# Tagged Questions

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 [tag:probability-...

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### Algebra involved in computations with (or extensions of) a probability measure on a lattice

Suppose we have a probability measure $\gamma$ defined on the $d$-dimensional lattice $\mathbb{N}^d$. Let us say $d = 5$ for simplicity. I will also write $\gamma_{1:5}$ for this measure and write say ...
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### Derivative of measure-valued function

I have a measure $\mu^x$ which is the law of a random variable and depends on $x$. The specific situation I am thinking of is $\mu^x$ is the law of $X_t$, the solution of an SDE with $X_0=x$. If I ...
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### The expectation after Bayesian inference of a Normal r.v

I'm confusing myself with this question. Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$. We known the variance $\sigma^2$ however don't know the mean $\mu$, and choose to use ...
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### Random walk with increasing step size

A random walk in the plane with step size $f(n)$ is cool if there is some constant $C$ such that it returns within a distance $C$ of the origin infinitely many times with probability $1$. At the $n$-...
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### Green kernel of Hunt processes

Let $X$ be a regular Hunt process on $R^+$ with starting at $x$ and $T_y :=\inf\{t>0: X_t=y\}, y\in R^+$. $G_q(\cdot,\cdot), q >0$ denotes Green kernel of the process $X$. We have the following ...
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### Alternative way of showing convergence of central moments

I woud like to show for a random variable $X$ $${1\over n} \sum_{i=1}^n (X_i-\bar{X}_n)^q\to E(X-EX)^q \quad (\text{convergence in probability)}$$ My approach was to define $Y_i:= (X_i-\bar{X}_n)$ ...
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### Convergence rate for the p.d.f. of a normalized mean to Gaussian (i..e Berry-Esseen for pdfs)

Berry-Esseen Theorem states that the rate of convergence of the probability distribution of normalized sample mean converges to Gaussian at rate $O(1/\sqrt{n})$ (given that certain conditions are met, ...
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### Questions about regular conditional probabilities

In Section 9.9. Regular conditional probabilities and pdfs of David Williams' Probability and Martingales: By linearity and (cMON), we can show that for a fixed sequence $(F_n)$ of disjoint ...
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### Show that the stopped sequence is a martingale

This is a series problem and I'm struggling with the last part. I assume that the last part has nothing to do with previous ones, so i won't put up the other parts. Question is : Let $\tau$ be a ...
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### Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
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### Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
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### Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
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### Convergence in law and sequence of reals

How to show that if I have $Y_n$ a sequence of random variables and $a_n$ a sequence of reals such that (i) $Y_n\to Y$ in law (ii) $a_n\to a$ i have $a_nY_n\to aY$ in law? I know that if I have (i)...
### One question regarding independence of $\pi$ systems.
Let G and H be sub-sigma-algebras of F and I and J are $\pi$ systems such that $\sigma(I)=G$ and $\sigma(J)=H$ Can anyone explain the following quote? Suppose I and J are independent, for fixed ...