define the maximum process $(M_t)_{t \ge 0}$ of $(B_t)_{t \ge 0}$ to be such that $M_t = \max_{0\le s \le t} B_s$. Prove that the ...
0
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1answer
36 views

Existence and uniqueness of SDE, is the independence requirement needed?

In Bernt Øksendals Stochastic differential equations he has this theorem in chapter 5: $\\\\\\$ However, in the proof I can not see where he uses the independence condition I marked in red. Do you ...
1
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1answer
30 views

random variable probability problem

I am trying to find the answer to a mathematical probability problem. let a box contain $5$ balls : $2$ balls white, $2$ balls green, and $1$ red ball (we can't differentiate between the balls by ...
1
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1answer
20 views

A formula for an expected value

We have a Markov chain with $X_0 = z$, the return time $\tau_z$ of the first time at which we return to $z$, and some other state $y$. A proof I'm reading states: $$\operatorname{E}(\text{number of ...
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0answers
22 views

Verifying data came from a Wiener Process

From the Wiki article a Wiener Process has the properties that $$E[W_t] = 0$$ $$Var[W_t] = t$$ According to A Standard Wiener Process the Wiener Process is given by: $$W(t) - W(s) \tilde{} ...
3
votes
1answer
38 views

Limit Brownian Bridge Integral

As a solution of the Brownian Bridge SDE, we arrive at the solution \begin{align} X_t = (1-t) \int_0^t \frac{1}{1-s}\ dB_S \end{align} defined for $0 \leq t <1$. In order to show that for any $g ...
2
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0answers
29 views

The relationship between random variables, distribution functions and probability measures

Given a probability space $(\Omega,\mathcal{F},P)$, and a random variable $X\colon\Omega\to\Bbb{R}$, we can associate with it its distribution function $F\colon \Bbb{R}\to[0,1]$ defined as ...
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0answers
46 views

Version of the CLT

It is well known, that for a sequence of i.i.d. rv. $X_i$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^{2}$ that $$ ...
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0answers
21 views

Conditional Distributions vs. Stochastic Processes

Is the concept of a version of a stochastic process related to the concept of a version of a conditional distribution? And is a regular version of a stochastic process somehow the same thing as the ...
0
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1answer
15 views

Almost sure convergence of the inverse

If a sequence of non-negative random variables $X_1, X_2, \dots$ converges almost surely to a random variable $X$, that is $X_n \xrightarrow{a.s} X$ or equivalently ...
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0answers
31 views

Problem that may use Borel Cantelli Lemma

So, there is a sequence of identically distributed independent random variables taking values on the integers, and they have a positive expectation. The problem is to prove that with probability 1 the ...
2
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1answer
31 views

Buyer's price in terms of risk-neutral measures

Let us consider a finite arbitrage-free market model $(B,S)$, where $B$ is a bank account and $S$ is a share. Let $X$ be a claim. We define a buyer's price of $X$ as follows:$$\Pi^b_0(X)=\sup \lbrace ...
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0answers
33 views

$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
3
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0answers
46 views

Find $P(B_3>0,B_6>0)$ where $(B_t)$ is a Brownian motion

Suppose that $B_{t}$ is a standard Brownian Motion. What is the probability that both $B_{3}$ and $B_{6}$ take positive values? This is what I've tried but then I get stuck and I'm not sure how to ...
0
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2answers
29 views

Integration by parts - Brownian motion and non-random function

Let $B$ be a standard one-dimensional Brownian motion. I want to show for a continuously differentiable non-random function $\phi$ that, \begin{align} \int_0^t \phi(s) dB_s = \phi(t) B_t - \int_0^t ...
1
vote
1answer
22 views

Derivation of a property of standard Wiener processes

I am reading A Standard Wiener Process and am struggling to piece together how they arrived at their conclusion. The major properties of any Wiener Process are: $W(t) = 0$ $W(t) - W(s) \sim N(0, ...
1
vote
1answer
32 views

Question regarding local martingales.

In the definition of a local martingale I have that for a filtered probasbility space $(\Omega, \mathcal{F},P,\mathbb{F})$. A local martingale is an adapted process M, such that there exists a ...
5
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0answers
49 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
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0answers
35 views

Suppose $x_n$ is a sequence of positive monotonically increasing random variables converging to $X$. Show $\lim_{n \rightarrow\infty}E(x_n)=E(X)$

I am hoping to get some verification of the below proof. I am worried that I am missing something conceptually. That $\lim_{n\rightarrow \infty}E(x_n)\leq E(X)$ is clear since $E(x_n)\leq x$ for any ...
0
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1answer
25 views

Combinatorics of vectors in plus minus 1

How many vectors of length n are there with entries in {-1,+1} such that the sum of all entries from 1 to k is positive for all k between 1 and n.