Olbers' Paradox, Wireless Telephones, and Poisson Random Sets: Is the Universe Finite?

Olbers' paradox is that if the universe is either infinite in age or extent, then the night sky should be bright. A related problem exists in terms of gravity, that an infinite universe full of stars should collapse on itself. Olbers' paradox has been used to support the "big bang" hypothesis. We find there is a simpler resolution of Olbers' paradox, that perhaps ought to be considered. We show that a standard theorem on convergence of infinite series of zero-mean independent random variables, due to Kolmogorov, can be viewed as saying that the sum of all the above contributions, allowing destructive interference, produces a finite sum even if the universe is infinite in age and extent. Thus our explanation of Olbers' paradox, is that if one thinks in terms of waves rather than of particles, the paradox disappears. A new model for interference noise in wireless telephony has been given in the thesis of one of us⁷ which is closely related to Chandrasekhar's model for stellar gravity. If we assume that the telephones which are sending signals to the base station at the origin are at the points of a planar Poisson random set, and that the n<underline>th</underline> most distant telephone from the origin is sending a signal X_n, then the total signal at any time received at the base station is [ T = sumlimits_{n = 1}^{∞} {{X_n}/{R_n^γ}} ] where R_n is the distance, 0 < R₁ < R₂ < …. The exponent, γ, is taken as 3.9 or 4 in engineering practice. In cosmology the situation is similar but the Poisson points are in three dimensions rather than two and represent the stars or the galaxies, and T now represents either the total gravitational force on the earth or the total radiation at a frequency. In cosmology, γ is usually taken as 2 corresponding to an inverse square loss. We will show first that under very general conditions the series for T is perfectly well convergent mathematically in both cases even if the sum extends over infinitely many summands. This seems to say that Olbers' paradox has an alternative explanation, namely that even if there are infinitely many sources of gravity and radiation the total contribution on earth has a specific finite value due to cancellation of the terms (interference). The sum of the squares of the terms above also converges although the series does not necessarily converge absolutely unless γ is large enough (one would need γ > 3) in 3 dimensions. Remarkably, (a special case is due to Chandrasekhar) the distribution of T is always stable and there is only a two-parameter family of stable distributions, S(α, β), that T can possibly have. Not all stable distributions appear since 0 < α < 2 so, for example, the normal distribution is not among them.