# Tagged Questions

1answer
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### Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$\|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
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### Find a asymptotic upper bound for $\sum_{n=N}^{\infty}p_{ii}^{(n)}$ for a asymetric one-dimensional simple random walk

For asymmetric one-dimensional simple random walk, that is $$P(X_n = X_{n-1} + 1) = p = 1 - P(X_n = X_{n-1} - 1)$$ for some $p \ne 1/2$, provide an asymptotic upper bound for ...
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### Set Difference Probability [duplicate]

Here is the question: Prove that for every $\epsilon>0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$. --I have ...
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### The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
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### Relation between convergence in distribution and weak convergence

If $(X_n: n\in \mathbb{N}), X$ are a sequence of random variables in $\mathbb{R}$, I wish to show that $X_n \to X$ weakly if and only if $X_n \to X$ in distribution. By 'converging weakly' I mean that ...
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### A naive question about “random” probability distributions

So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable $X$ taking values in the interval $[-1,1]$. Then, say, ...
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### Expected value, I do not get this “wikipedia triviality”

on wikipedia are two definitions of the expected value for some random variable $E(X)=\int t f_X(t) dt$ and $E(X)=\int X dP$. I do not see how they are equivalent. In cases, that $X$ would be ...
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### Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
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### supremum norm and convergence.

Suppose $f:[0,\infty) \rightarrow \mathbb{R}$ is continuous. Suppose that for some $\epsilon$ > 0, $\max_{t \in [0,n]} |f(t)|$ < $\epsilon$ for all $n \in \mathbb{N}$. Is it then true that ...
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38 views

### IFF conditions for convergence in probability and almost surely

I am working on a bunch of problems in preparation for an exam in Probability Theory. I have come across two similar questions that I need some assistance with. Suppose we have a sequence of ...
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### Law of large numbers for Brownian Motion

Let $\{B_t: 0 \leq t < \infty\}$ be standard Brownian motion and let $T_n$ be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold? ...
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### Definition of a random variable in the context of a hypergeometric distribution

We defined a random variable in a probability space $(\Omega, E, P)$ as a map $X: \Omega \rightarrow \mathbb{R}$. Unfortunately, I somehow have the impression that this term "random variable is used ...
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114 views

### Convergence of events in a probability space with respect to $L^2$

Define for events $X, Y$ that $d(X,Y) = P((X-Y) \cup (Y-X))$ = $P(X \bigtriangleup Y)$, show that $d(X_n,X) \rightarrow 0$ if and only if $\chi_{X_n}$ converges in $L^2$ to $\chi_X$ (these are ...
1answer
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### Does convergence of $\lim_{n \rightarrow \infty} \int h d\mu_n$ for all continuous, bounded $h$ imply weak convergence of $(\mu_n)$?

Let $(\mu_n)$ be a sequence of positive finite Borel measures on $\mathbb{R}$. Suppose $\lim_{n \rightarrow \infty} \int h d\mu_n$ converges for every bounded, continuous function $h$ on $\mathbb{R}$. ...
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### Does weak convergence with uniformly bounded densities imply absolute continuity of the limit?

Suppose $(f_n)$ is a sequence of probability density functions on $\mathbb{R}$ such that \begin{align*} f_n &\leq M \text{ for all }n \\ f_n(x) &= 0 \text{ for all } |x| > 1 \end{align*} ...
1answer
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### Proving $\int^{\infty}_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0$

I've been asked to prove that $$\int^\infty_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0, \space \forall x > n \geq 2.$$ My approach so far has been to use a theorem proved in class ...
1answer
50 views

### Techniques for determining whether the weak limit of an absolutely continuous sequence of probability measures is absolutely continuous?

Let $(\mu_n)$ be a sequence of probability measures on $\mathbb{R}$ that converges weakly to a probability measure $\mu$ on $\mathbb{R}$. So $$\lim \int hd\mu_n = \int h d\mu$$ whenever $h$ is a ...
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### Help with an asymptotic proof?

I am having difficulty with a proof of an asymptotic result in maximum likelihood estimation theory. I don't believe any statistical theory however is needed to solve this problem, rather I think ...
0answers
42 views

### When is the Fourier Transform Cyclic?

This is kind of a continuation of Questions about stable laws but I did not want the subject to be too broad in one question. The only stable laws that are also symmetric and nondegenerate (and hence ...
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### Brownian motion recurrence theorems and Hausdorff Dimension

I need help with proving: 1.If $d>1$ then d-dimensional Brownian motion starting at $x$ has 0 probability to actually hit $y$. Note that this is different from the usual notion of recurrence, ...
0answers
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### Question in ergodic theory

Again the source is http://www.math.ucla.edu/~biskup/275c.1.13s/PDFs/HW1.pdf this time I'm looking at #6 the part that is left as an open-ended question. If $f \in L^1$ and $\phi$ is a measure ...