Mixing
Feller Markov chains with a stationary distribution are equicontinuous
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Suppose we have a locally compact separable space $X$, on which we define a Markov kernel $P:Xtimesmathcal B(X)to[0,1]$, which satisfies the weak Feller property, that is to say, for any $fin C_b(X)$, the function
$$
Pf(y)=int_X! P(y,mathrm dx)f(x)
$$
is continuous in $y$. Now suppose there exists an invariant measure $piinmathcal P(X)$, i.e. $P^n(x,cdot)topi$ weakly for all $x$.
Then the book by Meyn and Tweedie claims in Prop 6.4.2 that $$P^nftopi(f)$$ uniformly on compact sets for any $fin C_b(X)$. However, I don't see how to get anything better than pointwise convergence and uniform boundedness here.
probability-theory markov-chains
asked Jul 31 at 15:05
Monstrous Moonshine
2,6611530
add a comment |Â
up vote
1
down vote
favorite
Suppose we have a locally compact separable space $X$, on which we define a Markov kernel $P:Xtimesmathcal B(X)to[0,1]$, which satisfies the weak Feller property, that is to say, for any $fin C_b(X)$, the function
$$
Pf(y)=int_X! P(y,mathrm dx)f(x)
$$
is continuous in $y$. Now suppose there exists an invariant measure $piinmathcal P(X)$, i.e. $P^n(x,cdot)topi$ weakly for all $x$.
Then the book by Meyn and Tweedie claims in Prop 6.4.2 that $$P^nftopi(f)$$ uniformly on compact sets for any $fin C_b(X)$. However, I don't see how to get anything better than pointwise convergence and uniform boundedness here.
probability-theory markov-chains
asked Jul 31 at 15:05
Monstrous Moonshine
2,6611530
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose we have a locally compact separable space $X$, on which we define a Markov kernel $P:Xtimesmathcal B(X)to[0,1]$, which satisfies the weak Feller property, that is to say, for any $fin C_b(X)$, the function
$$
Pf(y)=int_X! P(y,mathrm dx)f(x)
$$
is continuous in $y$. Now suppose there exists an invariant measure $piinmathcal P(X)$, i.e. $P^n(x,cdot)topi$ weakly for all $x$.
Then the book by Meyn and Tweedie claims in Prop 6.4.2 that $$P^nftopi(f)$$ uniformly on compact sets for any $fin C_b(X)$. However, I don't see how to get anything better than pointwise convergence and uniform boundedness here.
probability-theory markov-chains
asked Jul 31 at 15:05
Monstrous Moonshine
2,6611530
Suppose we have a locally compact separable space $X$, on which we define a Markov kernel $P:Xtimesmathcal B(X)to[0,1]$, which satisfies the weak Feller property, that is to say, for any $fin C_b(X)$, the function
$$
Pf(y)=int_X! P(y,mathrm dx)f(x)
$$
is continuous in $y$. Now suppose there exists an invariant measure $piinmathcal P(X)$, i.e. $P^n(x,cdot)topi$ weakly for all $x$.
Then the book by Meyn and Tweedie claims in Prop 6.4.2 that $$P^nftopi(f)$$ uniformly on compact sets for any $fin C_b(X)$. However, I don't see how to get anything better than pointwise convergence and uniform boundedness here.
probability-theory markov-chains
asked Jul 31 at 15:05
Monstrous Moonshine
2,6611530
asked Jul 31 at 15:05
Monstrous Moonshine
2,6611530
asked Jul 31 at 15:05
Monstrous Moonshine
2,6611530
asked Jul 31 at 15:05
Monstrous Moonshine
2,6611530
2,6611530
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2868148%2ffeller-markov-chains-with-a-stationary-distribution-are-equicontinuous%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password