Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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202 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
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470 views

Compact set of probability measures

I think I can solve the following exercise if X is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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280 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
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307 views

How do you compute numerically the Earth mover's distance (EMD)?

I was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...
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100 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
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98 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
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150 views

Recurrence of a certain class of $2$-$d$ random walks

As is well known, a symmetric random walk on $\mathbb{Z}^d$ (the lattice of $d$ dimensional vectors with integer components) is recurrent if and only if $d=1,2$. In particular it is transient for ...
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144 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
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170 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...
6
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59 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
6
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81 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: ...
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497 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
6
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1k views

Are vague convergence and weak convergence of measures both weak* convergence?

For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due ...
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150 views

Proving existence of limit by Martingale.

I'm thinking about a question: Suppose $y_n > −1$ for all $n$ and $\sum |y_n| < \infty$. Show that $\prod_{m=1}^\infty (1 + y_m)$ exists. Since $\sum |y_n| < \infty$, we must be able ...
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186 views

Question on Conditional expectation

Let $X_1$ and $X_2$ be two random variables on $(\Omega,\mathcal{B},P)$. Suppose there is a function $g:\mathcal{B}\times\mathbb{R}\rightarrow[0,1]$ such that for any $x$, $g(\cdot,x)$ is a ...
6
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391 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
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195 views

Regular Version of Conditional Gaussian Distribution

Let $Z_{1}$ and $Z_{2}$ be two independent normally distributed random variables with expectations $\mu_{1},\mu_{2}\in\mathbb{R}$ and variances $\sigma_{1}^2,\sigma_{2}^2\in (0,\infty)$ . I would ...
6
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352 views

Calculating probability of some event using geometric considerations

I want to estimate exponentially the following probability: Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius ...
6
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164 views

Closure in the Space of Probability Measures with the Prohorov metric

I have seen this result stated countless times: assume the metric space $(\theta,d)$ is separable; then $(\theta,d)$ is complete if and only if the space $(\mathcal{P}(\Theta),\rho)$ (the space of ...
6
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153 views

Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
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171 views

Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
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919 views

Monotone class theorem

I have some question about the Monotone Class Theorem and its application. First I state the Theorem: Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that ...
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40 views

Standard machine in measure theory

Step 1.Prove the property for $h$ which is an indicator function. Step 2.Using linearity, extend the property to all simple positive functions. Step 3. Using Monotone property extend the ...
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118 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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43 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
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69 views

Upper bounding a Poisson Process with indicators of exponentials

Define $E_1,E_2,\ldots, E_i,\ldots E_n$ as i.i.d. exponentials with parameter $\lambda$. These define processes on some interval $[0,\delta]$ (think of $\delta$ as very small, it will come into play ...
5
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80 views

representation theorem on the path space

I'm working on a project and have done some work. However, there are some point where I'm unsure if my thoughts are correct. It would be appreciated if someone could share their thoughts about it. ...
5
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130 views

Fractal dimension of Gaussian white noise is infinite?

I read in this paper that the fractal dimension of Gaussian white noise is infinite. The paper does not prove it nor give a reference to support it. I failed to find a reference from online searching. ...
5
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559 views

the parametrization of a Gumbel in terms of a Gaussian

Extreme Value Distribution From a Gaussian. I was wondering how the parametrization of $\alpha$ and $\beta$ of a Gumbel $e^{-e^{-\frac{x-\alpha }{\beta }}}$ was done in terms of a cumulative Gaussian ...
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146 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is ...
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76 views

Regularity of test functions in the definition of the total variation distance

As is well-known, the total variation distance between (the laws of) two random variables $X$ and $Y$ defined on $\mathbb{R}$ is given by $\sup |E[g(X)]-E[g(Y)]|$, where the supremum is taken over all ...
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86 views

Confusion in the proof of properties for $\psi$-irreducibility

Let $P$ be a stochastic kernel on a measurable space $(\mathsf X,\mathfrak B(\mathsf X))$. The kernel $P$ is called $\varphi$-irreducible if for a positive measure $\varphi$ and for all measurable ...
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423 views

weak law of large numbers: proof using characteristic functions vs proof using truncation

In Varadhan's lecture notes on Probability Theory (they are online here and on Amazon here) in Exercise 3.6 he writes: The weak law may hold sometimes, even if the mean does not exist. If we ...
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209 views

Do probability distributions form a comonad?

$\def\unit{{\rm unit}}\def\join{{\rm join}}$It's well known that (discrete) probability distributions form a monad. Specifically, if we let $PX$ be the set of discrete probability distributions on ...
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215 views

Where does directed random walk hit the boundary?

I have a problem that I more or less know the answer to, but would really like to see it done in a systematic, rather than ad hoc way. In spite of this, I will pose the question in a very concrete ...
5
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316 views

Topological necessary and sufficient condition for tightness

Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$: For each $\varepsilon>0$, we can find a compact subset $K$ of ...
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211 views

Weak convergence of stochastic process

For a stochastic process with trajectories in $C[0,1]$ why is it that convergence of the finite dimensional distributions is not sufficient for weak convergence,unless we also have relative ...
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2k views

Dunford-Pettis Theorem

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now ...
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500 views

Idempotence and the Rao–Blackwell theorem

Original question: In the Wikipedia article on the Rao–Blackwell theorem, we read: In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
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125 views

Probability Constructions

$y_n$ is a sequence of probability measures on $\mathbb{R}$ such that $y_n\rightarrow y$ where $y$ is another probability measure on $\mathbb{R}$. Construct an example where: $\int x \; dy_n$ ...
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35 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
4
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0answers
105 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
4
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0answers
38 views

Generalized Binomial Model independent in the limit

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ With $0\leq \theta_n+ ...
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51 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
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89 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
4
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0answers
104 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
4
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0answers
73 views

Conditional expectation proof

I hope this is the right place to ask my question - my question comes from some reading I'm doing in mathematical finance, but my question is really a question in probability theory, and is about how ...
4
votes
0answers
94 views

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
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127 views

Convergence of mutual information

Let $P_n (x,y)$ be a sequence of (cumulative) probability distributions defined on $\mathcal{X}\times \mathcal{Y}$ (of arbitrary cardinality), that weakly converges to $P(x,y)$: $$ P_n (x,y) ...
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49 views

Probability measure on the set of probability measures

Let $X$ be a metric space, $\mathcal{P}(X)$ be the the set of all Borel probability measures on $X$. I endow $\mathcal{P}(X)$ with the weak* topology. Question 1: Do I need to impose some further ...