An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of jth-generation individuals with birth times
$\leq t$
, when
$j,t\to\infty$
and
$j(t)={\textrm{o}}\big(t^{2/3}\big)$
. According to our terminology, such generations form a subset of the set of intermediate generations.
A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.
Let (ξ k ,η k ) k∈N be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence (T k ) k∈N defined byFurther, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For j ∈ N and t ≥ 0, denote by N j (t) the number of the jth generation individuals with birth times ≤ t. In this article we prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell's theorem and the key renewal theorem) for N j (t) under the assumption that j = j(t) → ∞ and j(t) = o(t 2/3 ) as t → ∞. According to our terminology, such generations form a subset of the set of intermediate generations.
We prove a functional limit theorem for the number of visits by a planar random walk on
Z
2
\mathbb {Z}^2
with zero mean and finite second moment to the points of a fixed finite set
P
⊂
Z
2
P\subset \mathbb {Z}^2
. The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.
Probabilistic properties of vantage point trees are studied. A vp-tree built from a sequence of independent identically distributed points in [−1, 1] d with the ∞ -distance function is considered. The length of the leftmost path in the tree, as well as partitions over the space it produces are analyzed. The results include several convergence theorems regarding these characteristics, as the number of nodes in the tree tends to infinity.
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