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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154 views

An identity about a continuous local martingale

Let $M_t$ be a continuous local martingale with $M_0=0$ and define $I_t^0=1$ and $I_t^n= \int_0^t I_s^{n-1}\; dM_s$. Prove that we have $$n I_t^n= I_t^{n-1}- \int_0^t I_s^{n-2} \;d((M)_t) $$ where ...
3
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328 views

somewhere between sub-gaussian and central limit theorem

(problem statement revamped.) Definition: A random variable $X$ is a sub-gaussian if $$\Pr(|X|>t) \leq \exp(-t^2/k^2) \quad \text{for all $t\geq 0$ and for some constant $k$.}$$ I want to compute ...
3
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398 views

Quadratic variation of a Brownian motion up to time $T$ converges to $T$ in $L^2$?

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely. where $[W, ...
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77 views

Multiplicative functionals for Markov process: discrete time

I read a theorem, stating Let $X_t$ be a Markov process w.r.t. to its natural filtration $(\mathcal F_t)$ on the space of cadlag functions on $\mathbb R_{\geq 0}$ and $(Z_t)_{t\geq 0}$ be a ...
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18 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
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30 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
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82 views

Almost surely either $X_n=0$ for some $n$ or $\lim_{n\to\infty}X_n=\infty$

How I came to this: Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of non negative random variables, $\mathcal{F}_n=\sigma(X_l,l\leq n)$ the sequence of corresponding sigma-algebras and define ...
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33 views

Calculating $\mathbf{P}[X < Y]$ for $X, Y$ exponentially distributed?

This is exercise 2.2.1 from Achim Klenke: »Probability Theory — A Comprehensive Course« Let $X$ and $Y$ be independent random variables with $X \sim \exp_\theta$ and $Y \sim \exp_\rho$ for certain ...
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71 views

Exercise in Probability/Measure Theory

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let also $A_{n,j}\in\mathcal{F},n\in\mathbb{N}_0,j\in\{1,2,3,...,2^n\}$, be such that for all ...
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27 views

Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...
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36 views

Expected magnitude of a vector of $n$ i.i.d. random variables as $n\to\infty$

Suppose that $X_i$ are i.i.d. real valued random variables with probability distribution $f(x)$ for $i=1,2,3,\ldots$. Let $Y_n=\left(\sum_{i=1}^nX_i^2\right)^{1/2}$. Assuming that ...
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46 views

Rigorously, what is the goal of (machine/statistical) Learning and why is that the goal?

After some time doing machine learning and statistical learning theory, I decided to return to my foundations and make sure that the goal of what I am doing makes sense. First let me define $I(f)$ as ...
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33 views

Questions regarding Martingales

I'm trying to learn about Martingales with specific focus on combinatorial problems. However i'm far from an expert in algebra and am having some trouble understanding the basic idea. I will write the ...
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38 views

Donsker's Invariance Principle and Gambler's Ruin

Let $(S_{n})_{n\geq0}$ be a Random Walk (i.e. $S_{n}:=X_{1}+\cdots+X_{n}$, where $\mathbb{P}(X_{i}=1)=\mathbb{P}(X_{i}=-1)=1/2$). Define interpolated random walks $(S^{n}(t))_{t\in\left[0,1\right]}$ ...
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24 views

Regarding the expectation of a function of two random variables…

I'm trying to prove the following: If $X,Y$ have discrete p.m.f $p(x,y)$, then $\forall$ real-valued function $g$, $$E[g(X,Y)]=\sum_{x}\sum_{y}g(x,y)p(x,y))$$ I wasn't sure if my argument was ...
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26 views

Random signs - a remark of david williams:probability with martingales

This can be found in David Williams p.113-114. Suppose that $(a_n)$ is a sequence of real numbers and that $(\epsilon_n)$ is a sequence of IID random variables with $P(\epsilon_n=\pm1)=\frac{1}{2}$. ...
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35 views

Distributions of infinite sums of random variables

Let $(Z_i)_{i \in \mathbb{N}}$ be the arrival times of a Poisson process of intensity $1$ on the interval $[0,\infty)$, i.e. $Z_i \sim \text{Gamma} (i,1)$. We define $X = \sum _{i \in \mathbb{N}} ...
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24 views

Strong markov property in two dimensional Brownian motion

I don't understand the following claim from my book: Let $(B_t)$ be a standard Brownian motion. Let $u:\Omega \rightarrow \mathbb{R}$ be a continuous function, where $\Omega$ is a domain and $B(x, ...
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43 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
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40 views

Correlation of belief distributions from distinct signals

Anne and Bob are two Bayesians who initially share a non-degenerate prior about a binary state of the world. Anne observes some signal (i.e., an experiment in Blackwell's terminology) about the state ...
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14 views

Bounding $\int_{\infty}^{\infty}|g(s)v^3k(v)|dv$ where $k$ is a second-order kernel

Suppose $k$ is a nonnegative, bounded real-valued function that satisfies $$ \int_{-\infty}^\infty k(v)dv=1,\quad k(v)=k(-v),\quad \int_{-\infty}^\infty ...
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20 views

Estimating the discrete laplacian to prove recurrence of simple random walk for d=2

Given a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}$ we define the discrete laplacian of $f$, $\triangle_df$, by the following rule $\triangle_df(x,y)= \dfrac{f(x + 1, y)+f(x, y + ...
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35 views

What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion?

Let $S= \{x \in [0,1 ]: \lim_{n \to \infty } \frac {1 } {n } \sum _{i=1 } ^n d _i (x) = 2/3 \} $, where $d _i(x) $ gives the ith digit of the infinite base-2 expansion of $x $. What is the Lebesgue ...
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31 views

Which of the following is always true for A and B

Given that: $ P(A) = 0.5$ $P(B) = 0.7$ $P(A \cap B) = 0.3$ I have to choose one option that is true... However they all seem to be false which means I am possibly making a mistake.. The only option ...
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20 views

An application of Strassen's theorem

Recently I handed in a problem set containing the following question, but neither myself nor my classmates managed to find a satisfying solution. We were quite certain that a fruitful approach was to ...
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49 views

Some properties of a random variable

I have absolutely no idea how to show this: Let $X$ be a random variable whose distribution is not degenerate. By considering the function $F( \theta) = \mathbb{E} U( \theta X)$, $\theta \in ...
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50 views

Development of measure and probability theory

I am interested in a reference (article, maybe a book chapter) on the development of mathematical probability theory - that is, mostly starting from the beginning of the 20th century. It is surprising ...
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27 views

Identify the possible weak limit

Suppose $X_1, X_2, \ldots$ are independent random variables with distribution: $$ \mathbb{P}(X_n = 0) = \frac{1}{n}, \, \mathbb{P}(X_n = 2n) = 1 - \frac{1}{n} $$ Let $Y_n = \frac{X_1 + X_2 + \ldots + ...
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41 views

Invariant Probability of Discrete Time MC from Continuous Time Markov Chain

Given rates α of an irreducible continuous-time MC on finite state space and told that π is the invariant probability measure of this chain, we define a discrete time MC as having transition ...
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23 views

Find the density for the random variable $Y=X_1+X_2$

Problem: Let $X_1, X_2$ be independent variables with uniform density on (-1,1). Find the density for the random variable $Y=X_1+X_2$. My attempt using convolution: $y_1=x_1+x_2, y_2=x_2 \Rightarrow ...
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26 views

Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
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33 views

Challenging CDF of $\sup_t|B_t|$ ($B_t$ is a Brownian Bridge)

Question 1: Let $W_t$ be a Brownian motion. Then how could we prove that $$\Pr\left\{\sup_t|W_t|<b\right\}=1-\frac{4}{\pi}\sum_{j=1}^\infty \frac{(-1)^j}{2j+1} ...
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35 views

How the second form of following equation is derived form first form (i.e. given first line, what are the steps involved in writing second line

How the second form of following equation is derived form first form (i.e. what are the steps involved in writing second line)
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27 views

(Billingsley, 2nd ed, 1968) D space, (12.33) inequality proof

Convergence of probability measures, Billingsley, 2nd ed, p132, Theorem 12.4 This is what I want to prove where $x \in D \equiv$ the set of cadlag functions defined on $[0,1]$ and $w_x^{''}(\delta) ...
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72 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
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64 views

Show that a measure is a probability measure

I have trouble with this question: We define an arc segment $B(\theta, \eta, r, R)=\{x \in \mathbb{R}^2\vert \omega(x)\in [\theta,\eta], \Vert x \Vert_2 \in [r,R] \}$ where $0 \leq \theta \leq \eta ...
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0answers
10 views

Monte Carlo Markov Chain Simulation Issues

The Markov Chain is uniformly distributed across all $50$x$50$ matrices of entries $0$ and $$1 with no neighboring $1's$. I am supposed to run a MC simulation to check the probability that the ...
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34 views

Show that $\sum_n P(X_n>M) <\infty$ implies $P(\sup_n X_n <\infty)=1$

Show that $\sum_n P(X_n>M) <\infty$, for some M, implies $P(\sup_n X_n <\infty)=1$. What I did. By Borel-Cantelli $\sum_n P(X_n>M) <\infty$ implies that $P(\limsup_n ...
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62 views

Convergence of Expectations (cont'd)

The question is related to this question. Suppose $\{X_n\}$ is a sequence of indep. random variables with zero expectation. Consider their sum $S_n$ which have the following properties: $S_n$ ...
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26 views

The independence of random variables

Here is my question: Consider a homogeneous ergodic Markov chain on a finite state space $X=\{1,\ldots,r\} $. Define the random variables $\tau_n \,,n\ge1$ as the consecutive times when the Markov ...
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26 views

Finding a general form of the density function when we have a four dimentional random variable.

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
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61 views

Let $X_n \sim U[0,1]$. Let $A_n$ count the number of local maxima of the sequence unto $n$. Prove a suitable central limit theorem for $A_n$.

Let $X_n $ be uniformly distributed on $[0,1]$. We say $X_k$ is a local maximum if $X_k> X_{k\pm 1}$. Let $A_n$ count the number of local maxima of the sequence unto and including $n$. Find $a_n, ...
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88 views

How to derive the following from Azuma's inequality?

This is claimed in Proposition 1 in the paper http://arxiv.org/abs/1409.6110 Let $A$ be a $n \times d$ matrix. $A$ can have only $K$ different types of rows i.e. rows of $A$ are chosen from a set of ...
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31 views

Sum of $\{X_n\}$ iid random variables contained in a compact interval implies each $X_i=0$ a.s.?

I am working through an exercise that starts with a sequence i.i.d. random variables where for $a\leq b$, $$\Pr\left(\lim\sup_n \sum_{i=1}^{n} X_i \in [a,b] \right) \neq 0.$$ Does this require $X_i ...
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40 views

Convergence in the space $C([a,b],M_1(\mathbb R))$.

Let $M_1(\mathbb R)$ be the space of probability measures on $\mathbb R$ with the weak(-*-)topology: $\mu_n \rightarrow \mu$ iff $\int f(x) \mu_n(dx) \rightarrow \int f(x) \mu(dx)$ for all $f \in ...
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0answers
22 views

Where have I used the assumption that $X\in L_2$?

Let $X\in L_2$ be a random variable and $g$ a positive real function. Let $I$ be an interval and $b>0$, and suppose that $\forall x\in I\ g(x)>b$. I have to show that: $$\operatorname ...
2
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0answers
36 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
2
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0answers
35 views

Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
2
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0answers
70 views

Measure-preserving maps on probability spaces

A professor posed me a problem a few days ago, and I have not been able to find an answer to it. Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following ...
2
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0answers
38 views

to be 99% certain of making a profit? central limit theorem?

Let $X_i$ be the profit card $i$ makes when its sold. I let $S_n = X_1 + ... + X_n$ so total profit. I found the mean of $X$ to be $0.1$. and $E[X^2] = 25$ so variance $= 24.99$ Are these correct? ...