698 reputation
311
bio website sites.google.com/site/…
location Jaworzno, Poland
age 28
visits member for 3 years, 4 months
seen Aug 27 at 2:55

PhD student of Mathematics at Silesian University in Poland, Probability Theory


Jul
2
awarded  Curious
Jun
25
awarded  Yearling
Oct
26
awarded  Autobiographer
Oct
26
comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
I mean the Arzela-Ascoli theorem says that if $(f_n)$ is a sequence of continuos functions on a compact set then each subsequence of $(f_n)$ has a unformly convergent subsequence iff $\{f_n\}$ is equicontinous and pointwise bounded. So, you are right, if $(f_n)$ is pointwise convergent it equivalent to the uniform convergence of $(f_n)$.
Oct
26
accepted How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
Oct
26
comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
Under the strong Feller property, you have shown that $(P^n f)_n$ is uniformly convergent on compacts sets, which is even stronger that we need. In general, I have tried something similar to this. Namely I have tried to show that a the subsequence of $(|P^n f - \pi f|)_n$ is decreasing on the set $K_0=\{x_n: n\in\mathbb{N}\} \cup \{x_0\}$, where $x_0\in K$ and $x_n$ is an arbitrary sequence of the set $K$ convergent to $x_0$. The we would have the uniform convergence on the set $K_0$ by the Dini Theorem, and therefore, on the set $K$.
Oct
26
comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
The Dini Theorem was the first thing I have tried to use in poving this.
Oct
25
comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
The authors write "Since the limit in (6.20) is continuous (and in fact constant) it follows from Ascoli’s Theorem (...)". This sentence sounds a little strange for me.
Oct
25
comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
Thanks for the reply. Now, at least, I know it is not obvious. Maybe it should be assumed that the weak convergence of $P^n(x,\cdot)$ is uniform with respect to $x$ on compact sets.
Oct
25
revised How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
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Oct
25
revised How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
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Oct
25
revised How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
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Oct
25
asked How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets?
Jun
25
awarded  Yearling
Feb
2
accepted Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable?
Feb
2
comment Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable?
Thanks! I can not believe it could be so easy:)
Feb
1
revised Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable?
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Feb
1
asked Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable?
Dec
23
comment How to prove that $\lim_{n\to\infty} n^{-1} \ln (\|A^n\|)=\ln(\rho(A))$, where $A$ is a matrix.
Yes, you are right, a was looking everywhere but not in wikipedia :)
Dec
23
accepted How to prove that $\lim_{n\to\infty} n^{-1} \ln (\|A^n\|)=\ln(\rho(A))$, where $A$ is a matrix.