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Characteristics of asymmetric switch processes with independent switching times

Henrik Bengtsson, Krzysztof Podgorski

TL;DR

This work analyzes asymmetric switch processes driven by independent switching times $T_+$ and $T_-$, introducing non-stationary and stationary versions and studying their moments, covariances, and transform-domain characteristics. A key contribution is the establishment of an equivalence between the monotonicity of the expected-value functions $E_+(t), E_-(t)$ and a geometric-sum stochastic representation of the switching times, enabling recovery of $T_+$ and $T_-$ from $E_+(t)$ and $E_-(t)$ via Laplace transforms. It also clarifies identifiability limitations when using the stationary covariance to infer the underlying switching-time distributions, with implications for the independent interval approximation in excursion analysis. The paper provides concrete constructions and examples (including geometric-divisible and gamma-switch cases) and outlines future work to broaden applicability by relaxing independence assumptions and tying results more directly to IIA practice.

Abstract

The asymmetric switch process is a binary stochastic process that alternates between the values one and minus one, where the distributions of the time in these states may differ. Two versions of the process are considered: a non-stationary version that starts with a switch at time zero and a stationary one constructed from the non-stationary one. Characteristics of these two processes, such as the expected values and covariance, are investigated. The main results show an equivalence between the monotonicity of the expected value functions and the distribution of the intervals having a stochastic representation in the form of a sum of random variables, where the number of terms follows a geometric distribution. This representation has a natural interpretation as a model in which switching attempts may fail at random. From these results, conditions are derived when these characteristics lead to valid interval distributions, which is vital in applications.

Characteristics of asymmetric switch processes with independent switching times

TL;DR

This work analyzes asymmetric switch processes driven by independent switching times and , introducing non-stationary and stationary versions and studying their moments, covariances, and transform-domain characteristics. A key contribution is the establishment of an equivalence between the monotonicity of the expected-value functions and a geometric-sum stochastic representation of the switching times, enabling recovery of and from and via Laplace transforms. It also clarifies identifiability limitations when using the stationary covariance to infer the underlying switching-time distributions, with implications for the independent interval approximation in excursion analysis. The paper provides concrete constructions and examples (including geometric-divisible and gamma-switch cases) and outlines future work to broaden applicability by relaxing independence assumptions and tying results more directly to IIA practice.

Abstract

The asymmetric switch process is a binary stochastic process that alternates between the values one and minus one, where the distributions of the time in these states may differ. Two versions of the process are considered: a non-stationary version that starts with a switch at time zero and a stationary one constructed from the non-stationary one. Characteristics of these two processes, such as the expected values and covariance, are investigated. The main results show an equivalence between the monotonicity of the expected value functions and the distribution of the intervals having a stochastic representation in the form of a sum of random variables, where the number of terms follows a geometric distribution. This representation has a natural interpretation as a model in which switching attempts may fail at random. From these results, conditions are derived when these characteristics lead to valid interval distributions, which is vital in applications.
Paper Structure (6 sections, 15 theorems, 72 equations, 2 figures)

This paper contains 6 sections, 15 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Switch process
  3. Characteristics of the switch process
  4. Examples
  5. Conclusion
  6. Acknowledgement

Key Result

Proposition 1

The Laplace transform of $P_\delta(t)=\text{\sf P}{\left(D(t)=1 | \delta\right)}$, $t>0$, is given by where $\Psi_+$ and $\Psi_-$ are Laplace transformation of probability distributions corresponding to $T_+$ and $T_-$, respectively. Moreover, for the expected value function $E_\delta(t)=\text{\sf E}{( D(t) |\delta)}$, $t>0$ we have

Figures (2)

  • Figure 1: A realization of a non-stationary switch process (blue), $D(t)$, with exponential switching times and $\delta=1$. Together with $E_+(t)$ (red) and $E_-(t)$ (orange).
  • Figure 2: A realization of a stationary switch process, $\tilde{D}$ with exponential switching times and $\delta=1$. $A$ and $B$ are the forward and backward delays.

Theorems & Definitions (29)

  • Proposition 1
  • Corollary 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Remark 2
  • Proposition 5
  • Definition 1
  • Theorem 1
  • ...and 19 more