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Source. The Theorem, stated pp. 343--344 (section 2), proof pp. 344--346 (section 3), of A. Dvoretzky and P. Erdős, Divergence of random power series, Michigan Math. J. 6 (1959), 343--347, the edition named on the source card.

Read depth. Claims checked: the statement and the setting were read clause by clause on the page images of pp. 343--344. The proof was read in outline only; its estimates were not checked. Nothing here is independently reviewed.

Statement

Setting (p. 343). $\phi_n(t)$ ($n=0,1,2,\dots$) are the Rademacher functions: $\phi_n(t)=(-1)^j$ for $j/2^n\le t<(j+1)/2^n$, $j=0,1,\dots,2^n-1$. For a sequence of complex numbers $\{a_n\}=\{a_0,a_1,a_2,\dots\}$, $\mathscr F\{a_n\}$ is the family of power series

$$P(z;t)=\sum_{n=0}^{\infty}\phi_n(t)\,a_n z^n\qquad(0\le t<1).$$

"Almost all" refers to Lebesgue measure in $t$; "everywhere" means at every point $z$ of the circle $|z|=1$.

Theorem (pp. 343--344). Let $\{c_n\}_{n=0}^{\infty}$ be a monotone sequence of positive numbers tending to zero such that

$$\limsup_{n\to\infty}\frac{\sum_{j=0}^{n}c_j^2}{\log 1/c_n}>0 .$$

If $\{a_n\}_{0}^{\infty}$ is a sequence of complex numbers with $|a_n|\ge c_n$ for all $n$, then almost all series of $\mathscr F\{a_n\}$ diverge everywhere on $|z|=1$.

The paper adds (p. 344) that its proof gives the statement with "diverge" strengthened to "have unbounded partial sums", and that for any sequence $\{a_n\}$ of nonzero complex numbers the sequence $c_n=\min_{0\le k\le n}|a_k|$ is monotone and satisfies $|a_n|\ge c_n$, so only the limsup condition has to be checked. The classical fact it strengthens (p. 343) is that $\sum|a_n|^2=\infty$ makes almost all series of $\mathscr F\{a_n\}$ diverge at almost every point of $|z|=1$.

Proof pointer

Section 3, pp. 344--346, written here in outline. One may assume $a_n\to0$ and that the limsup exceeds $8$. The indices are cut into blocks $(n_{k-1},n_k]$ on which $\sum|a_j|^2>8\log 1/c_{n_k}$, and each block into $\gamma_k>4\log 1/c_{n_k}$ short runs with $\sum|a_j|^2$ between $1$ and $2$. At a fixed point $z_0$, Kolmogorov's inequalities bound the chance that a run's partial sums all stay small, independence across runs makes that chance at most $e^{-\gamma_k}$ for the whole block, and a net of $\lambda_k$ points on the circle (with $\lambda_k$ of order $c_{n_k}^{-3}$) passes from points to the whole circle. Since $\lambda_k e^{-\gamma_k}\to0$, almost every $t$ has infinitely many $k$ such that at every point $z$ of $|z|=1$ some sum $\sum_{j=\alpha}^{\beta}\phi_j(t)a_jz^j$ over a stretch of block $k$ with $\beta-\alpha<2/c_{n_k}^2$ exceeds $1/(2e)$ in modulus.

Dependencies

Kolmogorov's inequalities, cited from M. Loève, Probability theory (New York, 1955), p. 235. The Corollary is presented on p. 344 as a special case.

Bears on

• Problem 527: the problem asks whether, for real $a_n$ with $\sum|a_n|^2=\infty$ and $|a_n|=o(1/\sqrt n)$, almost every choice of signs gives a series that converges at some point of $|z|=1$. The Theorem gives a sufficient condition on the coefficient sizes for the opposite outcome, divergence at every point of the circle for almost all sign choices. The paper does not pose the problem.