# Tagged Questions

Questions about characteristic functions, of a set (which gives $1$ if the element is on the set and $0$ otherwise) or of a random variable (its Fourier transform). Do not use this tag if you are asking about the method of characteristics in PDE or the characteristic polynomial in linear algebra.

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### $p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
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### Aymptotic Convergence of Mean Estimator

I want to show: Let $x_{i}$ be an iid random variable with support $x_{i} \in [0,1]$. Prove $n^{1/3}\frac{1}{n} \sum\limits_{i=1}^{n} (x_{i} - \mathbb{E}[x_{i}] ) \xrightarrow{p} 0$. From ...
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### Calculation of infinite product

My question is to prove the identity: $$\prod_{n=1}^{\infty}\left(\frac{\cos t-1}{n}+1\right)=\exp\left(-\int_0^1x^{-1}(1-\cos xt)dx\right)$$ which arises as a product of characteristic functions of ...
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### Convolution of characteristic function

I am trying to figure out following problem. Let A ⊂ R. Then we can define the characteristic function: Let a be bigger than 0. I am trying to find a following convolution: \begin{align} \chi_{[-a,a]...
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### Exsistence of random variable $U[-1,1]$

I have problems with the following question: Let X be the random variable such that $X\sim U[-1,1]$. Does there exsist a random variable Y, independent to X, such that $X+Y\sim 2Y$? I thought that ...
How to find characteristic of the following equation $\mathbf{e^{-x}u_{xx}+2 \hspace{1pt} e^yu_{xy}+e^x \hspace{1pt} u_{x}=0}$ would the characteristic derived from the below equation? $e^{-x}(\... 1answer 49 views ### Abstract machines that compute primitive recursive functions What it the simplest (least powerful) abstract machine that can compute primitive recursive sets, i.e. sets whose characteristic or indicator function is primitive recursive? $$f:\mathbb{N}\... 2answers 54 views ### Show that if a_n(X_n-X) \overset{\mathcal{D}}\to Z then X_n \overset{P}\to X. Let (a_n)\subseteq \Bbb{R} be a sequence such that a_n \to \infty. Let (X_n) be a sequence of random variables such that a_n(X_n-X) \overset{\mathcal{D}}\to Z fore some random variables X ... 1answer 43 views ### Which random variable has the characteristic function f(t)=\frac{e^{it}}{1-it} Which random variable has the characteristic function$$f(t)=\frac{e^{it}}{1-it}$$This is quite important for me to know, I know I have seen it somewhere, but I cant remember which random variable. 1answer 30 views ### Derivatives of characteristic function Let$\phi$be the characteristic function for random variable$X$. I know that if$E [|X|] < \infty$, then dominated convergence implies existence of the first derivative, and in particular,$\phi'(...
I need to solve the following exercise. Assume that $X_\lambda$ is Poisson distributed with mean $\lambda$ . Show that $Y(\lambda) = \frac{X_\lambda - \lambda}{\sqrt{\lambda}}$ is asymptotic ...