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Yuval Peres
Yuval Peres
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The function you propose is related to the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70

The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].

Returning to the original question, Decompose $\mu=\mu_a+\mu_s$ where $\mu_a$ is absolutely continuous to $\nu$ with Radon-Nikodym derivative $f$, and $\mu_s$ is singular to $\nu$. Write $M_s$ for the total mass of $\mu_s$. If $M_s \ge x$ then $F(x)=0$. Otherwise consider the sets $A_c:=\{f>c\}$. If there is such a set with $\mu(A_c)=x-M_s$ then $F(x)=\nu(A_c)$. If there is no such $c$, find the infimmum $c_*$ of the constants $c$ such that $\mu(A_c)<x-M_s$. one needs to do some tie-breaking inside the level set where $f=c_*$ and take a subset there of suitable $\mu$ measure $x-M_s-\mu(A_{c_*})$. Thus $F(x)=\nu(A_{c_*})+(x-\nu(A_{c_*}))/c_*$, I can add details if needed.

[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.

[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.

[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.

[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.

[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.

The function you propose is related to the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70

The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].

Returning to the original question, Decompose $\mu=\mu_a+\mu_s$ where $\mu_a$ is absolutely continuous to $\nu$ with Radon-Nikodym derivative $f$, and $\mu_s$ is singular to $\nu$. Write $M_s$ for the total mass of $\mu_s$. If $M_s \ge x$ then $F(x)=0$. Otherwise consider the sets $A_c:=\{f>c\}$. If there is such a set with $\mu(A_c)=x-M_s$ then $F(x)=\nu(A_c)$. If there is no such $c$ find the infimmum $c_*$ of the constants $c$ such that $\mu(A_c)<x-M_s$. one needs to do some tie-breaking inside the level set where $f=c_*$ and take a subset there of suitable $\mu$ measure $x-M_s-\mu(A_{c_*})$. Thus $F(x)=\nu(A_{c_*})+(x-\nu(A_{c_*}))/c_*$, I can add details if needed.

[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.

[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.

[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.

[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.

[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.

The function you propose is related to the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70

The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].

Returning to the original question, Decompose $\mu=\mu_a+\mu_s$ where $\mu_a$ is absolutely continuous to $\nu$ with Radon-Nikodym derivative $f$, and $\mu_s$ is singular to $\nu$. Write $M_s$ for the total mass of $\mu_s$. If $M_s \ge x$ then $F(x)=0$. Otherwise consider the sets $A_c:=\{f>c\}$. If there is such a set with $\mu(A_c)=x-M_s$ then $F(x)=\nu(A_c)$. If there is no such $c$, find the infimmum $c_*$ of the constants $c$ such that $\mu(A_c)<x-M_s$. one needs to do some tie-breaking inside the level set where $f=c_*$ and take a subset there of suitable $\mu$ measure $x-M_s-\mu(A_{c_*})$. Thus $F(x)=\nu(A_{c_*})+(x-\nu(A_{c_*}))/c_*$.

[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.

[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.

[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.

[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.

[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.

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Yuval Peres
Yuval Peres
  • 14.2k
  • 1
  • 28
  • 49

The inverse of the function function you propose, that is $$L(y) = \sup\{\mu(A): A\in\mathscr{B},\nu(A)\le y \}.$$ is known as is related to the L'evy concentration function, studied studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70

The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].

Returning to the original question, Decompose $\mu=\mu_a+\mu_s$ where $\mu_a$ is absolutely continuous to $\nu$ with Radon-Nikodym derivative $f$, and $\mu_s$ is singular to $\nu$. Write $M_s$ for the total mass of $\mu_s$. If $M_s \ge x$ then $F(x)=0$. Otherwise consider the sets $A_c:=\{f>c\}$. If there is such a set with $\mu(A_c)=x-M_s$ then $F(x)=\nu(A_c)$. If there is no such $c$ find the infimmum $c_*$ of the constants $c$ such that $\mu(A_c)<x-M_s$. one needs to do some tie-breaking inside the level set where $f=c_*$ and take a subset there of suitable $\mu$ measure $x-M_s-\mu(A_{c_*})$. Thus $F(x)=\nu(A_{c_*})+(x-\nu(A_{c_*}))/c_*$, I can add details if needed.

[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.

[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.

[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.

[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.

[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.

The inverse of the function you propose, that is $$L(y) = \sup\{\mu(A): A\in\mathscr{B},\nu(A)\le y \}.$$ is known as the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70

The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].

[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.

[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.

[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.

[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.

[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.

The function you propose is related to the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70

The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].

Returning to the original question, Decompose $\mu=\mu_a+\mu_s$ where $\mu_a$ is absolutely continuous to $\nu$ with Radon-Nikodym derivative $f$, and $\mu_s$ is singular to $\nu$. Write $M_s$ for the total mass of $\mu_s$. If $M_s \ge x$ then $F(x)=0$. Otherwise consider the sets $A_c:=\{f>c\}$. If there is such a set with $\mu(A_c)=x-M_s$ then $F(x)=\nu(A_c)$. If there is no such $c$ find the infimmum $c_*$ of the constants $c$ such that $\mu(A_c)<x-M_s$. one needs to do some tie-breaking inside the level set where $f=c_*$ and take a subset there of suitable $\mu$ measure $x-M_s-\mu(A_{c_*})$. Thus $F(x)=\nu(A_{c_*})+(x-\nu(A_{c_*}))/c_*$, I can add details if needed.

[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.

[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.

[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.

[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.

[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.

Source Link
Yuval Peres
Yuval Peres
  • 14.2k
  • 1
  • 28
  • 49

The inverse of the function you propose, that is $$L(y) = \sup\{\mu(A): A\in\mathscr{B},\nu(A)\le y \}.$$ is known as the L'evy concentration function, studied by Kolmogorov, Rogozin, Esseen and others. See the special volume [1] https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70

The classic book [2] has a chapter devoted to concentration functions with many references and the paper [3] has a quite sharp estimate; connection to combinatorics are in [4]. Also related is the study of small-ball probabilities, see the survey [5].

[1] Kruglov, V. M. "Concentration Functions (No. 45)." Selected Works of AN Kolmogorov. Springer, Dordrecht, 1992. 571-574.

[2] Petrov, Valentin Vladimirovich. Sums of independent random variables. Vol. 82. Springer Science & Business Media, 2012.

[3] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25.1 (1970): 133-144.

[4] Halász, Gábor. "Estimates for the concentration function of combinatorial number theory and probability." Periodica Mathematica Hungarica 8.3-4 (1977): 197-211.

[5] Li, Wenbo V., and Q-M. Shao. "Gaussian processes: inequalities, small ball probabilities and applications." Handbook of Statistics 19 (2001): 533-597.