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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Potentials and Markov Processes

Given a resistive electrical circuit $G$, i.e. a graph with nonzero weights attached to each edge whose reciprocal we call the "resistance," we can define a reversible markov chain on the graph, ...
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26 views

iid random variables and stopping time

This is Exercise 14.30 from Probability for Statistics and Machine Learning. Let $X_i$ be iid with $E|X| < \infty$, and let $T$ be a stopping time adapted to $\{ X_i \}$. Let $S_n = ...
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49 views

Monotone convergence theorem allows the limit to be infinity

Monotone convergence theorem(MCT) doesn't impose any restriction on the limit. For example, if $\{X_n\}$ satisfies $0 \le X_n \nearrow X$ with $EX=\infty$, then I still could use MCT to get ...
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37 views

Is set of product distributions compact under second moment constrains?

Please do not treat this question as duplicate of Definition of the set of independent r.v. with second moment contstraint which I didn't want to edit because of many useful comments. Also, in this ...
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23 views

Splitting up Variance over components

If $f\colon [0,1]^n\to\mathbb{R}$ and we take variance wrt the Lebesgue measure(s), do we have $ Var(f)\leq\sum_{k=1}^n \int_{[0,1]^{n-1}}Var f^k_{\tilde{x}}\;d\tilde{x} $ where ...
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28 views

ProbabilityTheory, precise definition of r.vs with same distribution

Studying probability theory, I wonder the definition of $X \overset{d}{=}Y$. However, it took me long time to search it to find nothing. Considering in the extension of limit law, i.e $X_n \to_d ...
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28 views

When is the joint density differentiable

My question is the following: given a real random vector $X = (X_1,...,X_k)$ with differentiable marginal densities $f_1,...,f_k$, what extra conditions on the marginals are needed to ensure that the ...
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35 views

Pollaczek–Khinchine formula for claims with expotential distribution - derivation

I am trying to understand ruin probability formula using Pollaczek–Khinchine formula described here: http://en.wikipedia.org/wiki/Ruin_theory $$\psi(x)=\left(1-\frac{\lambda ...
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29 views

What exactly do you mean by random variable converge almost surely?

If somebody claim that $\sum_{k=1}^n X_k$ converges almost surely to a random variable , does it imply that $Y$ is finite? If so, first of all, is it the point of proving Kolmogorov convergence ...
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25 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
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65 views

Weak convergence to a constant implies convergence in probability

If on some probability space the random variables $X_1, \dots, X_n$ with distributions $\mu_n$ convergence weakly to the constant random variable $c \in \mathbb{R}$, i.e. $ \int f \, d \mu_n \to ...
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53 views

Generalized Form of Fano's Inequality

The Wikipedia article on Fano's Inequality presents a generalization as follows: Let $\mathbf{F}$ be a class of probability densities with a subclass of $r+1$ densities denoted $f_{\theta^{(i)}}$ ...
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52 views

Function of probability measures representation

Let $\mu$ be a in $\mathbb M(\mathbb R)$, the space of probability measure on $\mathbb R$. Let $F$ be in $C_b (\mathbb M(\mathbb R), \mathbb R)$, the space of continuous bounded function on $\mathbb ...
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22 views

Can a randomized rule induce a random measure on the action space?

$D = \{d_i: X\to Y, i=1,\dots,n\}$ is a finite set of mappings from $X$ to $Y$, $(\Omega, \mathcal F, P)$ is a probability space, and $\delta: \Omega \to D$ is a measurable mapping. Can $\delta$ ...
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121 views

Product distribution function of two independent random Variables

Why, if $X $ and $Y $ are two independent'', continuous random variables, described by probability density functions $f_X $ and $f_Y $, then the distribution of $Z = XY$ is $$f_Z(z) = ...
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36 views

Show that the following family of hash functions is $2$-wise but not $3$-wise independent

Consider the following family of hash functions that map $w$-bit numbers to $l$-bit numbers (i.e. the range is $\{0,...,m-1\}$ where $m=2^l$): $\mathcal{H} = \{h_{A,b}|A\in \{0,1\}^{l\times w}, b \in ...
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21 views

Summation methods and entropy

I am aware of the theory of divergent series, but don't know much of it. If you have a text to recommend, I'd be glad to hear it. Suppose I have an infinite-dimensional probability vector $\mathbf{p} ...
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45 views

Inverse Markov inequality with additional information?

Let $X$ be a random variable with values in some finite subset of $\mathbb{N}$ containing 0. Let $a>0$ and $0<\epsilon<1$ be such that $\mathbb{P}(|X-\mathbb{E}[X]|\geq a)\leq2\epsilon\leq ...
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51 views

Let $X_n>0$ be iid and $P(X_n>t)\sim t^{-\alpha}$, show that $Y_n=n^{-1/\alpha}S_n$ and $1/Y_n$ are tight.

We are given that $X_n>0$ be iid with common distribtuon $X$, and $P(X>t)\sim t^{-\alpha}$, I need to show that the scale of $Y_n$ is $n^{1/\alpha}$. Or in other words show that ...
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80 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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98 views

Probability question involving stochastic process

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
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37 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
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32 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
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30 views

What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain ...
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34 views

On a problem of convergence of measure for Levy measures

I have a question that pertains to the Levy representation of infinitely divisible distributions. However, the technical item that is relevant to me right now is one that relates to weak or vague ...
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127 views

fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
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44 views

what will be the PDF of magnitude and phase of this random variable?

i have a random variable as shown in the figure and i tried to find the PDF of the magnitude and phase of this random variable using central limit theory as i mentioned , i know that if we have ...
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53 views

martingale convergence proof

This is out of Durrett 5.5.7. Let $X_n \in [0,1]$ be adapted to $\mathcal{F_n}$. Let $\alpha,\beta > 0$ such that $\alpha + \beta = 1$. Suppose that $$ P(X_{n+1} = \alpha + \beta X_n | ...
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20 views

What is the Fractional Functional Central Limit theorem?

What is the statement of the "Fractional Functional Central Limit theorem (FFCLT)"? There is a Functional Central Limit Theorem, also called Donsker's theorem. Which has a Wikipedia article . I ...
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21 views

Conditional Likelihood estimation

I'm reading a book called "Bayesian Reasoning and Machine Learning" and I have come across a question that gives me some problems. Unfortunately I neither have solutions, nor anyone else who I know ...
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40 views

Applying the Law of Large Numbers recursively

If I want to apply the LLN for an estimator that uses another estimator, can I apply the LLN inside the summation and after it simplify the outer summation by using the expected value of the inner ...
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46 views

What is the Vapnik-Chervonenkis dimension of sigmoidal functions?

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...
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170 views

Almost sure equality of two random variables

I have a homework problem where I'm asked to show that if $X$ and $Y$ are random variables with $E[X|\mathcal{G}] = Y$ and $E[X^2] = E[Y^2]$ then $X = Y$ a.s. My approach is to use Chebyshev's ...
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25 views

Difference between $E(y|X_1,X_2)$ and $E(y|\cup_{i=1,2}X_i)$

Suppose that $y$ is a random variable and $X_1$ and $X_2$ random vectors. Let $Z$ be the random vector that collects the distinct elements in $X_1$ and $X_2$. Can you please explain the difference (if ...
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47 views

A basic question on convergence in Skorohod metric

Consider a sequence of piecewise constant (positive) function in the space $D$ with Skorohod topology such that each of the functions is zero at the origin and any two discontinuities are at least ...
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26 views

Show that $f$ is a bivariate density

Let $g$ be a continuous univariate density, and let $f(x,y)=2g(x)g(y)$ for $x\leq y$ and $f(x,y)=0$ if $x>y$. Show that $f$ is a bivariate density. For some reason, nothing I attempt makes any ...
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75 views

What does the author mean here?

I am reading a paper and there is a short statement by Author for which I am not sure if I got him right or not. Could someone let me know if my understanding right or not? He says the following: ...
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34 views

Bivariate stopped processes

Take two dependent Levy processes $L_1(t)$ and $L_2(t)$ with law $\mathcal{L}(L_1(1),L_2(1)$. If we stop the first process at a general time $t=s_1$ and stop the second process at another general time ...
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47 views

How does one explicitly compute the rate function for a given probability distribution?

For a given probability distribution $\mu$, the associated rate function is $I_\mu(x) = sup \left \lbrace x \lambda - ln \left( \int e^{\lambda t} \mu (dx) \right) \right \rbrace$, and if there ...
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30 views

Level sets of a Wiener process

Assume we have a Wiener process $W$ starting at $W_0=0$. What can one tell about the Lebesgue measure of "level sets" $A_y = \{t>0; W_t=y\}, y \in \mathbb{R}$? I actually need to estimate these ...
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53 views

Proving that Levy-measures are $\sigma$-finite

I have a question that pertains to Levy measures and, more specifically, I would like to show that they are $\sigma$-finite. They are not finite in general and it is easy to provide examples of this. ...
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25 views

Existence of a sequence of random variables, provided weak convergence

I'm trying to prove the following statement: Let $ X_n, X_0 $ be such R.V.'s that $ X_n $ converge to $ X_0 $ in distribution (weakly). Prove that there exist $ Y_n, Y_0 $ on the probabilistic space ...
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26 views

Entropy of Sum vs Difference of Random Variable

I am looking for a proof of the following fact Let X and X' be i.i.d on {0,1,2}(not necessarily uniform). Prove that $H(X + X' mod\;3) \leq H(X - X' mod\;3)$ where $H()$ is the standard Shannon ...
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27 views

Is the following rule a stopping time in regards to reverse filtration?

Let $X_1, \dotsc, X_n \sim F$, where $F$ is a distribution function with support in $[0,1]$. For $t \in [0,1]$, define the sigma-algebra: $$ \mathscr{F}_t = \sigma(1_{\{X_i \leq s\}}\;,\; 1 \geq s ...
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Prove that $\text{Var} \tau = \frac{1 − (p − q)^2}{(p-q)^3} $ where $\tau$-stopping time

Let $S_n = \xi_1 + \dots + \xi_n$ be asimetric random walk such that $P(\xi_i = 1) = p > \frac{1}{2}$ and $P(\xi_i = -1) = q $. Let $\sigma^2 =1-(p-q)^2$ and let $X_n=(S_n-n-(p-q)n)^2 - \sigma^2n $ ...
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64 views

Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}$ is martingale.

Let $\xi_1 \neq const.$ be a random variable with moment-generating function $\phi(\theta) = Ee^{(\theta \xi_1)}$. Let $S_n = \xi_1 + \dots \xi_n$. Prove that $X_n = e^{(\theta S_n-n \ln ...
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29 views

Weak convergence of probability measure on $\mathscr{C[0,\infty]}$

What is the reason for considering the spaces of probability measures on the space of all continuous functions and then considering weak convergence there ? Is it that we can then use Skorohod's ...
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96 views

Rotational invariance and distributions

Let $k\leqslant n$ denote two positive integers, $A$ an $n \times k$ matrix with $A'A = I_k$, and $X$ and $Y$ two independent random variables on $\mathbb R^n$, each rotationally invariant (that ...
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39 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
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43 views

Does the system of equations always have a nontrivial solution?

$f:[0,1]^2\to R_+$ is a continuous conditional density function. For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq ...