You are not currently logged in.
Access JSTOR through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Random Variables with Independent Binary Digits
The Annals of Mathematical Statistics
Vol. 42, No. 6 (Dec., 1971), pp. 1922-1929
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2240118
Page Count: 8
You can always find the topics here!Topics: Perceptron convergence procedure, Random variables, Distribution functions, Infinite products, Absolute convergence, Mathematical theorems, Zero, Integers, Series convergence, Infinite series
Were these topics helpful?See something inaccurate? Let us know!
Select the topics that are inaccurate.
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Preview not available
Let X = · b1b 2b 3 ⋯ be a random variable with independent binary digits bn taking values 0 or 1 with probability pn and qn = 1 - pn. When does X have a density? A continuous density? A singular distribution? This note gives necessary and sufficient conditions for the distribution of X to be: discrete: $\Sigma\min (p_n, q_n) < \infty$; singular: Σ∞ m[log (pn/qn)]2 = ∞ for every m; absolutely continuous: $\Sigma^\infty_m\lbrack\log (p_n/q_n)\rbrack^2 < \infty$ for some m. Furthermore, X has a density that is bounded away from zero on some interval if and only if log (pn/qn) is a geometric sequence with ratio 1/2 for $n > k$, and in that case the fractional part of 2k X has an exponential density (increasing or decreasing with the uniform a special case).
The Annals of Mathematical Statistics © 1971 Institute of Mathematical Statistics