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Almost sure stability of some stochastic dynamical systems with memory
Infinite Bernoulli convolutions with different probabilities
1.  Institute of Mathematics, Budapest University of Technology and Economics, P.O.box 91, H1529, Budapest, Hungary 
$Y_{\lambda}^p$:=$\sum_{n=0}^{\infty}\pm \lambda ^n$
where the "$+$" and "$$" signs are chosen independently with
probability $p$ and $1p$. Let $\nu_\lambda^p$ be the distribution
of the random sum $\nu_\lambda^p(E)$:=$Prob(Y_{\lambda}^p \in
E)$ for every set $E$. The conjecture is that for every $p \in
(0,1)$ the measure $\nu_\lambda^p$ is absolutely continuous w.r.t.
Lebesgue measure and with the density in $L^2(R)$ for almost every
$\lambda\in (p^p\cdot(1p)^{(1p)},1).$
B. Solomyak and Y. Peres [3, [Corollary 1.4] proved that
for every $p \in (\frac{1}{3},\frac{2}{3})$ the distribution
$\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for
almost every $\lambda \in (p^2+(1p)^2,1).$ In this paper we
extend the parameter interval where a weakened version of the
conjecture still holds. Namely, we prove Corollary 3 that
for every $p \in (0,\frac{1}{3}]$ the measure
$\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for
almost every $\lambda\in(F(p),1)$, where
$F(p)=(12p)^{2\log41/\log 9}$, see Figure 3.
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