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Random Processes with Stationary Increments and Intrinsic Random Functions on the Real Line

Jongwook Kim

TL;DR

The paper addresses nonstationary stochastic processes by unifying two prominent frameworks: random processes with stationary increments (I(d)) and intrinsic random functions (IRF(d)) on the real line. It proves an equivalence between IRF(d) and I(d) under mild spectral conditions, showing that the differencing operator corresponds to an allowable measure and that the I(d) structure function and IRF(d) intrinsic covariance carry the same second-order information. This unification enables cross-pollination of methods, allowing spectral techniques from time series to inform geostatistics and vice versa, and provides a rigorous basis for universal kriging within either framework. The practical impact includes improved modeling of nonstationary phenomena, robust covariance modeling, and coherent estimation across time and space domains.

Abstract

Random processes with stationary increments and intrinsic random processes are two concepts commonly used to deal with non-stationary random processes. They are broader classes than stationary random processes and conceptually closely related to each other. This paper illustrates the relationship between these two concepts of stochastic processes and shows that, under certain conditions, they are equivalent on the real line.

Random Processes with Stationary Increments and Intrinsic Random Functions on the Real Line

TL;DR

The paper addresses nonstationary stochastic processes by unifying two prominent frameworks: random processes with stationary increments (I(d)) and intrinsic random functions (IRF(d)) on the real line. It proves an equivalence between IRF(d) and I(d) under mild spectral conditions, showing that the differencing operator corresponds to an allowable measure and that the I(d) structure function and IRF(d) intrinsic covariance carry the same second-order information. This unification enables cross-pollination of methods, allowing spectral techniques from time series to inform geostatistics and vice versa, and provides a rigorous basis for universal kriging within either framework. The practical impact includes improved modeling of nonstationary phenomena, robust covariance modeling, and coherent estimation across time and space domains.

Abstract

Random processes with stationary increments and intrinsic random processes are two concepts commonly used to deal with non-stationary random processes. They are broader classes than stationary random processes and conceptually closely related to each other. This paper illustrates the relationship between these two concepts of stochastic processes and shows that, under certain conditions, they are equivalent on the real line.
Paper Structure (9 sections, 3 theorems, 42 equations)

This paper contains 9 sections, 3 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Background and Definitions
  3. Random Processes with Stationary Increments
  4. Intrinsic Random Functions on the Real line
  5. Connection between Random Processes with Stationary Increments and Intrinsic Random Functions on $\mathbb{R}$
  6. Applications and Discussions
  7. Brownian Motion
  8. Universal Kriging
  9. Conclusion

Key Result

Lemma 1

Define Suppose that $X(t)$ is an IRF$(d)$. Then, $\lambda_{\Delta_{\iota,t}^d} \in \Lambda_d$, and $X(\lambda_{\Delta_{\iota,t}^d}) = \Delta_{\iota}^d X(t)$. That is, the finite difference operator $\Delta_{\iota}^d$, defined in Definition def:random_process_stationary_increment, corresponds to an allowabl

Theorems & Definitions (17)

  • Definition 2.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2.2
  • Remark 4
  • Definition 2.3
  • Remark 5
  • Definition 2.4
  • Lemma 1
  • ...and 7 more