  Let $d$ be a probability distribution. Under certain mild conditions we show
that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty
\frac{d^{*n}(x)}{n}=1,\qquad\text{where}\quad
d^{*n}:=\underbrace{\,d*d*\cdots*d\,}_{n\text{ times}}. $$ For a compactly
supported distribution $d$, we show that if $c>0$ is a given constant and the
function $f(k):=\widehat d(k)-1$ does not vanish on the line $\{k\in{\mathbb
C}:\Im\,k=-c\}$, where $\widehat d$ is the Fourier transform of $d$, then one
has the asymptotic expansion $$
\sum_{n=1}^\infty\frac{d^{*n}(x)}{n}=\frac{1}{x}\bigg(1+\sum_k m(k)
e^{-ikx}+O(e^{-c x})\bigg)\qquad (x\to +\infty), $$ where the sum is taken over
those zeros $k$ of $f$ that lie in the strip $\{k\in{\mathbb C}:-c<\Im\,k<0\}$,
$m(k)$ is the multiplicity of any such zero, and the implied constant depends
only on $c$. For a given distribution $d$ of this type, we briefly describe the
location of the zeros $k$ of $f$ in the lower half-plane $\{k\in{\mathbb
C}:\Im\,k<0\}$.
  For an odd prime $p$, let $n_0(p)$ be the least natural number such that
$(n|p)=-1$, where $(\cdot|p)$ is the Legendre symbol. As an application of our
work on probability distributions, in this paper we generalize a well known
result of Heath-Brown concerning the behavior of the Dirichlet $L$-function
$L(s,(\cdot|p))$ under the assumption that the Burgess bound $n_0(p)\ll
p^{1/(4\sqrt{e})+\epsilon}$ cannot be improved.
