[1] D. Blackwell :
“On a theorem of Lyapunov ,”
Ann. Math. Stat.
22 : 1
(March 1951 ),
pp. 112–114 .
MR
0039033
Zbl
0042.28502
article

Abstract
BibTeX
@article {key0039033m,
AUTHOR = {Blackwell, David},
TITLE = {On a theorem of {L}yapunov},
JOURNAL = {Ann. Math. Stat.},
FJOURNAL = {Annals of Mathematical Statistics},
VOLUME = {22},
NUMBER = {1},
MONTH = {March},
YEAR = {1951},
PAGES = {112--114},
URL = {http://www.jstor.org/pss/2236708},
NOTE = {MR:0039033. Zbl:0042.28502.},
ISSN = {0003-4851},
}
[2] R. Bellman and D. Blackwell :
“On moment spaces ,”
Ann. Math. (2)
54 : 2
(September 1951 ),
pp. 272–274 .
MR
0043866
Zbl
0044.12601
article

Abstract
People
BibTeX
@article {key0043866m,
AUTHOR = {Bellman, Richard and Blackwell, David},
TITLE = {On moment spaces},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {54},
NUMBER = {2},
MONTH = {September},
YEAR = {1951},
PAGES = {272--274},
DOI = {10.2307/1969527},
NOTE = {MR:0043866. Zbl:0044.12601.},
ISSN = {0003-486X},
}
[3] D. Blackwell :
“The range of certain vector integrals ,”
Proc. Am. Math. Soc.
2 : 3
(September 1951 ),
pp. 390–395 .
MR
0041195
Zbl
0044.27702
article

Abstract
BibTeX

Let \( u_1,\dots,u_n \) be completely additive set functions defined over a Borel field \( \mathcal{B} \) of subsets of a space \( X \) , and let \( A \) be any bounded subset of Euclidean \( n \) -space. With every \( \mathcal{B} \) -measurable function
\[ f=a(x) = [a_1(x),\dots,a_n(x)] \]
defined on \( X \) with range in \( A \) we associate the vector
\[ v(f) = \biggl(\int a_1(x)\,du_1,\dots,\int a_n(x)\,du_n\!\biggr) .\]
Our problem is to investigate the range \( R \) of the function \( v(f) \) .

@article {key0041195m,
AUTHOR = {Blackwell, David},
TITLE = {The range of certain vector integrals},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {2},
NUMBER = {3},
MONTH = {September},
YEAR = {1951},
PAGES = {390--395},
DOI = {10.2307/2031763},
NOTE = {MR:0041195. Zbl:0044.27702.},
ISSN = {0002-9939},
}
[4] D. Blackwell :
“A representation problem ,”
Proc. Am. Math. Soc.
5 : 2
(1954 ),
pp. 283–287 .
MR
0061653
Zbl
0055.28804
article

Abstract
BibTeX

The problem solved in this paper is the following. For a fixed number \( a \) , \( 0 < a < 1 \) , what functions \( f(x) \) on \( 0 \leq x \leq 1 \) have a representation
\begin{equation*}\tag{1} f(x) = \sum_1^{\infty}c_n\varphi_n(x), \end{equation*}
where \( c_n\geq 0 \) , \( \sum_1^{\infty}c_n \) converges, and each \( \varphi_n \) is the characteristic function of a subset of \( 0\leq x \leq 1 \) of Lebesgue measure \( a \) ? Clearly any \( f \) satisfying (1) satisfies
\begin{equation*}\tag{2} 0\leq f(x)\leq\frac{1}{a}\int_0^1 f(x)\,dx \quad\text{for all }x, \end{equation*}
since
\[ f(x)\leq\sum_1^{\infty}c_n=\frac{1}{a}\int_0^1 f(x)\,dx .\]
The result of this paper is that (2) is sufficient as well as necessary for a function \( f \) to admit a representation (1).

@article {key0061653m,
AUTHOR = {Blackwell, David},
TITLE = {A representation problem},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {5},
NUMBER = {2},
YEAR = {1954},
PAGES = {283--287},
DOI = {10.2307/2032235},
NOTE = {MR:0061653. Zbl:0055.28804.},
ISSN = {0002-9939},
}
[5] D. Blackwell :
“On discrete variables whose sum is absolutely continuous ,”
Ann. Math. Stat.
28 : 2
(1957 ),
pp. 520–521 .
MR
0088091
Zbl
0078.31602
article

Abstract
BibTeX

If \( \{Z_n\} \) , \( n = 1, 2, \ldots \) is a stationary stochastic process with \( D \) states
\[ 0, 1, \dots, D - 1 \quad\text{and}\quad X = \sum^\infty_1 \frac{Z_k}{D^n} ,\]
Harris [1955] has shown that the distribution of \( X \) is absolutely continuous if and only if the \( Z_n \) are independent and uniformly distributed over \( {}0 \) , \( 1, \dots \) , \( D - 1 \) , i.e., if and only if the distribution of \( X \) is uniform on the unit interval. In this note we show that if \( \{Z_n\} \) , \( n = 1 \) , \( 2, \dots \) is any stochastic process with \( D \) states \( {}0 \) , \( 1, \dots \) , \( D - 1 \) such that
\[ X = \sum^\infty_1 \frac{Z_n}{D^n} \]
has an absolutely continuous distribution, then the conditional distribution of
\[ R_k = \sum^\infty_{n = 1} \frac{Z_{k + n}}{D^n} \]
given \( Z_1, \dots \) , \( Z_k \) converges to the uniform distribution on the unit interval with probability 1 as \( k \to \infty \) . It follows that the unconditional distribution of \( R_k \) converges to the uniform distribution as \( k \to \infty \) . Since if \( \{Z_n\} \) is stationary the distribution of \( R_k \) is independent of \( k \) , the result of Harris follows.

@article {key0088091m,
AUTHOR = {Blackwell, David},
TITLE = {On discrete variables whose sum is absolutely
continuous},
JOURNAL = {Ann. Math. Stat.},
FJOURNAL = {Annals of Mathematical Statistics},
VOLUME = {28},
NUMBER = {2},
YEAR = {1957},
PAGES = {520--521},
DOI = {10.1214/aoms/1177706985},
NOTE = {MR:0088091. Zbl:0078.31602.},
ISSN = {0003-4851},
}
[6] D. Blackwell and L. E. Dubins :
“A converse to the dominated convergence theorem ,”
Illinois J. Math.
7 : 3
(1963 ),
pp. 508–514 .
MR
0151572
Zbl
0146.37503
article

Abstract
People
BibTeX

On a probability space \( (\Omega, \mathcal{B}, P) \) , let \( \{f_n \) , \( n=1 \) , \( 2,\dots\} \) be a sequence of nonnegative random variables in \( L_1 \) such that \( f_n \to f\in L_1 \) with probability 1 and define \( g=\sup_n f_n \) . If \( g\in L_1 \) , the Lebesgue dominated convergence theorem asserts that \( E(f_n)\to E(f) \) . More generally, as noted by [Doob 1953, p. 23], if \( g\in L_1 \) , then for any Borel field \( \mathcal{B}_0 \) contained in \( \mathcal{B} \) ,
\[ E(f_n\mid\mathcal{B}_0) \to E(f\mid\mathcal{B}_0) \quad\text{a.e.} \]
If one extends this result in a minor manner, Lebesgue’s condition \( g\in L_1 \) is not only sufficient but necessary, as the following converse to the dominated convergence theorem asserts.

If \( f_n \geq 0 \) , \( f_n \to f \) a.e., \( f_n\in L_1 \) , and \( g = \sup_n f_n \notin L_1 \) , there are, on a suitable probability space, random variables \( \{f_n^* \) , \( n=1 \) , \( 2,\dots\} \) , \( f^* \) , and a Borel field \( \mathcal{C} \) such that \( f^* \) , \( f_1^* \) , \( f_2^*,\dots \) have the same joint distribution as \( f \) , \( f_1 \) , \( f_2,\dots \) , and
\[ P\bigl\{E(f_n^*\mid\mathcal{C})\to E(f^*\mid\mathcal{C})\bigr\} = 0. \]

@article {key0151572m,
AUTHOR = {Blackwell, David and Dubins, Lester
E.},
TITLE = {A converse to the dominated convergence
theorem},
JOURNAL = {Illinois J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {7},
NUMBER = {3},
YEAR = {1963},
PAGES = {508--514},
URL = {http://projecteuclid.org/euclid.ijm/1255644957},
NOTE = {MR:0151572. Zbl:0146.37503.},
ISSN = {0019-2082},
}
[7] D. Blackwell and L. E. Dubins :
“Sharp bounds on the distribution of the Hardy–Littlewood maximal function ,”
Proc. Am. Math. Soc.
14 : 3
(1963 ),
pp. 450–453 .
MR
0148842
Zbl
0118.05401
article

Abstract
People
BibTeX

Somewhat tangentially to a recent study [Blackwell and Dubins 1963], we happened to notice an inequality which supplements one of Hardy and Littlewood [1930, Theorem 7, p. 95].

@article {key0148842m,
AUTHOR = {Blackwell, David and Dubins, Lester
E.},
TITLE = {Sharp bounds on the distribution of
the {H}ardy--{L}ittlewood maximal function},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {14},
NUMBER = {3},
YEAR = {1963},
PAGES = {450--453},
DOI = {10.2307/2033819},
NOTE = {MR:0148842. Zbl:0118.05401.},
ISSN = {0002-9939},
}
[8] D. Blackwell :
“A Borel set not containing a graph ,”
Ann. Math. Stat.
39 : 4
(1968 ),
pp. 1345–1347 .
MR
0229451
Zbl
0177.48401
article

Abstract
BibTeX

Examples of Borel sets \( X, Y, B \) such that

\( B \subset X \times Y \) ,

the projection of \( B \) on \( X \) is \( X \) , but

for no Borel-measurable \( d \) mapping \( X \) into \( Y \) is the graph of \( d \) a subset of \( B \) ,

have been given by [Novikoff 1931; Sierpiński 1931; Addison 1958]. Such examples are of interest in dynamic programming (see for instance [Blackwell 1965]), since if we interpret \( X \) as the set of states of some system, \( Y \) as the set of available acts, and \( I_B(x,y) \) where \( I_B \) is the indicator of \( B \) , as your income if the system is in state \( x \) and you choose act \( y \) , you can earn 1 in every state, but there is no Borel measurable plan, i.e. function \( d \) from \( X \) into \( Y \) , with \( d(x) \) specifying the act to be chosen when the system is in state \( x \) , that earns 1 in every state.

This note presents a new example \( X \) , \( Y \) , \( B \) , simpler than those previously given. The proof that it is an example uses ideas from Addison’s construction, and a theorem of Gale and Stewart [1953] on infinite games of perfect information, and is somewhat more complicated than Addison’s

@article {key0229451m,
AUTHOR = {Blackwell, David},
TITLE = {A {B}orel set not containing a graph},
JOURNAL = {Ann. Math. Stat.},
FJOURNAL = {Annals of Mathematical Statistics},
VOLUME = {39},
NUMBER = {4},
YEAR = {1968},
PAGES = {1345--1347},
DOI = {10.1214/aoms/1177698260},
NOTE = {MR:0229451. Zbl:0177.48401.},
ISSN = {0003-4851},
}
[9] D. Blackwell :
“Borel-programmable functions ,”
Ann. Probab.
6 : 2
(1978 ),
pp. 321–324 .
MR
0460573
Zbl
0398.28002
article

Abstract
BibTeX

A new class of functions, the BP (Borel-programmable) functions, is defined. It is strictly larger than the class of Borel functions, but has some similar properties, including closure under composition. All BP functions are absolutely measurable. The class of BP sets (those with BP indicators) is a Borel field and is closed under operation A. The relation of BP sets to the R-sets of Kolmogorov is not treated.

@article {key0460573m,
AUTHOR = {Blackwell, D.},
TITLE = {Borel-programmable functions},
JOURNAL = {Ann. Probab.},
VOLUME = {6},
NUMBER = {2},
YEAR = {1978},
PAGES = {321--324},
DOI = {10.1214/aop/1176995576},
NOTE = {MR:0460573. Zbl:0398.28002.},
}
[10] D. Blackwell :
“There are no Borel SPLIFs ,”
Ann. Probab.
8 : 6
(1980 ),
pp. 1189–1190 .
MR
602393
Zbl
0451.28001
article

Abstract
BibTeX

There is no Borel function \( f \) , defined for all infinite sequences of 0s and 1s, such that for every sequence \( X \) of 0–1 random variables that converges in probability to a constant \( c \) , we have \( f(x) = c \) a.s.

@article {key602393m,
AUTHOR = {Blackwell, D.},
TITLE = {There are no {B}orel {SPLIF}s},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {8},
NUMBER = {6},
YEAR = {1980},
PAGES = {1189--1190},
DOI = {10.1214/aop/1176994581},
NOTE = {MR:602393. Zbl:0451.28001.},
ISSN = {0091-1798},
CODEN = {APBYAE},
}
[11] D. Blackwell and A. Maitra :
“Factorization of probability measures and absolutely measurable sets ,”
Proc. Am. Math. Soc.
92 : 2
(1984 ),
pp. 251–254 .
MR
754713
Zbl
0554.60001
article

Abstract
People
BibTeX

We find necessary and sufficient conditions for a separable metric space \( Y \) to possess the property that for any measurable space \( (X,\mathcal{A}) \) and probability measure \( P \) on \( X \times Y \) , \( P \) can be factored.

@article {key754713m,
AUTHOR = {Blackwell, David and Maitra, Ashok},
TITLE = {Factorization of probability measures
and absolutely measurable sets},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {92},
NUMBER = {2},
YEAR = {1984},
PAGES = {251--254},
DOI = {10.2307/2045195},
NOTE = {MR:754713. Zbl:0554.60001.},
ISSN = {0002-9939},
CODEN = {PAMYAR},
}
[12] D. Blackwell and P. Diaconis :
“A non-measurable tail set ,”
pp. 1–5
in
Statistics, probability and game theory: Papers in honor of David Blackwell .
Edited by T. S. Ferguson, L. S. Shapley, and J. B. MacQueen .
IMS Lecture Notes–Monograph Series 30 .
Institute of Mathematical Statistics (Hayward, CA ),
1996 .
MR
1481768
incollection

Abstract
People
BibTeX
@incollection {key1481768m,
AUTHOR = {Blackwell, David and Diaconis, Persi},
TITLE = {A non-measurable tail set},
BOOKTITLE = {Statistics, probability and game theory:
{P}apers in honor of {D}avid {B}lackwell},
EDITOR = {Ferguson, T. S. and Shapley, L. S. and
MacQueen, J. B.},
SERIES = {IMS Lecture Notes -- Monograph Series},
NUMBER = {30},
PUBLISHER = {Institute of Mathematical Statistics},
ADDRESS = {Hayward, CA},
YEAR = {1996},
PAGES = {1--5},
DOI = {10.1214/lnms/1215453560},
NOTE = {MR:1481768.},
ISBN = {9780940600423},
}