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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
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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