Approximation in probability
$begingroup$
To prove a CLT, I need first to prove that the following approximation holds
$$
sqrt{n}sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty
$$
where $W$ is a Brownian motion independent from $nu$, which is an adapted continuous bounded stochastic process which has the property
$$
nu_{Delta} = nu_{0}+O_p(sqrt{Delta}),quadforallDelta>0.
$$
I have tried with the expansion (whose correctness is not so clear to me)
and also with the $L^2$-convergence. But, in both cases, I am able to prove only
$$
sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty,
$$
i.e without the $sqrt{n}$ factor.
stochastic-processes approximation-theory
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
$endgroup$
add a comment |
$begingroup$
To prove a CLT, I need first to prove that the following approximation holds
$$
sqrt{n}sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty
$$
where $W$ is a Brownian motion independent from $nu$, which is an adapted continuous bounded stochastic process which has the property
$$
nu_{Delta} = nu_{0}+O_p(sqrt{Delta}),quadforallDelta>0.
$$
I have tried with the expansion (whose correctness is not so clear to me)
and also with the $L^2$-convergence. But, in both cases, I am able to prove only
$$
sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty,
$$
i.e without the $sqrt{n}$ factor.
stochastic-processes approximation-theory
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
$endgroup$
add a comment |
$begingroup$
To prove a CLT, I need first to prove that the following approximation holds
$$
sqrt{n}sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty
$$
where $W$ is a Brownian motion independent from $nu$, which is an adapted continuous bounded stochastic process which has the property
$$
nu_{Delta} = nu_{0}+O_p(sqrt{Delta}),quadforallDelta>0.
$$
I have tried with the expansion (whose correctness is not so clear to me)
and also with the $L^2$-convergence. But, in both cases, I am able to prove only
$$
sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty,
$$
i.e without the $sqrt{n}$ factor.
stochastic-processes approximation-theory
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
$endgroup$
To prove a CLT, I need first to prove that the following approximation holds
$$
sqrt{n}sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty
$$
where $W$ is a Brownian motion independent from $nu$, which is an adapted continuous bounded stochastic process which has the property
$$
nu_{Delta} = nu_{0}+O_p(sqrt{Delta}),quadforallDelta>0.
$$
I have tried with the expansion (whose correctness is not so clear to me)
and also with the $L^2$-convergence. But, in both cases, I am able to prove only
$$
sum_{j=1}^nleft(left|int_{(j-1)/n}^{j/n}nu_s,dW_sright|,left|int_{j/n}^{(j+1)/n}nu_s,dW_sright|-nu_{(j-1)/n}^2,left|W_{j/n}-W_{(j-1)/n}right|,left|W_{(j+1)/n}-W_{j/n}right|,right)stackrel{p}{to} 0,quadtext{ as }nto+infty,
$$
i.e without the $sqrt{n}$ factor.
stochastic-processes approximation-theory
stochastic-processes approximation-theory
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
asked Jan 14 at 11:43
AlmostSureUserAlmostSureUser
326418
326418
add a comment |
add a comment |
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