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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51 views

Products of Random Matrices

I'm interested in the following process on the space of $d \times d$ real valued matrices, $M_d(\mathbb{R})$. Fix $n \in \mathbb{N}$ and consider the process $$X_{k,n} = \left( I + ...
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38 views

Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous?

Let $X$ be a standard Gaussian random variable. Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous ? I don't understand the question here. Now $X$ has density ...
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36 views

How to evaluate this expectation value?

How to evaluate this expected value: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$ where $\xi_i\overset{ind}{\sim} N(0,1), ...
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108 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
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11 views

invariant measures for the independently coupled Feller process

If I take countably many Markov processes and couple them independently, how does the collection of invariant measures for the coupled process relate to those of the marginals? I conjecture that it ...
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44 views

Existence of density

Suppose $ \{ D_i : i \geq 1 \} $ be a sequence of i.i.d. random variables taking values $ \{ 0, 1, 2, \dotsc, K-1 \} $ where $ K \geq 3 $ is a positive integer with probabilities $ \mathbb{P} ( D_1 = ...
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75 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
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49 views

The infinitessimal generator of Brownian Motion

Background: I know, and can almost prove, that the infinitessimal generator of BM is the Laplacian/2. For me (and I have never heard it done differently) we have Feller Processes, of which BM is one ...
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40 views

Implications of convergence in probability?

Let $||.||$ indicate the Euclidean norm. Let $\theta_0$ be a specific value of the parameter $\theta \in \Theta\subseteq \mathbb{R}^d$ and $G_n$ be a random vector-valued function $$ G_n : \Theta ...
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26 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
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284 views

applying iterated expectation when conditioning on multiple random variables

The law of iterated expectations tells us that ${\bf E}\big [{\bf E}[X\, |\, Y]\big ]={\bf E}[X]$. Suppose that we want apply this law in a conditional universe, given another random variable $Z$, in ...
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31 views

Shortest path length when edge length is limited

$N$ nodes are uniformly distributed in a square whose side length is $1$. There exists an undirected edge between two nodes, if and only if the distance between them is less than or equal to $r$. Here ...
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26 views

Growing of a score function

The argument that I'm dealing is very specific, I hope to make you understand the problem without going into detail. I have this score function: \begin{align} score = MargL^q + MargL^{\theta} ...
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65 views

Markov chain question

Consider an irreducible, recurrent Markov Chain ($X_n$) on a countable state space $S$ with transition probability $p(x,y).$ Pick a sigma-algebra $A \subset S$ and let $T_k=\inf\{n>T_{k-1}:X_n \in ...
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25 views

The “size” of a continuous uniform selection of points in the unit square

Let $\{X_r\}_{r\in[0,1]}$ be i.i.d. random variables, each distributed uniformly on $[0,1]$. Let $S\subseteq[0,1]^2$ be the random set defined as follows: $$S=\{(r,X_r)\mid r\in[0,1]\}$$ How would ...
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19 views

Which parameters are used for MGF?

While finding the mgf of the binomial distribution we allow for the X to cover all values from 0 to N, the number we "choose". Why don't we have to cover all possible values for the p and q as well? ...
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27 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
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47 views

two probability questions (related to lower bounding error probability)

Recently, I ran into the following $2$ problems. Say you have a fair coin which you toss $n$ times. Now, here is my first question. I want to check whether the lower bound on the probability that ...
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41 views

Conditioning on $\mathcal{F}_\sigma$ for $\sigma$ stopping time

I'm trying to show that $E[E[\ \cdot\mid \mathcal{F}_\sigma]\mid\mathcal{F}_\tau]=E[E[\ \cdot\mid \mathcal{F}_\tau]\mid\mathcal{F}_\sigma]$ for stopping times $\sigma$ and $\tau$, I've come to the ...
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50 views

Application of the Weak Law of Large Numbers.

I have in my problem that $X_1,\ldots,X_n$ is a random sample from a distribution with probability density $f(x; \theta)=\theta x^{\theta-1}, 0<x<1$. Furthermore, $-\log X_i ...
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154 views

Probability Density Function problem

$$ f(x) = \begin{cases} 0 & \text{if $x < 0$} \\ x^2 & \text{if $0 ≤ x < \mu$} \\ α + βx & \text{if $\mu ≤ x ≤ 10$} \\ 0 & \text{if $x ≥ 10$} \end{cases} $$ Considering ...
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111 views

Change of variables in Lebesgue-Stieltjes integral

Suppose $F$ is a probability distribution function with corresponding measure $\mu_F$ on $\mathbb{R}$. Suppose moreover that $f:\mathbb{R} \rightarrow\mathbb{R}$ is a continuous, increasing function. ...
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27 views

Prove that the empirical measure is a measurable fucntion

This problem came from Schervish, Theory of Statistics, Sec. 1.4 Prob. 24. Suppose that $X_1, \ldots, X_n$ are exchangeable and take values in the Borel space $(\mathcal{X}, \mathcal{B})$. Prove ...
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74 views

Is $Z_n$ a Martingale with mean $1$?

Consider a sequence of independent tosses of a coin, and let $P_h$ be the probability of a head on any toss. Let $A$ be the hypothesis that $P_h = a$, and let $B$ be the hypothesis that $P_h = b$. Let ...
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30 views

Probability kernel calculations

Suppose $\lambda \Phi(y,\cdot) = \delta_y$ and $\nu_t = \lambda\mu_t \Phi$ satisfies $\lambda \mu_t = \nu_t \lambda$ for each $t\geq 0$. Define $g: S\to \mathbb{R}_{++}$ as $$e^{-\alpha t} \int_S ...
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62 views

Criterion of independent random vectors

I would like to know weather the following statements are true or false. Given two random vectors $\boldsymbol{X} = (X_1,\cdots,X_m)$ and $\boldsymbol{Y} = (Y_1,\cdots,Y_n)$, $X_i$ is independent ...
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95 views

Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
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48 views

About one stoping time definition in Chung's book (A Course in Probability Theory)

In Chung's book, he defines the stopping time $ \alpha^k $ in the following way. $\alpha^1 = \alpha$; $\alpha^{k+1}(\omega) = \alpha^k(\tau^\alpha \omega)$; where $\tau^\alpha$ is the ...
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49 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
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36 views

Proving that $ (\mathbb E [X^n])^{1/n}\leq (\mathbb E [X^m])^{1/m}$ for $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$

How to prove that for a positive essentially bounded random variable $X$ satisfing $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$ and for any $m,n \in \mathbb N^*$ with $m\geq n$ we have $$ (\mathbb E ...
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92 views

How to obtain Black-Scholes from displaced diffusion process?

The displaced diffusion process is $$ d(F_t+a)=\sigma(F_t+a)dW_t $$ I have solved it and found it to be $$ F_t={F_0\over\beta} \exp\left( -\frac 12\beta^2\sigma^2t + \beta\sigma W_t \right) ...
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61 views

A question regarding a prefix code

Let $C=\{ c_1, c_2, \dots, c_m \}$ be a set of sequences over an alphabet $\Sigma$ and $|\Sigma|=\sigma$. Assume that $C$ is a prefix-free code, in the sense that no codeword in $C$ is a prefix of ...
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18 views

$L_1$ mean ergodic theorem for the action of compact group

Let $X$ be a Polish space with Borel probability measure $\mu$. A compact group $G$ acts on X continuously. It is right that for any $f\in C_b(X)$ exists a sequence $(g_k \in G)_{k\in \mathbb{N}}$ ...
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34 views

Normality of two sample z-test with different sample sizes

Let $X_1, X_2, ... , X_m$ be iid with $E X_i = \mu_1$ and $Y_1, Y_2, ... , Y_m$ be iid with $E Y_i = \mu_2$. They both have the same (finite and positive) variance $V X_i = \sigma^2$. I would like to ...
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38 views

Non - nearest neighbor random walk in $\mathbb{Z^{2}}$

$\textbf{Problem:}$ let {$X_{n} : n ≥ 0$} be any symmetric random walk on $\mathbb{Z^{2}}$ whose jumps have finite second moment. That is, $X_{0} = 0$ , {$X_{n} − X_{n−1} : n ≥ 1$} are mutually ...
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56 views

Notion of Independence

Given two random variables $X,Y: \Omega \rightarrow \mathbb{R} $, is there a notion of independence in the following sense: \begin{equation} \mathbb{P} \big( \{ \omega \in \Omega | X(\omega) \in A, ...
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24 views

independent random variables made from chi-square distribution

if $x_1$,...,$x_n$ are r.s from N(0,1), how can we prove $\frac{x_i^2}{\sum_ {j=1}^{n}x_j^2}$ and $\sum_ {j=1}^{n} x_j^2$ are independent?
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55 views

Cross variation not independent Brownian motions

How can I calculate the cross variation between a standard Brownian motion $(B_t)_{t\geq 0}$ and the process $(B_t^{\tau})_{t\geq \tau}$ defined as $B_t^{\tau}= B_t-B_{t-\tau}$? Here $\tau$ is just a ...
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91 views

Picking a random point on the surface of a unit sphere

It is mentioned in many places (e.g., in Wikipedia) that if $X_1, X_2,\dots, X_n$ are independent standard Gaussian random variables, then the vector $$Y:=\frac{1}{\sqrt {X_1^2+X^2_2+\dots+X_n^2}} ...
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48 views

Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
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28 views

A basic problem on distribution function

Let $F$ and $G$ be two distribution functions. Given that the largest square that can be inscribed between $F$ and $G$ has less than $\epsilon$ length I want to prove that for any $x$ there exist ...
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29 views

Why is $\varphi_{\mu_1\star\mu_2}=\varphi_{\mu_1}\varphi_{\mu_2}$?

(1) Let $X_1,\ldots,X_n$ be independent randiom variables with $\mathbb{P}(\left\{|X_i|<\infty\right\})=1, i=1,\ldots,n$. Then for the characteristical function $\varphi$ of ...
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19 views

How could we know the Dirichlet allocation is describing the topic rather than something else?

Dirichlet distribution is widely used in document modelling and document clustering. I tried to understand its rational. I read from this article that: Different Dirichlet distributions can be ...
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58 views

Markov Chain depicting unruly customer behavior

A store has 2 bins of balls. 1 bin is red, and contains 3 red balls. The other bin is gray and contains 2 gray balls. Every minute, on the minute, exactly one customer comes by the bins, picks up ...
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69 views

If $X$ is a random variable, under which conditions is $g(X)$ also a r.v.?

In many instances, functions of random variables appear, and we usually treat them as random variables also. In the 3d edition, pp. 85-86, of this well-known book (now in its 4th edition), we find the ...
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33 views

Regularity of measures proof

A probability measure $\mathbb P$ on a metric space $(S,d)$ is closed regular if $$ \mathbb P(A) = \sup \{ \mathbb P(F) : F \subseteq A, F \text{ - closed} \} \text{*}$$ with $A\in ...
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60 views

Pi-system generating a tail sigma-field

I have the following problem. Let $(X_n)_{n\in Z_+}$ be a sequence of random variables with values in $\{-1,1\}$. Let $\mathcal{F}:=\sigma(X_0,X_1,...)$ be the product sigma-field on $\{-1,1\}^{Z_+}$ ...
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104 views

Behavior of weighted sum when weights are functions

Let $w_n$ be a (deterministic) sequence of bounded real-valued functions on the interval $[a,b]$, and let $X_n$ be a sequence of IID random variables with mean $0$ and variance $1$. Define the random ...
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33 views

Events on different experiments

I have two questions for Probability theory and hope getting your help. Question 1: Consider the following two experiments: $T_1$: "Tossing a fair coin". $T_2$: "Tossing a fair six-sided die". ...
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43 views

A question about choice of signs in expectation

Let $ f_1,...,f_N $ a collection of fucntions, $ \epsilon_1,...,\epsilon_N$ randomized signs ( $\pm1$) with same probability and $ N\in\mathbb{N}$. If $ ...