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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333 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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253 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 ...
7
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168 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 ...
7
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131 views

Failure of Doob-Dynkin lemma in general measurable spaces

The version of the Doob-Dynkin lemma given in my textbook is as follows: Let $f: \Omega_1 \to \Omega_2$ be a function, let $\mathcal{F}$ be a $\sigma$-algebra on $\Omega_2$, and let $\sigma(f)$ be ...
7
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99 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 ...
7
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189 views

Uniqueness of the random variable from its distribution

When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf ...
6
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69 views

Covergenge of the sum of reciprocal random variable.

If $(X_n)_{n\in\mathbb{N}}$ are independent identically distributed random variables with density $f$ even, continuous in $0$ and such that $f(0)>0$, then $$\frac{1}{n}\left(\frac{1}{X_1}+\dots + ...
6
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107 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 ...
6
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123 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 ...
6
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0answers
285 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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849 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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122 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 ...
6
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287 views

A problem about strong law of large numbers of Shiryaev's Probability

This is a problem after the section "Strong Law of Large Numbers" of Shiryaev's Probability: Let $\xi_1,\xi_2,...$ denote independent and identically distributed random variables such thatt ...
6
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0answers
253 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 $H$. ...
6
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171 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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327 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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137 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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143 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 $b_1=(u,v_1)$ ...
6
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268 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 ...
5
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76 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. ...
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95 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. ...
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130 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 ...
5
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69 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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146 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 ...
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78 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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236 views

Existence of iid random variables

In probability theory we often used the existence of a sequence $(X_n)_n$ of independent and identically distributed random variables. This was already discussed here. One of the answers says: As ...
5
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236 views

Portmanteau theorem for vague convergence

I would like to investigate if an analog of the classical Portmanteau theorem holds for vague convergence of Radon measures. Here are the definitions I'm using. Let $X$ be a Hausdorff locally ...
5
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0answers
186 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 ...
5
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203 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
votes
0answers
178 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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432 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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725 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 ...
5
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0answers
122 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: $\int x \; dy_n$ ...
4
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114 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
4
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206 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
4
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74 views

Submartingality of generalized stochastic exponential of a BMO martingale

I attended a talk today on BMO martingales. It was my first encounter with the subject, and this may explain my inability to solve this myself. We take a continuous local martingale $L$, and say ...
4
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0answers
53 views

Is there a characterization of the shift-invariant ergodic measures?

Consider probability measures $\mu$ on the space $\{0,1\}^\mathbb{N}$ that are shift-invariant with respect to the left-shift map. Is there a nice characterization of the ergodic shift-invariant ...
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46 views

Convergence in distribution of bernoulli rv over square root of uniform rv

This is a question from an old comprehensive exam: Let $U$ be a $\operatorname{Uniform}[0,1]$ random variable and let $X$ be a $\operatorname{Bernoulli}(1/2)$ random variable independent of $U$. ...
4
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72 views

Identifying the joint distribution from some values of $t \cdot X$

Suppose that $S$ is a subset of $\mathbb{R}^n$ and $X, Y$ are $\mathbb{R}^n$ valued RVs. We already know that $X$ and $Y$ are equidistributed iff $t \cdot X=^d t\cdot Y$ for all $t \in \mathbb{R}^n$. ...
4
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0answers
91 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
4
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78 views

Could you please help me understand the discrepancy metric?

I am trying to understand the discrepancy metric and its properties. It is defined as $$d_D(\mu,\nu):=\sup_{\small \mbox{ all closed balls}\,\, B}|\mu(B)-\nu(B)|$$ for probability measures $\mu$ and ...
4
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72 views

Angle between $(X,Y)$ and $(E(X), E(Y)) $ where X and Y are independenyt random variables.

Suppose that X and Y are two independent random variables with known (different/same) probability distribution functions. Now consider the vector $(X,Y)$, I want to find the angle between $(X,Y)$ and ...
4
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0answers
71 views

Probability theory: Cheating husband and subway

Here is the problem: Man cheats on his wife, everyday after work. So when the work day ends he goes to a subway station and waits for a train. Here is a subway map: ...
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55 views

Reversing a diffusion bridge.

Suppose I have an $n$-dimensional Itô SDE $$dX_t = \sigma(X_t) dW_t + \lambda(X_t)dt$$ and I'm interested in diffusion bridges from $X_0=a\in\mathbb R^n$ to $X_T=b\in\mathbb R^n$. Now let $Y_t$ be a ...
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36 views

Intuition on continuty in probability/mean square of a process

How to explain that a process is continuous in probability? I know the definition, but what does it mean? The same with continuity in mean square.
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460 views

Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am ...
4
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0answers
126 views

probability of this event happening

Play $(n+1)t$ rounds of the same coin-tossing game and the coin is fair ($n$ is a fixed natural number). Please help me find the following probability: $P$(the number of rounds of tossing that ...
4
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0answers
215 views

An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.

We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
4
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187 views

Disintegration of Measures

I was thinking about this exercise and I can't see how to end it. I'm sorry about the long post and thank you for the attention. Before asking the question, I need some background. Let $(\Omega, ...
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0answers
81 views

Does Multiplicative Version of Azuma's Inequality Hold?

We know that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound. Chernoff bound: ...