A $\widetilde{Q}-$representation of real numbers is introduced as a generalization of the $p-$adic and $Q-$representations. It is shown that the $\widetilde{Q}-$representation may be used as a convenient tool for the construction and study of fractals and sets with complicated local structure. Distributions of random variables $ξ$ with independent $\widetilde{Q}-$symbols are studied in details. Necessary and sufficient conditions for the probability measures $μ_ξ$ associated with $ξ$ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the $\widetilde{Q}-$representation. In addition the metric-topological properties for the distribution of $ξ$ are investigated. A number of examples are presented.
Properties of the set $T_s$ of "particularly non-normal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose s-adic digits have the asymptotic frequencies in the nonterminating $s-$ adic expansion of $x$, and some do not). It is proven that the set $T_s$ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ($T_s$ is a superfractal set, i.e., its Hausdorff-Besicovitch dimension is equal to~1). A topological and fractal classification of sets of real numbers via analysi…
We give sharp two-sided estimates for the functions $g_M(t,x,y)$ and $g_M(t,x,y)-g(t,x,y)$, where $g_M(t,x,y)$ are the transition probability densities of the reflected Brownian motion on a $M$-complex of size $M \in \mathbb{Z}$ of an unbounded planar simple nested fractal and $g(t,x,y)$ are the transition probability densities of the `free' Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.
We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} ξ= \sum_{k=1}^\infty \frac{(-1)^{k+1}ξ_k}{q_k}, \end{equation} where $q_k$ is a sequence of positive integers with $q_{k+1}\geq q_k(q_k+1)$, and $\{ξ_k\}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $ξ$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_ξ)=0$) whose Fourier-Stieltjes transform does not tend to zero at infinity. …
Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p<p_c(M) and with positive probability there is a connected component intersecting opposite sides of the unit square for p\geq p_c(M). For all M\geq 2, the value of p_c(M) is unknown. In this paper we present ideas to find lower and upper bounds, significantly sharper than those alread…
We determine the decay rate of the bottom crossing probability for symmetric jump processes under the condition on heat kernel estimates. Our results are applicable to symmetric stable-like processes and stable-subordinated diffusion processes on a class of (unbounded) fractals and fractal-like spaces.
We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\ge 1$, $F_{n+2} = λF_{n+1} \pm F_{n}$ (linear case) and $\widetilde F_{n+2} = |λ\widetilde F_{n+1} \pm \widetilde F_{n}|$ (non-linear case), where each $\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\le 1$). Our main result is that, when $λ$ is of the form $λ_k = 2\cos (π/k)$ for some integer $k\ge 3$, the exponential growth of $F_n$ for $0<p\le 1$, and of $\widetilde F_{n}$ for $1/k < p\le 1$, is almost surely positive and given by $$ \int_0…
Let $\{B(t), t\ge 0\}$ be a Brownian motion. Consider the Brownian motion risk model with interest rate collection and tax payment defined by \begin{align}\label{Rudef} \widetilde{U}_γ^δ(t)=\widetilde{X}^δ(t)-γ\sup_{s\in[0,t]} \left(\widetilde{X}^δ(s)e^{δ(t-s)}-ue^{δ(t-s)}\right),t\ge0, \end{align} with $$\widetilde{X}^δ(t)=ue^{δt}+c \int_0^t e^{δ(t-v)}dv-σ\int_0^t e^{δ(t-v)}dB(v),$$ where $c>0,γ\in [0,1)$ and $δ\in \mathbb{R}$ are three given constants. When $δ=0$ and $γ\in (0,1)$ this is the risk model introduced from Albrecher and Hipp in \cite{AH2007} where the ruin probability in the infi…
We establish several new probabilistic, dynamical, dimensional and number theoretical phenomena connected with Ostrogradsky-Sierpiński-Pierce expansion. First of all, we develop metric, ergodic and dimensional theories of the Ostrogradsky-Sierpiński-Pierce expansion. In particular, it is proven that for Lebesgue almost all real numbers any digit $i$ from the alphabet $A= \mathbb{N} $ appears only finitely many times in the difference-version of the Ostrogradsky-Sierpiński-Pierce expansion. Properties of the symbolic dynamical system generated by a shift-transformation $T$ on the difference-ver…
Let $A$ be a limsup random fractal with indices $γ_1, ~γ_2 ~$and $δ$ on $[0,1]^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ with the condition $(\star)$$\colon$ $\dim_{\rm H}(G)>γ_2+δ$, where $\dim_{\rm H}$ denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan, Peres and Xiao [10] by relaxing the condition that the probability $P_n$ of choosing each dyadic hyper-cube is homogeneous and $\lim\limits_{n\to\infty}\frac{\log_2P_n}{n}$ exists. We also present some counterexamples to show the Hausdorff dimension in condition $(\st…
We analyze here different types of fractional differential equations, under the assumption that their fractional order $ν\in (0,1] $ is random\ with probability density $n(ν).$ We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process $N(t),t>0.$\ We prove that, for a particular (discrete) choice of $n(ν)$, it leads to a process with random time, defined as $N(% \widetilde{\mathcal{T}}_{ν_{1,}ν_{2}}(t)),t>0.$ The distribution of the random time argument $\widetilde{\mathcal{T}}_{ν_{1,}ν_{2}}(t)$ can be expressed, for any fixed $t$, in …
We study the probability distribution of the maximum $M_S $ of a smooth stationary Gaussian field defined on a fractal subset $S$ of $\R^n$. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution $¶(M_S>u)$ as $u\rightarrow +\infty.$ The basic tool is Rice formula for the moments of the number of local maxima of a random field.
In the trace reconstruction problem, an unknown bit string $x \in \{0,1\}^n$ is observed through the deletion channel, which deletes each bit of $x$ with some constant probability $q$, yielding a contracted string $\widetilde{x}$. How many independent copies of $\widetilde{x}$ are needed to reconstruct $x$ with high probability? Prior to this work, the best upper bound, due to Holenstein, Mitzenmacher, Panigrahy, and Wieder (2008), was $\exp(\widetilde{O}(n^{1/2}))$. We improve this bound to $\exp(O(n^{1/3}))$ using statistics of individual bits in the output and show that this bound is sharp …
We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{…
On the lattice $\widetilde{\mathbb Z}^2_+:={(x,y)\in \mathbb Z \times \mathbb Z_+\colon x+y \text{is even}}$ we consider the following oriented (northwest-northeast) site percolation: the lines $H_i:={(x,y)\in \widetilde {\mathbb Z}^2_+ \colon y=i}$ are first declared to be bad or good with probabilities $\de$ and $1-\de$ respectively, independently of each other. Given the configuration of lines, sites on good lines are open with probability $p_{_G}>p_c$, the critical probability for the standard oriented site percolation on $\mathbb Z_+ \times \mathbb Z_+$, and sites on bad lines are open wi…
In the trace reconstruction problem an unknown string ${\bf x}=(x_0,\dots,x_{n-1})\in\{0,1,...,m-1\}^n$ is observed through the deletion channel, which deletes each $x_k$ with a certain probability, yielding a contracted string $\widetilde{\bf X}$. Earlier works have proved that if each $x_k$ is deleted with the same probability $q\in[0,1)$, then $\exp(O(n^{1/3}))$ independent copies of the contracted string $\widetilde{\bf X}$ suffice to reconstruct $\bf x$ with high probability. We extend this upper bound to the setting where the deletion probabilities vary, assuming certain regularity condi…
We consider the second Ostrogradsky expansion from the number theory, probability theory, dynamical systems and fractal geometry points of view, and establish several new phenomena connected with this expansion. First of all we prove the singularity of the random second Ostrogradsky expansion. Secondly we study properties of the symbolic dynamical system generated by the natural one-sided shift-transformation $T$ on the second Ostrogradsky expansion. It is shown, in particular, that there are no probability measures which are simultaneously invariant and ergodic (w.r.t. $T$) and absolutely con…
Given a sequence $(M^n)^{\infty}_{n=1}$ of nonnegative martingales starting at $M^n_0=1$, we find a sequence of convex combinations $(\widetilde{M}^n)^{\infty}_{n=1}$ and a limiting process $X$ such that $(\widetilde{M}^n_τ)^{\infty}_{n=1}$ converges in probability to $X_τ$, for all finite stopping times $τ$. The limiting process $X$ then is an optional strong supermartingale. A counterexample reveals that the convergence in probability cannot be replaced by almost sure convergence in this statement. We also give similar convergence results for sequences of optional strong supermartingales $(X…
We study Mandelbrot's percolation process in dimension $d \geq 2$. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining or discarding each subcube with probability $p$ or $1-p$ respectively. This step is then repeated within the retained subcubes at all scales. As $p$ is varied, there is a percolation phase transition in terms of paths for all $d \geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$. For any $d \geq 2$, we consider the random fractal set produced at…
Attributes which are infrequently expressed in a population can require weeks or months of counting to reach statistical significance. But replacement in a stable population increases long-term counts to a degree determined by the probability distribution of lifetimes. If the lifetimes are in a Pareto distribution with shape factor $1-r$ between 0 and 1, then the expected counts for a stable population are proportional to time raised to the $r$ power. Thus $r$ is the fractal dimension of counts versus time for this population. Furthermore, the counts from a series of consecutive measurement in…
In the trace reconstruction problem, an unknown bit string ${\bf x}\in\{0,1 \}^n$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string $\widetilde{\bf x}$. How many i.i.d.\ samples of $\widetilde{\bf x}$ are needed to reconstruct $\bf x$ with high probability? We prove that there exist ${\bf x},{\bf y} \in\{0,1 \}^n$ such that at least $c\, n^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\bf x}$ and ${\bf y}$ for some absolute constant $c$, improving the previous lower bound of $c\,n$. Fur…
Doubly-intractable distributions appear naturally as posterior distributions in Bayesian inference frameworks whenever the likelihood contains a normalizing function $Z$. Having two such functions $Z$ and $\widetilde Z$ we provide estimates of the total variation and Wasserstein distance of the resulting posterior probability measures. As a consequence this leads to local Lipschitz continuity w.r.t. $Z$. In the more general framework of a random function $\widetilde Z$ we derive bounds on the expected total variation and expected Wasserstein distance. The applicability of the estimates is illu…
