1
vote
2answers
93 views

$L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...
6
votes
2answers
62 views

Writing $f\in L^2([-\pi,\pi])$ as a power series.

Consider the space $L^2([-\pi,\pi])$. I want to show that every function $f\in L^2([-\pi,\pi])$ can be written as a power series. I remember a result that polynomials are dense in $L^2([-\pi,\pi])$. ...
1
vote
2answers
216 views

The radius of convergence of the power series is a constant with probability one.

I came across this question. If $\{X_n , n\geq 1\}$ are independent random variables, show that the radius of convergence of the power series $\sum_{n=1}^\infty X_n z^n$ is a constant with probability ...
1
vote
1answer
40 views

Show that Laplace transform of measure belongs to $C^{\infty}(0,\mathbb{R}^n_{+})$

Let $\mu$ be an exponentially decreasing Borel measure on $\mathbb{R}^n_{+}$, i.e. there exists $r>0$ such that $$ \int\limits_{\mathbb{R}^n_{+}} e^{r|x|} \, \mu(dx) < \infty. $$ I want to ...
-2
votes
1answer
154 views

Radius of convergence of a series of random variables

Let $X_n$ be i.i.d. and (a.s.) bounded random variables.(none of them identically zero) Prove that the radius of convergence of the series with coefficients $X_n$, ...