Celebratio Mathematica — Blackwell

Probability

Works connected to Joseph Lawson Hodges, Jr.

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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.