Convergence of discrete-time Markov chain to Feller processes
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,73141429
add a comment |
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,73141429
add a comment |
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,73141429
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(X_t)_{tge0}$ be a Feller process on $(Omega,mathcal A,operatorname P)$
$(h_d)_{dinmathbb N}subseteq(0,infty)$ with $$h_dxrightarrow{ntoinfty}0$$
$left(Y^{(d)}_nright)_{ninmathbb N_0}$ be a time-homogeneous Markov chain on $(Omega,mathcal A,operatorname P)$ and $$X^{(d)}_t:=Y^{(d)}_{lfloorfrac t{h_d}rfloor};;;text{for }tge0$$ for $dinmathbb N$
$N$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with parameter $1$ independent of $Y^{(d)}$ for all $dinmathbb N$ and $$N^{(d)}_t:=N_{frac t{h_d}};;;text{for }tge0$$ as well as $$tilde X^{(d)}_t:=Y^{(d)}_{N^{(d)}_t};;;text{for }tge0$$ for $dinmathbb N$
Note that $N^{(d)}$ is a Poisson process with parameter $h_d^{-1}$ for all $dinmathbb N$.
How can we show that (in probability with respect to the Skorohod topology) $X^{(d)}xrightarrow{dtoinfty}X$ iff $tilde X^{(d)}xrightarrow{dtoinfty}X$?
In the book of Kallenberg, the author is mentioning that the claim follows from the following two theorems:
I don't get how we need to apply them. Clearly, for fixed $tge0$, we can consider $$frac1dsum_{i=1}^dleft(N^{(i)}_t-N^{(i-1)}_tright)$$ with $N^{(0)}_t:=0$. However, while independent, the $N^{(i)}_t-N^{(i-1)}$ are not identically distributed ...
If it's hard to prove in the general setting, it's okay for me to assume $h_d^{-1}=d$ for all $dinmathbb N$. In that case, the strong law of large numbers yields $$sup_{tin[0,:T]}left|frac1d N^{(d)}_t-tright|xrightarrow{dtoinfty}0;;;text{almost surely for all }T>0tag1.$$ Now, let $tau^{(d)}_0:=0$, $$tau_n^{(d)}:=infleft{t>tau^{(d)}_{n-1}:N^{(d)}_t-N^{(d)}_{tau^{(d)}_{n-1}}>0right};;;text{for }dinmathbb N$$ and $$lambda^{(d)}_t:=sum_{n=0}^infty1_{left[frac nd,:frac{n+1}dright)}(t)left(tau^{(d)}_n+(dt-n)left(tau^{(d)}_{n+1}-tau^{(d)}_nright)right);;;text{for }tge0$$ for $dinmathbb N$. Moreoverr, let $T>0$ and $rho_T$ denote the metric inducing the Skorohod $J_1$-topology on the space of càdlàg functions $[0,T]tomathbb R$. We should obtain $$rho_Tleft(X^{(d)},tilde X^{(d)}right)lesup_{tin[0,:T]}left|lambda^{(d)}_t-tright|+sup_{tin[0,:T]}left|X^{(d)}_t-tilde X^{(d)}_{lambda^{(d)}_t}right|tag2,$$ where the last term should be $0$. So, if we could show that the first term converges in probability to $0$ as $dtoinfty$, we should be able to conclude (since $T$ was arbitrary).
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
probability-theory stochastic-processes markov-chains markov-process stochastic-analysis
asked Nov 29 at 18:46
0xbadf00d
1,73141429
asked Nov 29 at 18:46
0xbadf00d
1,73141429
edited Dec 4 at 14:22
asked Nov 29 at 18:46
0xbadf00d
1,73141429
asked Nov 29 at 18:46
0xbadf00d
1,73141429
asked Nov 29 at 18:46
0xbadf00d
1,73141429
1,73141429
add a comment |
add a comment |
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