Herbert Robbins

A Remark on Stirling’s Formula

We shall prove Stirling’s formula by showing that for n=1, 2,…

Strong consistency of least squares estimates in multiple regression II

n! = \sqrt {2\pi } {n^{n + 1/2}}{e^{ - n}} \cdot {e^{{r_n}}}

Estimating the Total Probability of the Unobserved Outcomes of an Experiment

(1) where rn satisfies the double inequality

On optimal stopping rules

\frac{1}{{12n + 1}} < {r_n} < \frac{1}{{12n}}.

Consistency and asymptotic efficiency of slope estimates in stochastic approximation schemes

(2) .

A Martingale System Theorem and Applications

The central limit theorem has been extended to the case of dependent random variables by several authors (Bruns, Markoff, S. Bernstein, P. Levy, Loeve). The conditions under which these theorems are stated either are very restrictive or involve conditional distributions, which makes them difficult to apply. In the present paper we prove central limit theorems for sequences of dependent random variables of a certain special type which occurs frequently in mathematical statistics. The hypotheses do not involve conditional distributions.