Integrability of the quotient between two densities
$begingroup$
Consider probability measures $(P_n)_{n=1}^infty$ and $P$, all absolutely continuous with respect to the Lebesgue measure of $mathbb{R}^d$ - say $lambda$ - and all having the same support, $E subset mathbb{R}^d$. Assume that $P_n$ converges weakly to $P$, i.e. for every Lebesgue measurable continuous and bounded function on $E$,
$$int_E f dP_n to int_E f dP.$$
Denote by $p_n= dP_n/dlambda$ and $p=dP/dlambda$ the densities associated to $P_n$ and $P$, respectively.
My question is the following: are there $mathbf{sufficient , conditions}$ such that the quotients $p_n/p$ satisfy the following integrability condition:
$$
int_E left(frac{p_n}{p}right)^delta p_n < infty
$$
for some $deltain (0,1]$ and large enough $n$?
ADDENDUM: of course asking that, for all $n>n_0$, $sup_{x in E} p_n(x)/p(x) < M_0<infty$, would give us a sufficient condition, but I would find something weaker.
probability functional-analysis density-function
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
$endgroup$
add a comment |
$begingroup$
Consider probability measures $(P_n)_{n=1}^infty$ and $P$, all absolutely continuous with respect to the Lebesgue measure of $mathbb{R}^d$ - say $lambda$ - and all having the same support, $E subset mathbb{R}^d$. Assume that $P_n$ converges weakly to $P$, i.e. for every Lebesgue measurable continuous and bounded function on $E$,
$$int_E f dP_n to int_E f dP.$$
Denote by $p_n= dP_n/dlambda$ and $p=dP/dlambda$ the densities associated to $P_n$ and $P$, respectively.
My question is the following: are there $mathbf{sufficient , conditions}$ such that the quotients $p_n/p$ satisfy the following integrability condition:
$$
int_E left(frac{p_n}{p}right)^delta p_n < infty
$$
for some $deltain (0,1]$ and large enough $n$?
ADDENDUM: of course asking that, for all $n>n_0$, $sup_{x in E} p_n(x)/p(x) < M_0<infty$, would give us a sufficient condition, but I would find something weaker.
probability functional-analysis density-function
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
$endgroup$
add a comment |
$begingroup$
Consider probability measures $(P_n)_{n=1}^infty$ and $P$, all absolutely continuous with respect to the Lebesgue measure of $mathbb{R}^d$ - say $lambda$ - and all having the same support, $E subset mathbb{R}^d$. Assume that $P_n$ converges weakly to $P$, i.e. for every Lebesgue measurable continuous and bounded function on $E$,
$$int_E f dP_n to int_E f dP.$$
Denote by $p_n= dP_n/dlambda$ and $p=dP/dlambda$ the densities associated to $P_n$ and $P$, respectively.
My question is the following: are there $mathbf{sufficient , conditions}$ such that the quotients $p_n/p$ satisfy the following integrability condition:
$$
int_E left(frac{p_n}{p}right)^delta p_n < infty
$$
for some $deltain (0,1]$ and large enough $n$?
ADDENDUM: of course asking that, for all $n>n_0$, $sup_{x in E} p_n(x)/p(x) < M_0<infty$, would give us a sufficient condition, but I would find something weaker.
probability functional-analysis density-function
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
$endgroup$
Consider probability measures $(P_n)_{n=1}^infty$ and $P$, all absolutely continuous with respect to the Lebesgue measure of $mathbb{R}^d$ - say $lambda$ - and all having the same support, $E subset mathbb{R}^d$. Assume that $P_n$ converges weakly to $P$, i.e. for every Lebesgue measurable continuous and bounded function on $E$,
$$int_E f dP_n to int_E f dP.$$
Denote by $p_n= dP_n/dlambda$ and $p=dP/dlambda$ the densities associated to $P_n$ and $P$, respectively.
My question is the following: are there $mathbf{sufficient , conditions}$ such that the quotients $p_n/p$ satisfy the following integrability condition:
$$
int_E left(frac{p_n}{p}right)^delta p_n < infty
$$
for some $deltain (0,1]$ and large enough $n$?
ADDENDUM: of course asking that, for all $n>n_0$, $sup_{x in E} p_n(x)/p(x) < M_0<infty$, would give us a sufficient condition, but I would find something weaker.
probability functional-analysis density-function
probability functional-analysis density-function
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
asked Dec 26 '18 at 10:38
Jack LondonJack London
34018
34018
add a comment |
add a comment |
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