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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Probability Generating Function of Compound Poisson Process

Let $(N_t)_{t\ge 0}$ be a poisson process with intensity $\alpha > 0$. Let $(X_n)_{n \in \mathbb N}$ be iid real valued random variables that are independent of $N_t$. Let $Y_t = ...
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38 views

Estimation of the Hermite Polynomials using Plancherel-Rotach asymptotics

Suppose $H_n(x)$ is a Hermite Polynomial such that $$\int_{\mathbb{R}} H_n(x) H_m(x) e^{-x^2} dx = \delta_{m,n}.$$ I want to show for $ \phi_n(x) = H_n(x)e^{-\frac{X^2}{2}}$ $$ \left( ...
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61 views

Confidence Interval of Information Entropy?

Information entropy, $IE$, is defined as: $$IE = \sum_{i} p_i log\frac{1}{p_i}$$ Where $p_i$ is the probability of event $i$ (and we are summing over all possible events). Let's say I have data ...
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199 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
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58 views

Support of distribution functions in copula theory

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
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45 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
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86 views

Image of probability measure

Let $(\Omega,\Sigma,P)$ be a probability space. What is known about $P(\Sigma)$, the set of probabilities of events in $\Sigma$ by $P$? Clearly, $P(\Sigma)$ contains $0$ and $1$ since ...
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35 views

Showing that the space of bounded measurable functions can't be used to characterize convergence in distribution

An exercise I am doing from Chung's book asks me to find some asbsolutely continuous probability measures $\mu_n, \mu$ and some measurable and bounded (but not continuous!) function $f: \mathbb{R} ...
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52 views

probability measure on space of sequences

Let $\Omega=\{0,1\}^\infty$. For some $n$, let $B\subset \{0,1\}^n$. I have seen these two statements which make me confused little bit. (1) If $A\subset \Omega$, $A=B\times \{0,1\}^\infty$, and ...
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If $X_1,X_2$ are independent beta then showing $\sqrt{X_1X_2}$ is beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities ...
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181 views

Right-continuity of the augmented filtration

In most books about stochastic processes, the authors write about "the usual augmentation of a filtration". I am having trouble with proving that their construction is correct, i.e. that the augmented ...
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44 views

Supermartingales and optimal strategies for a game

Your winnings per unit stake on game $n$ are given by independent random variables $\epsilon_n$ such that $P(\epsilon_n=1)=p$, $P(\epsilon_n=-1)=q$ with $1/2<p=1-q<1$. Let $C_n$ be your stake on ...
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51 views

Aid to understand conditional expectation

I have a hard time to understand conditional expectation, that is $\mathbf{E}[X| \mathcal{G}]$ where $X$ is a random variable on the probability space $(\Omega, \mathcal{A}, \mathbf{P})$ and ...
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46 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...
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45 views

Measurable functions on complete space.

$(\Omega, \mathcal{E}, \mathbb{P})$ a probability space, $\mathcal{F}$ is $\mathcal{E}$ completed by the $\mathbb{P}-$null sets. Which conditions should meet a topological space $X$ (taken with its ...
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80 views

Sufficient condition for martingale property

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ ...
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64 views

Weak convergence of order statistics

I've encountered the following problem: Let $U_1,...,U_n$ be iid uniformly distributed on $[0,1]$ and let $U_{n(k_n)}$ denote the $k_n$-th order statistic where $k_n$ is chosen, s.t. ...
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67 views

An irreducible Markov chain is a martingale

Let $\{X_n\}$ be an irreducible Markov chain. Does exist example of such $\{X_n\}$ which is also a martingale given that: a. $\{X_n\}$ is recurrent with finite number of states (but bigger ...
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97 views

Existence of regular conditional distribution of random variable given the value of another variable

Let $(\Omega, \mathcal{A}, \mathbf{P})$ be a probability space with a measurable function $Y: (\Omega, \mathcal{A}) \rightarrow (E, \mathcal{E})$ and another measurable function $X: (\Omega, ...
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85 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
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130 views

Proof of Wald's Identity, is this valid?

So Wald says that assuming that $T$ a stopping time and $X_i$ i.i.d. variables are $L^1$, that $E[S_{T}] = E[T]E[X_1]$ given that $S_n = \sum_{i=1}^n X_i$. Consider the following proof that is ...
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76 views

Condition for $L^p$ convergence of backwards martingale

Is there any condition that is known to be sufficient for $L^p, 1<p<\infty$ convergence of a backwards martingale (and why is it sufficient)? I couldn't find anything else than the normal $L^1$ ...
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2k views

Proof of the Box-Muller method

This is Exercise 2.2.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise (Box–Muller method): Let $U$ and $V$ be independent random variables that are uniformly ...
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42 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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122 views

How to prove product of measurable set is measurable using dynkin's $\pi-\lambda$ theorem?

My question: If $E_1$ and $E_2$ are measurable subsets of $R^1$, I want to show that $E_1 \times E_2$ is a measurable subset of $R^2$ and $|E_1 \times E_2|=|E_1||E_2|$. My attempt: First I tried to ...
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106 views

Estimating the support of a probability density function

The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ...
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102 views

quadratic SDE solution

I have this SDE $dX_t=[a+bX_t+sX_t(1-X_t)]dt+\frac{1}{2}X_t(1-X_t)dW_t, \, X_0=0,$ where $a,b \in(0,1)$ and let's say that $s$ is a real constant (it's actually a function of $X$, but I think I can ...
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209 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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34 views

A basic question on spaces of probability measures

This problem is regarding the space of probability measures. For $N \geq 1$, let $\{e_i^N(.), i\geq 1\}$ denote a complete orthonormal basis for $L_2[0,N]$. Let $\{f_j\}$ be countable dense in the ...
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43 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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58 views

Stronger version of strong law of large numbers

Let $(X_i)_{i\in\mathbb{N}}$ be pairwise independent random variables where $E[X_i]=0$ for all $i\in\mathbb{N}$ and $\sup_{n}E[X_n^2]\lt\infty$. Then for $S_n=\sum_{i=1}^n X_i$ and ...
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53 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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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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101 views

finding n in binomial distribution

A player tosses a coin $2n$ times. The probability of head is $0.48$. The player wins if he gets head more than $n$ times. But he can choose the number $n$ before the game. What $n$ should he choose ...
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50 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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88 views

Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
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106 views

Laplace transform and Fourier transform of kernel

Suppose $p_t(x)=\frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$ is the Gaussian density. $p_t(x)$ is also the Green kernel of the heat equation in 1D: $(\partial_t-\frac12\Delta)u=0$. The Fourier transform ...
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115 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
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158 views

Probability that a real number is bigger than another

I am not a mathematician; I lack formal training in probability theory, but the following problem came to my mind. I take two random real numbers $a,b > 0$. What is the probability that $a > b$ ...
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105 views

An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
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63 views

A basic measure theory question on Stochastic Process

Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process, so it is measurable map. Let $S^T$ be the collection of all ...
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77 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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186 views

the continuity theorem with respect to Laplace transform

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{N}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit ...
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65 views

“Uniform” Convergence in Distribution (bounded Lipschitz metric)

I have been thinking about the following problem. Let me know if the notation below makes sense. Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ...
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103 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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57 views

Brownian motion, modifications vs indistinguishablity

In Protters book Stochastic Integration and Differential Equations And in uncountable other sources, they mention the continuous sample paths of the brownian motion. That is: It holds that ...
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46 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 "$\Rightarrow$" ...
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92 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 ...
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48 views

Does $(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp (Y,W) \mid Z$ hold?

I know that $ X \perp (Y,W) \mid Z \implies (X \perp Y\mid Z) \ \& \ (X \perp W \mid Z)$ but does the converse hold? i.e. does: $$(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp ...
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86 views

Is there a continuous version of the Borel-Cantelli lemma?

Given a sequence of events $A_n$ for $n\in \mathbb N$, the first Borel Cantelli lemma states that, if the sum over all probabilities $\sum_{i=1}^n P(A_n)$ is finite, then the probability of the limes ...