Modern Stochastics: Theory and Applications

Self-similar processes are crucial for modeling complex systems that exhibit scale-invariant or fractal-like behavior. These processes have been studied extensively due to their theoretical significance and practical applications in various fields, including network traffic modeling, financial time series analysis, geophysics, and hydrology. Significant contributions to this area include the works of Mandelbrot and Van Ness [14], Samorodnitsky and Taqqu [19], and Embrechts and Maejima [9]. A key feature of self-similar processes is their capacity to capture long-range dependence, where correlations persist over large time intervals.

On the other hand, processes exhibiting family-wise scaling are vital in modern stochastic modeling because they offer a more flexible and realistic framework for systems where scaling behavior isn’t simple and fixed. While traditional self-similarity describes a system that looks statistically the same at all scales, family-wise scaling accounts for a deeper, more complex relationship where the very nature of the process changes when it is scaled, all while remaining part of the same “family” of processes. Jørgensen, Martínez and Demétrio [12] introduced a framework for self-similarity and the Lamperti convergence in one-parameter families of processes. Also, Borgnat, Amblard and Flandrin [5] discussed scale invariances and Lamperti transformations as tools for understanding how time rescaling interacts with parameter changes. These perspectives justify distinguishing “family-wise scaling” from the standard notion of self-similarity of a fixed-parameter process. Processes exhibiting family-wise scaling have importance and practical benefits. From a theoretical point, studying processes with family-wise scaling expands the scope of stochastic analysis beyond the well-understood self-similar processes like fractional Brownian motion. It allows researchers to develop new mathematical tools and theories to understand a broader range of complex, nonstandard behaviors. This contributes to the development of more sophisticated models that can handle a wider array of real-world data. For instance, in financial markets, volatility and other statistical properties of asset returns can change over different time scales. A traditional self-similar model might miss these nuances. A family-wise scaling model, however, can capture this by allowing a parameter (e.g., related to volatility) to change with the time scale, providing a more accurate and robust model for risk assessment. These processes enable us to deal with non-Gaussian phenomena. For example, many natural and human processes, such as network data traffic and seismic activity, exhibit sudden jumps or shocks that cannot be modeled by Gaussian (natural) processes.

Long-memory and scaling phenomena play a central role in stochastic processes and time series. Classical models often rely on self-similar processes and their stationary increments, such as fractional Brownian motion and fractional Gaussian noise. Extensions driven by Poisson-type or Lévy-type noises provide a richer framework for modeling discontinuities and heavy-tailed behavior.

Stationary long-range dependent (LRD; this abbreviation will also stand for ‘long-range dependence’) series can be broadly classified into three main types, each with distinct mechanisms for LRD, jumps, and scaling. Lévy-based series, including fractional Lévy noise and Lévy-driven moving averages, exhibit LRD via slowly decaying kernel correlations, with jumps intrinsic to the Lévy process, and show approximate scaling: $X(ct)\sim {c^{H}}X(t)$ for large c [6, 7]. Time-changed Gaussian processes generate LRD through memory in the stochastic time-change $\tau (t)$; if $\tau (t)$ is self-similar with index ν, then $X(ct)\stackrel{d}{=}{c^{\nu /2}}X(t)$ [8]. Counting-based Poisson-type series, such as fractional Poisson and mixed Poisson models, produce LRD through heavy-tailed interarrival times or correlated random intensity, with jumps intrinsic to the counting mechanism, but generally lack scaling except in certain asymptotic regimes [2, 13, 15].

In this paper, we introduce a family-type scaling process, where scaling the time variable by a factor $a\gt 0$ produces a scaled version of another process from the same family, rather than the same process. That is, the process parameters themselves are affected by scaling, distinguishing this construction from classical self-similar processes. Based on this process, we construct, under certain conditions, a stationary time series exhibiting long-range dependence. This framework extends classical results on a fractional Gaussian noise and Lévy or Poisson type series, providing a versatile tool for spectral analysis of one-sided long-memory processes.

The paper is organized as follows. In Section 1, we review the concepts of self-similarity and long-range dependence, followed by an overview of classical models such as fractional Gaussian noise (fGn). We then introduce the notion of pseudo-Poisson type processe, which serve as the driving process in our construction. In Section 2, we introduce our approach to family-type scaling processes, and then present the construction of such a process as the integral of a pseudo-Poisson type process. The Gamma distribution of the random intensity in the underlying Poisson process provides a natural mechanism for heavy-tailed behavior and significantly influences the statistical properties of the model. We refer to this process as fractional PSI-process. Its covariance structure, as given in Proposition 1, reflects long-range dependence with a power-law decay governed by the exponent parameter $H\in \left(\frac{1}{2},1\right)$. This implies persistent correlations over long time intervals, making the model suitable for capturing memory effects observed in various real-world phenomena such as network traffic, financial time series, and environmental data.

The fractional PSI-process ${\Psi _{\theta }}(t),t\in {\mathbb{R}_{+}}$, inherits several important characteristics from both the underlying fractional Brownian motion and the pseudo-Poisson time change. In Section 3, we study a time series defined on ${\mathbb{Z}_{+}}$, constructed from the stationary increments of a fractional PSI-process. The main contribution of this paper is to establish the long-memory behavior of the resulting stationary time series. Specifically, we show that its autocovariance function exhibits a power-law decay of the form where $L(h)$ is a slowly varying function at infinity. Also, we prove the functional central limit theorem for LRD series ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$ in the Skorokhod space $\mathcal{D}[0,1]$. Finally, in Section 4, we derive the spectral density and discuss its implications for the long-memory behavior. Although a stationary time series defined on ${\mathbb{Z}_{+}}$ is not strictly causal, its one-sided domain allows for the application of the unilateral Fourier transform to derive its spectral density. Nevertheless, the lack of symmetry and causality introduces additional technical challenges in both the spectral representation and its interpretation. By using distributional Fourier analysis, we show that the resulting spectral density is, in general, a complex-valued and nonsymmetric function. As a result, the corresponding spectral measure is not classical but must be understood in the sense of a generalized spectral measure, specifically, as a tempered distribution.

In this section, we will briefly review key concepts and results that are essential for our construction and analysis. All concepts and definitions presented here can be found in [16]. If a different source is referenced, it will be indicated explicitly.

Typically, one selects $k=1$ when examining the properties of the stationary time series ${Y_{n}}=X(n+1)-X(n)$.

As is known, the most famous self-similar process with stationary increments is a fractional Brownian motion (fBm). The fBm, denoted by ${B_{H}}={\left\{{B_{H}}(t)\right\}_{t\in \mathbb{R}}}$, is Gaussian SSSI process with Hurst parameter $H\in (0,1)$, zero mean and covariance function, for $s,t\in \mathbb{R}$, given by

A stochastic process ${\left\{X(t)\right\}_{t\in T}}$ is called weakly(or second-order) stationary if for all $s,t\in T$, where $T=\mathbb{R}$ or $\mathbb{Z}$.

The autocovariance function of a weakly stationary time series ${\left\{{X_{n}}\right\}_{n\in \mathbb{Z}}}$ is defined as By weak stationarity, ${\gamma _{X}}(h)$ depends only on the lag h. The autocovariance function serves as a fundamental measure of dependence in time series analysis and it has a symmetry property, i.e., ${\gamma _{X}}(h)={\gamma _{X}}(-h),\hspace{0.1667em}h\in \mathbb{Z}$.

The spectral density function is a fundamental tool in analyzing weakly stationary time series, providing a frequency-domain perspective of their dependence structure. It is especially important for detecting and characterizing long-range dependence, as it reveals how the variance of the series is distributed over low-frequency components. This insight helps identify persistent correlations over long time horizons, which are crucial for accurate modeling and forecasting in many fields of application.

The parameter $d\in (0,1/2)$ is called a long-range dependence parameter.

Thus, although the four conditions capture different perspectives – covariance decay, spectral density, variance of partial sums, and nonsummability – they are tied together through a network of implications, providing a unified but flexible framework for analyzing long-memory behavior. This is shown in Table 1.

The fractional Gaussian noise is a zero-mean stationary process with variance and the autocovariance function given by According to [16, 20], the spectral density of fGn is given by where ${c_{H}}=\frac{\mathbb{E}[{B_{H}}{(1)^{2}}]}{2\pi }\Gamma (2H+1)\sin (\pi H)$.

Moreover, the spectral density exhibits a power-law divergence near the origin and behaves asymptotically as

In his foundational monograph [11], Feller introduced the concept of a pseudo-Poisson process, through what he called Poissonian randomization of time. This approach involves subordinating the index set of a sequence of random variables ${({\xi _{n}})_{n\ge 0}}$ using a Poisson process, typically assumed to be independent of the underlying sequence, which itself exhibits the Markov property.

In this paper, we consider a generalization of this construction in which the index time of the sequence ${({\xi _{n}})_{n\ge 0}}$ is subordinated by a Poisson process ${\Pi _{\lambda (\omega )}}$ with a random intensity $\lambda (\omega )$. This type of model has been studied by Rusakov, Yakubovich and Baev [17], as well as by Rusakov, Yakubovich, and Laskin [18].

Note that the standardized Poisson process with rate $\lambda (\omega )$ can be obtained by a random time-scaling of the unit-rate process:

Throughout this work, we assume that ${({\xi _{n}})_{n\ge 0}}$ is a sequence of independent and identically distributed (i.i.d.) random variables with zero mean and finite variance $\mathbb{E}{\xi _{0}^{2}}={\sigma ^{2}}\lt \infty $ and that $\lambda (\omega )$ is a positive random variable. As shown in [17, 18], the mean and covariance function of the PSI-process are given by where ${L_{\lambda }}(\cdot )$ denotes the Laplace transform of the random variable $\lambda (\omega )$.

For simplicity, we denote $\lambda (\omega )$ by λ.

In this paper, we introduce a new class of stochastic models that extends the classical notion of self-similarity, referred to as family-wise scaling processes. Up to now, there has been no rigorous or unified definition for processes whose scaling property involves not only time rescaling but also a simultaneous rescaling of an additional parameter. The existing literature mainly provides precise definitions for self-similar processes (characterized by a single Hurst exponent). However, none of these frameworks captures the family-wise relation that links the dynamics of the process with the simultaneous transformation of its family parameter. Therefore, the definition we introduce here provides the formulation of such a family-wise scaling processes.

Instead of the classical self-similarity condition, here the scaling also acts on the parameter θ.

In this section, we introduce a stochastic model that combines two important features: family-wise scaling and randomness in the time index through a pseudo-Poisson-type process. This model introduced by Azzo and Rusakov in [1] is defined by where $\theta \in (0,\infty )$ and $\kappa \in (0,1)$.

The process ${\Psi _{\theta }}(t)$ can be interpreted as a randomly time-changed integral of a scaled discrete-time noise sequence ${\{{\xi _{n}}\}_{n\ge 0}}$, where the time distortion is governed by a Poisson process ${\Pi _{\theta \lambda }}(s)$ with random intensity λ, scaled by the parameter θ. The parameter $\theta \gt 0$ controls the degree of aggregation or dispersion in the time deformation, while $\kappa \in (0,1)$ determines the scaling behavior of the integrand. Moreover, since the Poisson process and the noise sequence are independent, the process ${\Psi _{\theta }}(t)$ is well defined and integrable under suitable assumptions on ${\xi _{0}}$ and λ.

This model can be applied to the analysis of stochastic systems in which random events generate workload or transmit information over time. Examples include telecommunications networks, financial systems, and queueing models. The process reflects the total load that is the accumulated effect or quantity of events occurring within a specified period – offering important insights into system efficiency, resource allocation, and risk evaluation.

Moreover, the stochastic process defined in (11), indexed by parameter θ, exhibits a family-wise scaling property with exponent parameter $H\in (\frac{1}{2},1)$, as opposed to the classical H-self-similarity. Specifically, a linear scaling of time by a factor a corresponds, in the distributional sense, to a deterministic rescaling of another process from the same family, indexed by $a\theta $, i.e., for any $a\gt 0$, This property can be verified through a straightforward calculation. where the scaling exponent $H=1-\frac{\kappa }{2}\in \left(\frac{1}{2},1\right)$.

We are interested in analyzing the covariance structure of the process defined in (11), under the assumption that the random intensity λ of the Poisson process follows a gamma distribution with unit scale and shape parameter $\kappa \in (0,1)$.

Denote the covariance function of ${\Psi _{\theta }}(t)$ by ${K_{{\Psi _{\theta }}}}(s,t):=\mathrm{Cov}\left({\Psi _{\theta }}(s),{\Psi _{\theta }}(t)\right)$.

Observe that for $H\in \left(\frac{1}{2},1\right)$, the constant factor $\frac{{\sigma ^{2}}}{2H(2H-1)}$ is strictly positive. Consequently, we refer to the process ${\Psi _{\theta }}(t),\hspace{3.33333pt}t\in {\mathbb{R}_{+}}$, defined in (11) and satisfying the assumptions of Proposition 1, as the fractional PSI-process with exponent parameter $H=1-\frac{\kappa }{2}\in \left(\frac{1}{2},1\right)$.

The following corollary follows directly from the stationary increments property of the fractional PSI-process and highlights that the increment process is weakly stationary, with constant variance and an autocovariance function depending solely on the time lag.

The construction of the fractional PSI-process captures essential features of long-range dependent systems by incorporating both a nontrivial time structure and a non-Gaussian or Lévy source of randomness. Since the fractional PSI-process ${\Psi _{\theta }}(t),t\in {\mathbb{R}_{+}}$, has stationary increments, it is well suited to the analysis of LRD phenomena on ${\mathbb{Z}_{+}}$. In practice, many empirical time series (e.g., in network traffic, finance, and climate data) are naturally indexed by nonnegative time. For instance, fractional ARIMA processes, which are classical examples of LRD models, are typically defined on ${\mathbb{Z}_{+}}$. Accordingly, we define the stationary increment series as The resulting stationary time series ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$ has key properties, as detailed below.

Figure 1 illustrates that when θ is sufficiently large, the effect of $m=\frac{1}{\theta }$ in the autocovariance of the series derived by the fractional PSI-process becomes negligible, and it matches the fGn autocovariance. Consequently, for large θ the studied series behaves very similarly to fGn, with $\mathbb{E}[{B_{H}}{(1)^{2}}]=\frac{{\sigma ^{2}}}{H(2H-1)}$, regarding long-range correlations.

For the following result, we are interested in verifying that the stationary series ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$ satisfies condition III of Definition 4. From 3, we have where Note that ${L_{3}}(N)$, defined in (25), is a slowly varying function at infinity. To analyze this, define Let $x=\frac{h}{N}$, so that $h=Nx$. Then Since $2H-2\lt 0$, as $N\to \infty $ we have ${\left(\frac{m}{N}\right)^{2H-2}}\to 0$ and ${\left(x+\frac{m}{N}\right)^{2H-2}}\to {x^{2H-2}}\hspace{1em}\text{for}\hspace{3.33333pt}x\gt 0$. So Thus which is a constant as $N\to \infty $. Thus the function ${L_{3}}(N)$, defined by (25), is slowly varying at infinity for $H(\frac{1}{2},1)$.

Then the stationary series ${\{{X_{n}}\}_{n\in {\mathbb{Z}_{+}}}}$ satisfies condition III of Definition 4, i.e., where $d=H-\frac{1}{2}$, with $d\in \left(0,\frac{1}{2}\right)$, and ${L_{3}}(N)$ is the slowly varying function defined in (25).

Now, we consider the following model and then state and prove the functional central limit theorem for LRD series ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$, constructed from the increments of a fractional PSI-process. The convergence is understood in distribution in the Skorokhod space $\mathcal{D}[0,1]$ towards a fractional Brownian motion.The assumptions below follow the classical framework of [21].

For $N\ge 1$ and $t\in [0,1]$, define the partial-sum process

The (classical) spectral density of a weakly stationary time series $X={\left\{{X_{n}}\right\}_{n\in \mathbb{Z}}}$, defined by (1), is always real and nonnegative. This is because the autocovariance function ${\gamma _{X}}(h)$ of a stationary process is always symmetric, satisfying ${\gamma _{X}}(h)={\gamma _{X}}(-h)$. Consequently, the Fourier representation reduces to a cosine series, and no imaginary component appears.

In contrast, in our framework we consider the nonsymmetric autocovariance function of stationary series ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$ constructed by increment process of the fractional PSI-process ${\Psi _{\theta }}(t),t\in {\mathbb{R}_{+}}$. Since the series is defined only on ${\mathbb{Z}_{+}}$ and the autocovariance function is defined only for $h\ge 0$, it is nonsymmetric. Hence, the spectral density can be defined using a unilateral (one-sided) Fourier series which is not the classical spectral density but rather a one-sided distributional extension.

By Corollary 2, since the autocovariance series diverges for $H\in \left(\frac{1}{2},1\right)$, the unilateral Fourier series of ${\gamma _{X}}(h)$ may not converge pointwise. Nevertheless, the spectral density can still be rigorously defined as a tempered distribution.

To obtain a rigorous representation of the spectral density ${f_{X}}(\lambda )$, defined by (31), in terms of a unilateral Fourier series, one must establish the convergence of the unilateral Fourier transform of ${\gamma _{X}}(h)$ in the distributional sense within the space of tempered distributions ${\mathcal{S}^{\prime }}(\mathbb{R})$.

Let $\mathcal{S}(\mathbb{R})$ denote the Schwartz space of rapidly decreasing smooth functions, i.e., Its topological dual, denoted by ${\mathcal{S}^{\prime }}(\mathbb{R})$, is the space of tempered distributions. A tempered distribution $T\in {\mathcal{S}^{\prime }}(\mathbb{R})$ is a continuous linear functional on $\mathcal{S}(\mathbb{R})$, i.e., for each $\varphi \in \mathcal{S}(\mathbb{R})$,

Thus, the spectral density ${f_{X}}(\lambda )$, defined by (31), can be rigorously defined as a tempered distribution by where $\widehat{\varphi }(h)={\textstyle\int _{\mathbb{R}}}{e^{-ih\lambda }}\varphi (\lambda )\hspace{0.1667em}d\lambda $ is the Fourier transform of a test function φ. This ensures that ${f_{X}}\in {\mathcal{S}^{\prime }}(\mathbb{R})$ and provides a rigorous basis for the unilateral Fourier transform in the distributional sense.

Furthermore, by Corollary 2, the autocovariance function ${\gamma _{X}}(h)$ exhibits long-range dependence of the form where ${L_{1}}$ is a slowly varying function at infinity. Estrada and Kanwal in [10] prove that if coefficients ${\gamma _{X}}(h)$ satisfy at most polynomial growth, then ${f_{X}}(\lambda )$ defines a tempered distribution (via its partial sums). Accordingly, we state the following result.

Lemma 2 shows that the unilateral Fourier expansion of ${\gamma _{X}}(h)$ is well defined in ${\mathcal{S}^{\prime }}(\mathbb{R})$, and its spectral interpretation remains valid.

Now, we present a typical result used in the study of long-memory processes and fractional behavior.

Thus, a key distinction between general stationary LRD models and our model lies in the availability of closed-form covariance functions and spectral distributions. As we have shown, our LRD model admits a closed-form covariance function and tractable one-sided spectral distributions, whereas other LRD models behave differently. Figure 2 illustrates the spectral structure of the stationary time series constructed by the increments of the fractional process. We observe that as $\lambda \to \pm \pi $, the spectral density decays smoothly, indicating diminishing contribution from high-frequency components. Notably, the magnitude of $|{f_{X}}(\lambda )|$ sharply peaks at zero frequency, which is a characteristic signature of LRD processes, where low frequencies dominate the spectrum. Additionally, while the imaginary part of ${f_{X}}(\lambda )$ is small for large θ (i.e., small $m=\frac{1}{\theta }$), it demonstrates the asymmetry of the spectral density caused by the exponential factor ${e^{i\lambda /\theta }}$. This reflects the lack of symmetry in the autocovariance function. In spectral analysis and many applications, we are interested in how much energy or variation is present at each frequency. This is accurately represented by $|{f_{X}}(\lambda )|$ which, represents the spectral energy at the frequency. The upper panel of Figure 3 illustrates the effect of Hurst parameter H for $H=0.6,0.7,0.9$ with fixed $\theta =1000$. The spectral energy $|{f_{X}}(\lambda )|$ shows a clear peak at zero frequency, which sharpens and increases in height as H rises.

The lower panel of Figure 3 shows the effect of varying $\theta =0.3,0.5,10,100$ on $|{f_{X}}(\lambda )|$, with fixed $H=0.8$. As θ increases, the spectral energy decays more smoothly near $\lambda =\pm \pi $, indicating reduced contributions from high-frequency components. Furthermore, large values of θ produce a spectrum that closely resembles that of classical fractional Gaussian noise.

In this simulation, we generate and analyze a stationary time series constructed from the increments of a fractional PSI-process with $\kappa =0.4$ and $\theta =50$ over a fixed total duration of 1000, representing the complete time interval across which the process is observed and the series is constructed. Figure 4 shows a sample path of the simulated series ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$ over the first 500 observations. The amplitude of changes between successive points is influenced by the fractional PSI-process parameters, particularly $\theta =50$ and $H=1-\frac{\kappa }{2}=0.8$. The upper panel of Figure 5 displays the empirical autocorrelation function of ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$. The slow decay of correlations across lags indicates the presence of long-range dependence, consistent with the exponent parameter $H=0.8$. The lower panel of Figure 5 presents the low-frequency spectral density of the sequence ${\left\{{X_{n}}\right\}_{n\in {\mathbb{Z}_{+}}}}$. It can be observed that the spectral power attains its highest values near the origin $(\lambda \approx 0)$, indicating that most of the energy of the process is concentrated at low frequencies. This feature reflects the dominance of slow-varying components in the signal, which evolve gradually over time. Such a concentration of spectral power near zero frequency is a well-known signature of long-range dependence and is consistent with the slow decay observed in the empirical autocorrelation function shown in the upper panel. As the spectral density decays smoothly at higher frequencies with no additional dominant peaks (Figure 2), only the low-frequency range is shown in Figure 5 to highlight the slow-varying components and the long-memory behavior of the sequence.