Celebratio Mathematica

[1]D. Blackwell and M. A. Girshick: “On functions of sequences of independent chance vectors with applications to the problem of the ‘random walk’ in \( k \) dimensions,” Ann. Math. Stat. 17 : 3 (September 1946), pp. 310–317. MR 0017898 Zbl 0060.29007 article

Consider a sequence \( \{x_i\} \) of independent chance vectors in \( k \) dimensions with identical distributions, and a sequence of mutually exclusive events \( S_1 \), \( S_2, \dots \), such that \( S_i \) depends only on the first \( i \) vectors and \( \sum P(S_i) = 1 \). Let \( \varphi_i \) be a real or complex function of the first \( i \) vectors in the sequence satisfying conditions:

\( E(\varphi_i) = 0 \) and
\( E(\varphi_j \mid X_1, \dots, X_i) = \varphi_i \) for \( j \geq i \).

Let \( \varphi = \varphi_i \) and \( n = i \) when \( S_i \) occurs. A general theorem is proved which gives the conditions \( \varphi_i \) must satisfy such that \( E\varphi = 0 \). This theorem generalizes some of the important results obtained by Wald for \( k = 1 \). A method is also given for obtaining the distribution of \( \varphi \) and \( n \) in the problem of the ”random walk” in \( k \) dimensions for the case in which the components of the vector take on a finite number of integral values.

[2]D. Blackwell: “On an equation of Wald,” Ann. Math. Stat. 17 : 1 (March 1946), pp. 84–87. MR 0019902 Zbl 0063.00422 article

[3]D. Blackwell: “A renewal theorem,” Duke Math. J. 15 : 1 (1948), pp. 145–150. MR 0024093 Zbl 0030.20102 article

Let \( x_i \) be independent non-negative chance variables with identical distributions. The asymptotic behavior of the expected number \( U(T) \) of sums \[ s_k = x_1 + \dots +x_k \] lying in the interval \( (0,T) \) has been studied by Feller [1941], using the integral equation of renewal theory and the method of Laplace transforms. Recently Doob [1948] has obtained as a consequence of general theorems on stationary Markov processes the following result: if the distribution of some \( s_k \) is non-singular, then \[ U(T + h) - U(T) \to \frac{h}{E(x_1)} \] as \( T\to\infty \) for every \( h > 0 \). Täcklind [1945] has obtained an excellent estimate for \( U(T) \) itself: when the \( k \)-th moment of \( x_1 \) exists for some \( k > 2 \) and the values of \( x_1 \) are not all integral multiples of some fixed constant, his estimate shows at once that \[ U(T + h) - U(T) \to \frac{h}{E(x_1)} .\]

In this paper we shall prove the following

Unless all values of \( x_1 \) are integral multiples of some fixed constant, \[ U(T + h) - U(T) \to \frac{h}{E(x_1)} \qquad (T\to\infty) \] for every \( h > 0 \). (If \( E(x_1) = \infty \), then \( h/E(x_1) \) is to be interpreted as zero.)

[4]D. Blackwell: “Extension of a renewal theorem,” Pacific J. Math. 3 : 2 (1953), pp. 315–320. MR 0054880 Zbl 0052.14104 article

A chance variable \( x \) will be called a \( d \)-lattice variable if

\( \sum_{-\infty}^{\infty}Pr\{x=nd\} = 1 \), and
\( d \) is the largest number for which (1) holds.

If \( x \) is not a \( d \)-lattice variable for any \( d \), \( x \) will be called a nonlattice variable. The main purpose of this paper is to give a proof of:

Let \( x_1 \), \( x_2, \dots \) be independent identically distributed chance variables with \( E(x_1) = m > 0 \) (the case \( m = +\infty \) is not excluded); let \[ S_n = x_1 + \dots + x_n ;\] and, for any \( h > 0 \), let \( U(a,h) \) be the expected number of integers \( n\geq 0 \) for which \( a\leq S_n < a+h \). If the \( x_n \) are nonlattice variables, then \[ U(a,h)\to \frac{h}{m}, 0 \quad\textit{as }a\to +\infty,-\infty .\] If the \( x_n \) are \( d \)-lattice variables, then \[ U(a,d)\to \frac{d}{m}, 0 \quad\textit{as }a\to +\infty,-\infty .\] (If \( m=+\infty \), \( h/m \) and \( d/m \) are interpreted as zero.)

[5]D. Blackwell: “On optimal systems,” Ann. Math. Stat. 25 (1954), pp. 394–397. MR 0061776 Zbl 0055.37002 article

For any sequence \( x_1, x_2, \dots \) of chance variables satisfying \( | x_n | \leq 1 \) and \[ E(x_n\mid x_1, \dots, x_{n-1}) \leq -u(\max | x_n| \mid x_1, \dots, x_{n-1}) ,\] where \( u \) is a fixed constant, \( 0 < u < 1 \), and for any positive number \( t \), \[ \mathrm{Pr} \bigl\{ \sup_n (x_1 + \dots + x_n) \geq t\bigr\} \leq \Bigl(\frac{1 - u}{1 + u}\Bigr)^t. \] Equality holds for integral \( t \) when \( x_1 \), \( x_2, \dots \) are independent with \begin{align*} \mathrm{Pr} \{x_n = 1\} &= (1 - u)/2,\\ \mathrm{Pr} \{x_n = -1\} &= (1 + u)/2 . \end{align*} This has a simple interpretation in terms of gambling systems, and yields a new proof of Levy’s extension of the strong law of large numbers to dependent variables, with an improved estimate for the rate of convergence.

[6]D. Blackwell: “On a class of probability spaces,” pp. 1–6 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. II. Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1956. MR 0084882 Zbl 0073.12301 inproceedings

[7]D. Blackwell and J. L. Hodges, Jr.: “The probability in the extreme tail of a convolution,” Ann. Math. Stat. 30 : 4 (1959), pp. 1113–1120. MR 0112197 Zbl 0099.35105 article

Let \( X_1, X_2, \dots \) be independent and identically distributed random variables with possible values that are integers whose differences have g.c.d. one. Assume the m.g.f. of \( X_1 \) exists in an interval about 0, let \( a \) be any number such that \[ E(X_1) < a < \sup X_1 ,\] and let \[ \varphi(a, t) = E\,e^{t(X_{1-a})} .\] There exists a unique value \( t^{\ast}(a) \) of \( t \) which minimizes \( \varphi(a, t) \) with respect to \( t \); write \[ m(a) = \varphi[ a, t^{\ast}(a)] \quad\text{and}\quad z = e^{-t^{\ast}(a)} .\] Let \( Y_1,Y_2, \dots \) be independent and identically distributed random variables such that \( Y_1 \) and \( X_1 \) have the same range and \[ \Pr(Y_1 = x) = \Pr(X_1 = x) \cdot \frac{e^{t^{\ast}(a)\,(x-a)}}{m(a)} ,\] and let \( \mu_2 = \sigma^2, \mu_3, \mu_4 \) be central moments of \( Y_1 \). We show that \[ \Pr \{X_1 + \dots + X_n = na\} = [ m(a) ]^n \Pr \{Y_1 + \dots + Y_n = na\} ,\] and use this to establish the approximation \[ \Pr \{X_1 + \dots + X_n = na\} = \pi^{\ast\ast}_n[ 1 + 0(n^{-2})] ,\] where \( na \) is a possible value of \( X_1 + \dots + X_n \) and \[ \pi^{\ast\ast}_n = \frac{[ m(a)]^n}{\sigma\sqrt{2\pi n}} \Bigl[ 1 + \frac{1}{8n} \Bigl(\frac{\mu_4}{\mu^2_2} - 3 - \frac{5}{3} \frac{\mu^3_2}{\mu^3_2}\Bigr)\Bigr]. \] Similarly we find that \[ \Pr \{X_1 + \dots + X_n \geq na\} = \Pi^{\ast\ast}_n[ 1 + 0(n^{-2})] ,\] where \[ \Pi^{\ast\ast}_n = \pi^{\ast\ast}_n \cdot \frac{1}{1 - z}\Bigl\{1 - \frac{1}{2n}\Bigl[\frac{(z\mu_3/\mu_2) + z(1 + z)/(1 - z)}{(1 + z)\mu_2}\Bigr]\Bigr\}. \] We provide some numerical illustrations of the accuracy of these approximations, and give a conjectured analog of the leading term of \( \Pi^{\ast\ast}_n \) for nonlattice variables.

[8]D. Blackwell and L. Dubins: “Merging of opinions with increasing information,” Ann. Math. Stat. 33 : 3 (1962), pp. 882–886. MR 0149577 Zbl 0109.35704 article

One of us [Blackwell 1957] has shown that if \( Z_n \), \( n = 1 \), \( 2,\dots \) is a stochastic process with \( D \) states, \( {}0 \), \( 1,\dots \), \( D-1 \) such that \[ X=\sum_{n=1}^{\infty} \frac{Z_n}{D^n} \] has an absolutely continuous distribution with respect to Lebesgue measure, then the conditional distribution of \[ R_k = \sum_{n=1}^{\infty} \frac{Z_{k+n}}{D^n} \] given \( Z_1,\dots \), \( Z_k \) converges with probability one as \( k\to\infty \) to the uniform distribution on the unit interval, in the sense that for each \( \lambda \), \( 0 < \lambda \leq 1 \), \[ P(R_k < \lambda\mid Z_1,\dots,Z_k)\to \lambda \] with probability 1 as \( k\to \infty \). It follows that the unconditional distribution of \( R_k \) converges to the uniform distribution as \( k\to\infty \). If \( \{Z_n\} \) is stationary, the distribution of \( R_k \) is independent of \( k \), and hence uniform, a result obtained earlier by Harris [1955]. Earlier work relevant to convergence of opinion can be found in [Savage, 1954, chap. 3, sect. 6].

Here we generalize these results and also show that the conditional distribution of \( R_k \) given \( Z_1,\dots \), \( Z_k \) converges in a much stronger sense. All probabilities in this paper are countably additive.

[9]D. Blackwell and C. Ryll-Nardzewski: “Non-existence of everywhere proper conditional distributions,” Ann. Math. Stat. 34 : 1 (1963), pp. 223–225. MR 0148097 Zbl 0122.13202 article

[10]D. Blackwell and D. Freedman: “A remark on the coin tossing game,” Ann. Math. Stat. 35 : 3 (1964), pp. 1345–1347. MR 0169257 Zbl 0129.31502 article

[11]D. Blackwell, P. Deuel, and D. Freedman: “The last return to equilibrium in a coin-tossing game,” Ann. Math. Stat. 35 : 3 (1964), pp. 1344. MR 0169256 Zbl 0129.31501 article

David Freedman	Related
P. Deuel	Related

[12]D. Blackwell and D. Freedman: “On the amount of variance needed to escape from a strip,” Ann. Probab. 1 : 5 (1973), pp. 772–787. MR 0356214 Zbl 0293.60041 article

[13]D. Blackwell: “Discreteness of Ferguson selections,” Ann. Statist. 1 : 2 (1973), pp. 356–358. MR 0348905 Zbl 0276.62009 article

[14]D. Blackwell and J. B. MacQueen: “Ferguson distributions via Pólya urn schemes,” Ann. Statist. 1 : 2 (1973), pp. 353–355. MR 0362614 Zbl 0276.62010 article

[15]D. Blackwell and L. E. Dubins: “On existence and non-existence of proper, regular, conditional distributions,” Ann. Probab. 3 : 5 (1975), pp. 741–752. MR 0400320 Zbl 0348.60003 article

If \( \mathscr{A} \) is the tail, invariant, or symmetric field for discrete-time processes, or a field of the form \( \mathscr{F}_{t+} \) for continuous-time processes, then no countably additive, regular, conditional distribution given \( \mathscr{A} \) is proper. A notion of normal conditional distributions is given, and there always exist countably additive normal conditional distributions if \( \mathscr{A} \) is a countably generated sub \( \sigma \)-field of a standard space. The study incidentally shows that the Borel-measurable axiom of choice is false. Classically interesting subfields \( \mathscr{A} \) of \( \mathscr{B} \) possess certain desirable properties which are the defining properties for \( \mathscr{A} \) to be “regular” in \( \mathscr{B} \).

[16]D. Blackwell and L. E. Dubins: “An extension of Skorohod’s almost sure representation theorem,” Proc. Am. Math. Soc. 89 : 4 (1983), pp. 691–692. MR 718998 Zbl 0542.60005 article

[17]D. Blackwell: Approximate normality of large products. Preprint 54, U.C. Berkeley Statistics Department (Berkeley, CA), 1985. techreport

[18]D. Blackwell and R. D. Mauldin: “Ulam’s redistribution of energy problem: Collision transformations,” pp. 149–153 in In memory of Stan Ulam, published as Lett. Math. Phys. 10 : 2–3 (1985). MR 815237 Zbl 0582.60035 incollection

Stanisław Marcin Ulam	Related
Richard Daniel Mauldin	Related

[19]D. Blackwell and S. Ramakrishnan: “Stationary plans need not be uniformly adequate for leavable, Borel gambling problems,” Proc. Am. Math. Soc. 102 : 4 (1988), pp. 1024–1027. MR 934886 Zbl 0658.60072 article

[20]D. Blackwell: “Large deviations for martingales,” pp. 89–91 in Festschrift for Lucien Le Cam. Edited by D. Pollard, E. N. Torgersen, and G. L. Yang. Springer (New York), 1997. MR 1462940 Zbl 0883.60041 incollection

Erik Nikolai Torgersen	Related
Grace Lo Yang	Related
David Pollard	Related
Lucien Marie Le Cam	Related

David H. Blackwell

Probability