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Inequality for exponential sum
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I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has to do with the remainder I guess...
$$|E[exp(itX_n,k)|F_n,k-1]-1-frac12t^2E[X_n,k^2|F_n,k-1]|\
leq frac16|t|^3E[|X_n,k|^3mathrm1_leq epsilonbigF_n,k-1]+t^2E[X_n,k^21_X_n,k>epsilon|F_n,k-1]$$
Where $E[X_n,k|F_n,k-1]=0$ for all $k,n in mathbbN$
Especially this is taken from
Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES
and can be found in the proof of theorem 2.1 equality (4.4). Thanks in advance.
probability probability-theory inequality central-limit-theorem exponential-sum
asked Jul 23 at 14:41
DrShredz
82
add a comment |Â
up vote
0
down vote
favorite
I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has to do with the remainder I guess...
$$|E[exp(itX_n,k)|F_n,k-1]-1-frac12t^2E[X_n,k^2|F_n,k-1]|\
leq frac16|t|^3E[|X_n,k|^3mathrm1_leq epsilonbigF_n,k-1]+t^2E[X_n,k^21_X_n,k>epsilon|F_n,k-1]$$
Where $E[X_n,k|F_n,k-1]=0$ for all $k,n in mathbbN$
Especially this is taken from
Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES
and can be found in the proof of theorem 2.1 equality (4.4). Thanks in advance.
probability probability-theory inequality central-limit-theorem exponential-sum
asked Jul 23 at 14:41
DrShredz
82
It looks like $e^itX_n,k$ has been expanded into a power series in $t$, and the error term after $t^2$ term is used.
– herb steinberg
Jul 23 at 21:49
Yeah but how does one obtain the Indicators?
– DrShredz
Jul 24 at 12:03
Please define what you mean by "indicators".
– herb steinberg
Jul 24 at 16:17
I meant the conditioning of both expectations on $leq epsilon $ respectively $$
– DrShredz
Jul 24 at 19:24
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has to do with the remainder I guess...
$$|E[exp(itX_n,k)|F_n,k-1]-1-frac12t^2E[X_n,k^2|F_n,k-1]|\
leq frac16|t|^3E[|X_n,k|^3mathrm1_leq epsilonbigF_n,k-1]+t^2E[X_n,k^21_X_n,k>epsilon|F_n,k-1]$$
Where $E[X_n,k|F_n,k-1]=0$ for all $k,n in mathbbN$
Especially this is taken from
Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES
and can be found in the proof of theorem 2.1 equality (4.4). Thanks in advance.
probability probability-theory inequality central-limit-theorem exponential-sum
asked Jul 23 at 14:41
DrShredz
82
I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has to do with the remainder I guess...
$$|E[exp(itX_n,k)|F_n,k-1]-1-frac12t^2E[X_n,k^2|F_n,k-1]|\
leq frac16|t|^3E[|X_n,k|^3mathrm1_leq epsilonbigF_n,k-1]+t^2E[X_n,k^21_X_n,k>epsilon|F_n,k-1]$$
Where $E[X_n,k|F_n,k-1]=0$ for all $k,n in mathbbN$
Especially this is taken from
Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES
and can be found in the proof of theorem 2.1 equality (4.4). Thanks in advance.
probability probability-theory inequality central-limit-theorem exponential-sum
asked Jul 23 at 14:41
DrShredz
82
asked Jul 23 at 14:41
DrShredz
82
asked Jul 23 at 14:41
DrShredz
82
asked Jul 23 at 14:41
DrShredz
82
82
It looks like $e^itX_n,k$ has been expanded into a power series in $t$, and the error term after $t^2$ term is used.
– herb steinberg
Jul 23 at 21:49
Yeah but how does one obtain the Indicators?
– DrShredz
Jul 24 at 12:03
Please define what you mean by "indicators".
– herb steinberg
Jul 24 at 16:17
I meant the conditioning of both expectations on $leq epsilon $ respectively $$
– DrShredz
Jul 24 at 19:24
add a comment |Â
It looks like $e^itX_n,k$ has been expanded into a power series in $t$, and the error term after $t^2$ term is used.
– herb steinberg
Jul 23 at 21:49
Yeah but how does one obtain the Indicators?
– DrShredz
Jul 24 at 12:03
Please define what you mean by "indicators".
– herb steinberg
Jul 24 at 16:17
I meant the conditioning of both expectations on $leq epsilon $ respectively $$
– DrShredz
Jul 24 at 19:24
It looks like $e^itX_n,k$ has been expanded into a power series in $t$, and the error term after $t^2$ term is used.
– herb steinberg
Jul 23 at 21:49
It looks like $e^itX_n,k$ has been expanded into a power series in $t$, and the error term after $t^2$ term is used.
– herb steinberg
Jul 23 at 21:49
Yeah but how does one obtain the Indicators?
– DrShredz
Jul 24 at 12:03
Yeah but how does one obtain the Indicators?
– DrShredz
Jul 24 at 12:03
Please define what you mean by "indicators".
– herb steinberg
Jul 24 at 16:17
Please define what you mean by "indicators".
– herb steinberg
Jul 24 at 16:17
I meant the conditioning of both expectations on $leq epsilon $ respectively $$
– DrShredz
Jul 24 at 19:24
I meant the conditioning of both expectations on $leq epsilon $ respectively $$
– DrShredz
Jul 24 at 19:24
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
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Somebody on MO just pointed out to me that the inequality, especially the first term in the second line comes from the remainder in integral form of the corresponding Taylor series.
answered Jul 24 at 19:29
DrShredz
82
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Somebody on MO just pointed out to me that the inequality, especially the first term in the second line comes from the remainder in integral form of the corresponding Taylor series.
answered Jul 24 at 19:29
DrShredz
82
add a comment |Â
up vote
0
down vote
Somebody on MO just pointed out to me that the inequality, especially the first term in the second line comes from the remainder in integral form of the corresponding Taylor series.
answered Jul 24 at 19:29
DrShredz
82
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Somebody on MO just pointed out to me that the inequality, especially the first term in the second line comes from the remainder in integral form of the corresponding Taylor series.
answered Jul 24 at 19:29
DrShredz
82
Somebody on MO just pointed out to me that the inequality, especially the first term in the second line comes from the remainder in integral form of the corresponding Taylor series.
answered Jul 24 at 19:29
DrShredz
82
answered Jul 24 at 19:29
DrShredz
82
answered Jul 24 at 19:29
DrShredz
82
answered Jul 24 at 19:29
DrShredz
82
82
add a comment |Â
add a comment |Â
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It looks like $e^itX_n,k$ has been expanded into a power series in $t$, and the error term after $t^2$ term is used.
– herb steinberg
Jul 23 at 21:49
Yeah but how does one obtain the Indicators?
– DrShredz
Jul 24 at 12:03
Please define what you mean by "indicators".
– herb steinberg
Jul 24 at 16:17
I meant the conditioning of both expectations on $leq epsilon $ respectively $$
– DrShredz
Jul 24 at 19:24