Hitting probabilities of constrained random walks representing tandem networks
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arXiv
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| Natura: | Preprint |
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2021
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| _version_ | 1866918308548182016 |
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| author | Sezer, Ali Devin |
| author_facet | Sezer, Ali Devin |
| contents | Let $X$ be the constrained random walk on $\mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $λ$, $μ_1$, $μ_2$,...,$μ_d$, where $\{e_1,e_2,..,e_d\}$ are the standard basis vectors. The process $X$ is assumed stable, i.e., $λ< μ_i$ for all $i=1,2,3,...,d.$ Let $τ_n$ be the first time the sum of the components of $X$ equals $n$. We derive approximation formulas for the probability ${\mathbb P}_x(τ_n < τ_0)$. For $x \in \bigcup_{i=1}^d \Big\{x \in {\mathbb R}^d_+: \sum_{j=1}^{i} x(j)$ $> \left(1 - \frac{\log λ/\min μ_i}{\log λ/μ_i}\right) \Big\}$ and a sequence of initial points $x_n/n \rightarrow x$ we show that the relative error of the approximation decays exponentially in $n$. The approximation formula is of the form ${\mathbb P}_y(τ< \infty)$ where $τ$ is the first time the sum of the components of a limit process $Y$ is $0$; $Y$ is the process $X$ as observed from a point on the exit boundary except that it is unconstrained in its first component (in particular $Y$ is an unstable process); $Y$ and ${\mathbb P}_y(τ< \infty)$ arise naturally as the limit of an affine transformation of $X$ and the probability ${\mathbb P}_x(τ_n < τ_0).$ The analysis of the relative error is based on a new construction of supermartingales. We derive an explicit formula for ${\mathbb P}_y(τ< \infty)$ in terms of the ratios $λ/μ_i$ which is based on the concepts of harmonic systems and their solutions and conjugate points on a characteristic surface associated with the process $Y$; the derivation of the formula assumes $μ_i \neq μ_j$ for $i\neq j.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_05474 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hitting probabilities of constrained random walks representing tandem networks Sezer, Ali Devin Probability 60G50, 60G40, 60F10, 60J45 Let $X$ be the constrained random walk on $\mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $λ$, $μ_1$, $μ_2$,...,$μ_d$, where $\{e_1,e_2,..,e_d\}$ are the standard basis vectors. The process $X$ is assumed stable, i.e., $λ< μ_i$ for all $i=1,2,3,...,d.$ Let $τ_n$ be the first time the sum of the components of $X$ equals $n$. We derive approximation formulas for the probability ${\mathbb P}_x(τ_n < τ_0)$. For $x \in \bigcup_{i=1}^d \Big\{x \in {\mathbb R}^d_+: \sum_{j=1}^{i} x(j)$ $> \left(1 - \frac{\log λ/\min μ_i}{\log λ/μ_i}\right) \Big\}$ and a sequence of initial points $x_n/n \rightarrow x$ we show that the relative error of the approximation decays exponentially in $n$. The approximation formula is of the form ${\mathbb P}_y(τ< \infty)$ where $τ$ is the first time the sum of the components of a limit process $Y$ is $0$; $Y$ is the process $X$ as observed from a point on the exit boundary except that it is unconstrained in its first component (in particular $Y$ is an unstable process); $Y$ and ${\mathbb P}_y(τ< \infty)$ arise naturally as the limit of an affine transformation of $X$ and the probability ${\mathbb P}_x(τ_n < τ_0).$ The analysis of the relative error is based on a new construction of supermartingales. We derive an explicit formula for ${\mathbb P}_y(τ< \infty)$ in terms of the ratios $λ/μ_i$ which is based on the concepts of harmonic systems and their solutions and conjugate points on a characteristic surface associated with the process $Y$; the derivation of the formula assumes $μ_i \neq μ_j$ for $i\neq j.$ |
| title | Hitting probabilities of constrained random walks representing tandem networks |
| topic | Probability 60G50, 60G40, 60F10, 60J45 |
| url | https://arxiv.org/abs/2105.05474 |