Random series

March 2, 2013

A little while ago, a question was posted on MathOverflow on why lacunary series are “badly behaved”, in the sense that they their circle of convergence is their natural boundary. As the answers indicate, it is actually the opposite: This behavior is typical, in the sense that a random power series will have this property. I posted a short note pointing this out as an answer to a related question on Math.StackExchange. Here it is, with very minor edits:

Consider $\sum_n X_n z^n$, where the $X_n=X_n(\omega)$ are independent random (complex) variables and $z$ is complex.

First of all, the radius of convergence of the series (at a given $\omega$ in the underlying measure space) is $r(\omega)=(\limsup_{n\to\infty}|X_n(\omega)|^{1/n})^{-1}.$

Note that $r$ is a measurable function of $\omega$, and its value does not depend on the values of a finite number of the $X_n$. Kolmogorov’s zero-one law then gives us that $r(\omega)$ is a constant, say $R$, almost surely. This $R$ lies in ${}[0,\infty]$, and both $0$ and $+\infty$ are possible values, depending on the distribution of the $X_n$, though the most interesting cases to study are perhaps when $0.

There is a nice book that presents the relevant theory:

Jean-Pierre Kahane. Some random series of functions. Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. MR0833073 (87m:60119).

(“Random Taylor series” is Chapter 4. What follows is based on Kahane’s presentation. Kahane’s book includes proofs of all the results below.)

Consideration of random series seems to have been first suggested by Borel, in

Emile Borel. Sur les séries de Taylor. C. r. hebd. Séanc. Acad. Sci., Paris 123, (1896), 1051-2.

(The journal is available here.)

Part of the problem was that at the time the concepts of probability theory were not quite formalized yet, so going from Borel’s remarks to actual theorems took some time.  Borel wrote

Si les coefficients sont quelconques, le cercle de convergence est une coupure.

What Borel is saying is that if the coefficients of a series $\sum_n X_n z^n$ are “arbitrary”, then the circle of convergence is a natural boundary for the function. What this means is that there is no way to extend $F(z)=\sum_n X_n z^n$ analytically beyond the circle of convergence (because the singular points are dense on the boundary).

The first actual result in this regard is due to Steinhaus in 1929: If the $r_n$ are positive, and $0<\limsup_n r_n^{1/n}<\infty$, and the $\omega_n$ are independent random variables equidistributed on ${}[0,1]$, then $\sum_n r_ne^{2\pi i\omega_n}z^n$ has the circle of convergence as natural boundary, almost surely. A different formalization was found later by Paley and Zygmund, in 1932, in terms of Rademacher sequences.

On the other hand, Borel’s statement cannot quite be translated as “the coefficients are independent random variables”. Kahane’s example is the series $\sum_n (2^n\pm1)z^n,$

which has radius of convergence $1/2$, and $1/2$ is the only singular point on the circle of convergence.

Kahane mentions a conjecture of Blackwell that the general situation should be that one of the two scenarios above applies: Either

1. $F(z)=\sum_n X_n z^n$ has the circle of convergence as natural boundary, or
2. There is a series $\sum_n c_n z^n$ (the $c_n$ being constants, not random variables; Kahane calls it a sure series) that added to $F$ results on a (random) Taylor series with a strictly larger circle of convergence which is its natural boundary.

The conjecture was proved in 1953 by Ryll-Nardzewski, see

Czesław Ryll-Nardzewski. D. Blackwell’s conjecture on power series with random coefficients. Studia Math. 13, (1953). 30–36. MR0054882 (14,994e).

Kahane also wrote a nice survey of these and related matters, in

Jean-Pierre Kahane. A century of interplay between Taylor series, Fourier series and Brownian motion. Bull. London Math. Soc. 29 (3), (1997), 257–279. MR1435557 (98a:01015).