Renewal Theorem For Random Walks Research Articles

Fourier transform of self-affine measures

Suppose F is a self-affine set on Rd, d≥2, which is not a singleton, associated to affine contractions fj=Aj+bj, Aj∈GL(d,R), bj∈Rd, j∈A, for some finite A. We prove that if the group Γ generated by the matrices Aj, j∈A, forms a proximal and totally irreducible subgroup of GL(d,R), then any self-affine measure μ=∑pjfjμ, ∑pj=1, 0<pj<1, j∈A, on F is a Rajchman measure: the Fourier transform μˆ(ξ)→0 as |ξ|→∞. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of Γ is connected real split Lie group in the Zariski topology, then μˆ(ξ) has a power decay at infinity. Hence μ is Lp improving for all 1<p<∞ and F has positive Fourier dimension. In dimension d=2,3 the irreducibility of Γ and non-compactness of the image of Γ in PGL(d,R) is enough for power decay of μˆ. The proof is based on quantitative renewal theorems for random walks on the sphere Sd−1.

Basic renewal theorems for random walks with widely dependent increments

In this paper, we derive some basic renewal theorems for random walks with widely dependent increments, which contain some common negatively dependent random variables (r.v.s.), some positively dependent r.v.s. and some others. For this purpose, we investigate uniform integrability for related counting processes and the strong law of large numbers for the widely dependent r.v.s.

A Renewal Theorem for Random Walks in Multidimensional Time

Let $X, {X_1}, {X_2}, \ldots$ be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let ${S_n} = {X_1} + {X_2} + \cdots + {X_n}$. The asymptotic behavior of the weighted sum $R(k) = \sum {a_n}P({S_n} = k)$, with summation over $n \geq 1$, is investigated as $k \to + \infty$. In the special case ${a_n} = {d_r}(n)$, the number of solutions of the equation $n = {n_1}{n_2} \cdots {n_r}$ in positive integers ${n_j}, 1 \leq j \leq r, R(k)$ becomes the renewal function $Q(k)$ for a random walk in $r$-dimensional time whose terms are distributed as $X$. Under some assumptions on the magnitude of ${a_n}$ and of $A(x) = \sum \nolimits _{n \leq x} {{a_n}}$, (i) it is shown that $R(k)$ is asymptotically distribution free as $k \to + \infty$, (ii) the proper order of magnitude of $R(k)$ is determined, and under some further restrictions on $A(x)$, (iii) a simple asymptotic formula is given for $R(k)$. From (i), the known asymptotic formula for $Q(k)$ with $r = 2$ or 3 is deduced under the sole assumption of finite variance. The relaxation of previous moment assumptions requires a new inequality for the sum of the divisor function ${d_r}(n), 1 \leq n \leq x$, which by itself is of interest.

A renewal theorem for random walks in multidimensional time

Let $X, {X_1}, {X_2}, \ldots$ be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let ${S_n} = {X_1} + {X_2} + \cdots + {X_n}$. The asymptotic behavior of the weighted sum $R(k) = \sum {a_n}P({S_n} = k)$, with summation over $n \geq 1$, is investigated as $k \to + \infty$. In the special case ${a_n} = {d_r}(n)$, the number of solutions of the equation $n = {n_1}{n_2} \cdots {n_r}$ in positive integers ${n_j}, 1 \leq j \leq r, R(k)$ becomes the renewal function $Q(k)$ for a random walk in $r$-dimensional time whose terms are distributed as $X$. Under some assumptions on the magnitude of ${a_n}$ and of $A(x) = \sum \nolimits _{n \leq x} {{a_n}}$, (i) it is shown that $R(k)$ is asymptotically distribution free as $k \to + \infty$, (ii) the proper order of magnitude of $R(k)$ is determined, and under some further restrictions on $A(x)$, (iii) a simple asymptotic formula is given for $R(k)$. From (i), the known asymptotic formula for $Q(k)$ with $r = 2$ or 3 is deduced under the sole assumption of finite variance. The relaxation of previous moment assumptions requires a new inequality for the sum of the divisor function ${d_r}(n), 1 \leq n \leq x$, which by itself is of interest.

Some renewal theorems for random walks in multidimensional time

Let Kr denote the set of r-tuples n = (n1n2, …, nr) with positive integers as coordinates (r ≥ 1) and {X, Xn, n ε Kr} be a family of independent, identically distributed random variables with positive mean 0 &lt; EX ≡ μ &lt; ∞ and finite positive variance 0 &lt; var X ≡ σ2 ∞. The notation m ≤ n, where m = (mi) and n = (ni), means that mi ≤ ni (i = 1, 2,…, r) and |n| = n1n2 … nr. Denote Sn =Σj ≤ nXj (j, n ε Kr). When r = 1, {Xn, n ε Kr) reduces to the sequence {Xj, j ε 1} of independent random variables each distributed as X, and Sn becomes the ordinary partial sum .

Marches al�atoires sur un groupe nilpotent

Under the assumption that there exist moments of order four we give a proof of the renewal theorem for random walks on connected compactly generated nilpotent groups. In three dimension, we get a bound for the potential kernel.