# Questions tagged [infinite-divisibility]

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15
questions

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243 views

### Can we show that the characteristic function of an infinitely divisible probability measure has no zeros

Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a probability measure on $\mathcal B(E)$ and $\varphi_\mu$ denote the characteristic function$^1$ of $\mu$.
Assume $\mu$ is infinitely divisible, ...

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**1**answer

81 views

### How can we show this estimate for the convolution of two probability measures?

Let $(\delta_k)_{k\in\mathbb N}\subseteq(0,\infty)$ be nonincreasing with $\delta_k\xrightarrow{k\to\infty}0$ and $(\varepsilon_k)_{k\in\mathbb N}\subseteq(0,\infty)$ with $\sum_{k\in\mathbb N}\...

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**2**answers

119 views

### Characterization of the generator of a Lévy process using martingale problems

Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...

**1**

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**1**answer

119 views

### Is this statement of the Lévy–Khintchine formula ill-posed?

Please take a look at the following statement of the Lévy–Khintchine formula given in Probability Theory: A Comprehensive Course (2nd edition)$^1$:
Am I missing something or is this an ill-posed ...

**1**

vote

**1**answer

79 views

### Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$

Let $E$ be a $\mathbb R$-Banach space, $\mathcal M_1(E)$ (resp. $\mathcal M_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi_\mu$ ...

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63 views

### Define the convolution root of probability measures on a measurable group

Let $(G,\mathcal G)$ be a measurable group and $\nu^{\ast k}$ denote the $k$th convolution power of a probability measure $\nu$ on $(G,\mathcal G)$ for $k\in\mathbb N$.
Remember that a probability ...

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**2**answers

214 views

### If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$

If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...

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86 views

### Infinite divisiblity of log-normals

What is the law of a piece of a log-normal distribution?
We know that log-normals are infinitely divisible. What would be the law of a root of log-normal?
More specifically, suppose that $X$ is a ...

**2**

votes

**1**answer

256 views

### Questions about Levy measure in the canonical representation of infinitely divisible distributions

Let $X$ be a random variable with infinitely divisible and symmetric distribution $F$ distributed on $\mathbb{R}$.
It is well known that the characteristic function of $X$ has a canonical ...

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220 views

### Any counter example for this: ${\phi(2^n-1)} \bmod \tau(2^n-1)=0$ for every integer $n \geq 1$? [closed]

I asked this question here In S.E but i don't received any resposnes for it, I would like to know if it is appropriate for M.O.
I'm always interesting for properties of the following series : $ \...

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86 views

### Boudedness of linear operator between generalized Orlicz spaces

I am using the notations, definitions, and results of the Section X of [1] on generalized Orlicz spaces.
We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is ...

**3**

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**1**answer

113 views

### Uncountable divisible groups and the existence of order-preserving isomorphisms of their subsets

Let $(G,+,0,<)$ be an ordered divisible group of uncountable dimension. Consider the subset $G^{<0}$ of $G$.
Question: Are $G$ and $G^{<0}$ isomorphic as ordered sets? Does there exist an ...

**1**

vote

**2**answers

754 views

### divisibility of uniform distribution [closed]

Let $X$ and $Y$ be independent and identically distributed random variables.
Can $X+Y$ be a uniform distribution?
(Please prove.)
In other words, is a uniform distribution divisible?
The meaning of "...

**7**

votes

**1**answer

771 views

### Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer? [closed]

After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for $...

**2**

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273 views

### What is the Blumenthal-Getoor index of Student's distributions?

For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes.
For a symmetric infinitely ...