11institutetext: Department of Brain and Cognitive Sciences, MIT, USA
22institutetext: Department of Computer Science, ETH Zurich, Switzerland
33institutetext: Department of Information Technology and Electrical
Engineering, ETH Zurich, Switzerland 44institutetext: Department of Biosystems
Science and Engineering, ETH Zurich, Switzerland
✉ 44email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Shared first authorship.

# M(otion)-mode Based Prediction of Ejection Fraction using Echocardiograms

Ece Ozkan* (✉) 11 0000-0002-9889-6348 Thomas M. Sutter* (✉) 22
0000-0001-7503-4473 Yurong Hu 33 0009-0008-8997-0543 Sebastian Balzer 44 Julia
E. Vogt 22 0000-0002-6004-7770

###### Abstract

Early detection of cardiac dysfunction through routine screening is vital for
diagnosing cardiovascular diseases. An important metric of cardiac function is
the left ventricular ejection fraction (EF), where lower EF is associated with
cardiomyopathy. Echocardiography is a popular diagnostic tool in cardiology,
with ultrasound being a low-cost, real-time, and non-ionizing technology.
However, human assessment of echocardiograms for calculating EF is time-
consuming and expertise-demanding, raising the need for an automated approach.
In this work, we propose using the M(otion)-mode of echocardiograms for
estimating the EF and classifying cardiomyopathy. We generate multiple
artificial M-mode images from a single echocardiogram and combine them using
off-the-shelf model architectures. Additionally, we extend contrastive
learning (CL) to cardiac imaging to learn meaningful representations from
exploiting structures in unlabeled data allowing the model to achieve high
accuracy, even with limited annotations. Our experiments show that the
supervised setting converges with only ten modes and is comparable to the
baseline method while bypassing its cumbersome training process and being
computationally much more efficient. Furthermore, CL using M-mode images is
helpful for limited data scenarios, such as having labels for only 200
patients, which is common in medical applications.

###### Keywords:

Echocardiography M-mode Ultrasound Ejection Fraction Computer Assisted
Diagnosis (CAD)

##  1 Introduction

Cardiovascular diseases (CVD) are the leading cause of death worldwide,
responsible for nearly one-third of global deaths [29]. Early assessment of
cardiac dysfunction through routine screening is essential, as clinical
management and behavioral changes can prevent hospitalizations and premature
deaths. An important metric for assessing cardiac (dys)function is the left
ventricular (LV) ejection fraction (EF), which evaluates the ratio between LV
end-systolic and end-diastolic volumes [3, 21].

Echocardiography is the most common and readily available diagnostic tool to
assess cardiac function, ultrasound (US) imaging being a low-cost, non-
ionizing, and rapid technology. However, the manual evaluation of
echocardiograms is time-consuming, operator-dependent, and expertise-
demanding. Thus, there is a clear need for an automated method to assist
clinicians in estimating EF.

M(otion)-mode is a form of US, in which a single scan line is emitted and
received at a high frame rate through time to evaluate the dynamics to assess
different diseases [23]. M-mode is often utilized in clinical practice e. g.
in lung ultrasonography [1, 25] or echocardiography [6, 7, 26, 10]. Since
cardiac function assessment relies on heart dynamics, M-mode images can be an
excellent alternative to B(rightness)-mode image- or video-based methods.
However, little effort is directed toward exploiting M-mode images in an
automated manner.

Data collection and annotation are expensive for most applications. Therefore,
learning from limited labeled data is critical in data-limited problems, such
as in healthcare. To overcome this data bottleneck, self-supervised learning
(SSL) methods have been recently proposed to learn meaningful high-level
representations from unlabeled data [16, 24].

Related Work A few existing works [14, 18] reconstruct M-mode images from
B-mode videos to detect pneumothorax using CNNs. Furthermore, authors in [27]
propose an automatic landmark localization method in M-mode images. A more
related method using M-mode images in an automated manner to estimate EF is
[22], which uses single M-mode images in parasternal long-axis view to measure
chamber dimensions for calculating EF.

For automated EF prediction, some previous works exploit either still-images
[17, 31, 8] or spatio-temporal convolutions on B(rightness)-mode
echocardiography videos [21]. However, still-image-based methods have a high
variability [20], and video-based methods rely on a complex pipeline with
larger models. Furthermore, [19] uses vision transformers and CNNs to tackle
the problem of estimating the LV EF, and [15] uses geometric features of the
LV derived from ECG video frames to estimate EF. The authors in [28] evaluate
ML-based methods in a multi-cohort setting using different imaging modalities.
In the SSL setting, [5] propose a contrastive learning framework for deep
image regression, which consists of a feature learning branch via a novel
adaptive-margin contrastive loss and a regression prediction branch using
echocardiography frames as input.

Our Contribution We propose to extract images from readily available B-mode
echocardiogram videos, each mimicking an M-mode image from a different scan
line of the heart. We combine the different artificial M-mode images using
off-the-shelf model architectures and estimate their EF to diagnose
cardiomyopathy in a supervised regime. Using M-mode images allows the model to
naturally observe the motion and sample the heart from different angles while
bypassing cumbersome 3D models. Secondly, we propose an alternative scheme for
predicting EF using generated M-mode images in a self-supervised fashion while
extending contrastive learning. We design a problem-specific contrastive loss
for M-mode images to learn representations with structure and patient
awareness. We evaluate both regimes on the publicly available EchoNet-Dynamic
dataset ([20]) and demonstrate both models’ effectiveness.

To the best of our knowledge, this is the first work on image-based and
temporal information incorporating cardiac function prediction methods to
estimate EF. Furthermore, our method can easily be applied to other problems
where cardiac dynamics play an essential role in the diagnosis. To ensure
reproducibility, we made the code available:
https://github.com/thomassutter/mmodeecho.

##  2 Methods

This work aims to create a pipeline with as little intervention as possible;
thus, our method consists of two parts, as shown in Figure 1. The first part
is extracting M-mode images from readily available B-mode videos. The second
part includes representation learning, which are lower-level information that
preserves more information of the input image and are used to predict EF from
M-mode images, including two schemes: supervised and self-supervised learning.

Figure 1: Overview of our proposed method. (a) Generate M-mode images from
B-mode echocardiography videos at different scan lines. (b) Learn
representations from the generated M-mode images using supervised and self-
supervised learning schemes. (c) Evaluate EF prediction to diagnose
cardiomyopathy.

###  2.1 From B-mode Videos to M-mode Images

Assume our dataset contains $N$ patients. For each patient
$i=\\{1,2,\cdots,N\\}$, the label $y_{i}$ indicates its EF. Furthermore, the
B-mode echocardiogram video of each patient $i$ is given of size $h\times
w\times t$ with $h$ being height, $w$ width, and $t$ number of frames of the
video. The $m$-th M-mode image of patient $i$ is given as
$\boldsymbol{x}_{i}^{m}$ with $m=\\{1,2,\cdots,M\\}$. It is a single line of
pixels through the center of the image with an angle $\theta_{m}$ over frames,
assuming LV is around the center throughout the video, as in Figure 1(a). This
image, corresponding to $\theta_{m}$, is then of size $s_{m}\times t$, with
$s_{m}$ as the length of the scan line. For simplicity, we set $s_{m}=h\hskip
2.84544pt\forall\hskip 2.84544ptm$ independent of its angle $\theta_{m}$. For
generating multiple M-mode images, a set of $M$ angles
$\boldsymbol{\theta}=[\theta_{1},\ldots,\theta_{M}]$ is used to generate $M$
M-mode images, where the angles $\boldsymbol{\theta}$ are equally spaced
between $0^{\circ}$ and $180^{\circ}$.

While the proposed approach for generating M-mode images is intuitive and
works well (see Section 3.3), other approaches are also feasible. For
instance, the center of rotation in the middle of the image in our M-mode
generation process could be changed. Like that, we could mimic the behavior of
the data collection process as every generated M-mode image would resemble a
scan line of the US probe. However, the main goal of this work is to highlight
the potential of M-mode images for the analysis of US videos. Given our
convincing results, we leave the exploration of different M-mode generation
mechanisms for future work.

###  2.2 Learning Representations from M-mode Images

####  2.2.1 Supervised Learning for EF Prediction

We aim to learn supervised representations using off-the-shelf model
architectures to estimate EF. Instead of using a single M-mode, one can
aggregate the information of M-mode images from the same patient to increase
robustness. We evaluate two fusion methods for aggregating information among
the $M$ M-mode images: early-fusion and late-fusion [2]. With early fusion, we
construct a $M\times s\times t$ image with the $M$ M-mode images being the $M$
channels of the newly created image. In late-fusion, we exploit three
different methods. For all of the late-fusion schemes, we first infer an
abstract representation $\boldsymbol{z}_{i}^{m}$ for every M-mode image
$\boldsymbol{x}_{i}^{m}$. The representations $\boldsymbol{z}_{i}^{m}$ are
then aggregated to a joint representation $\boldsymbol{\tilde{z}}_{i}$ using
an LSTM cell [11], averaging, or concatenating.

We utilize a standard ResNet architecture [9] with 2D-convolutional layers
independent of the fusion principle. With 2D-convolutions, we assume a single
M-mode image as a 2D gray-scale image with two spatial dimensions, $s$ and
$t$.

####  2.2.2 Self-Supervised Learning for EF Prediction

This part aims to learn meaningful representations from unlabeled data to
estimate EF using echocardiograms. To this end, we propose an SSL scheme for
M-mode images based on contrastive learning, where M-mode images from the same
patient can naturally serve as positive pairs since they share labels for many
downstream tasks. As discussed by [30], bio-signal data is inherently highly
heterogeneous; thus, when applying learning-based methods to patient data, we
need to consider both the similarity and the difference between samples
originating from the same patient. Thus, we propose a problem-specific
contrastive loss with patient and structure awareness, as shown in Figure 2.

Figure 2: Overview of our proposed SSL method. The contrastive loss includes
(a) patient awareness to attract similarity between data from the same patient
while discouraging it between different patients and (b) structure awareness
to take the (possible) dissimilarity from the same patient into account.

##### Contrastive Learning Framework

The framework contains training and evaluation stages and the overview is
illustrated in Figure 3. In the training stage, we optimize the model with the
contrastive loss leveraging the information from underlying structures of the
unlabeled images. In the evaluation stage, a multilayer perceptron (MLP) head
is trained on top of the learned representations in a supervised manner.

For each generated M-mode image $\boldsymbol{x}_{i}^{m}$, we generate its
augmented view $\boldsymbol{x}_{i}^{v(m)}$ using the $Aug(\cdot)$ module. So
the augmented dataset is represented as $\\{(\boldsymbol{x}_{i}^{m},\
\boldsymbol{x}_{i}^{v(m)},\ y_{i})\\}$. The encoder network $Enc(\cdot)$ maps
each image $\boldsymbol{x}_{i}^{m}$ to a feature vector
$\boldsymbol{z}_{i}^{m}$. We utilize a standard ResNet architecture [9].

In the training stage, $\boldsymbol{z}_{i}^{m}$ is normalized to the unit
hyper-sphere before being passed to the projection network. Following the work
[4], we introduce a learnable non-linear projection network between the
representation and the contrastive loss. The projection network $Proj(\cdot)$
takes the normalized lower-level representation $\boldsymbol{z}_{i}^{m}$ as
input and outputs the higher-level representation $\boldsymbol{p}_{i}^{m}$. We
use a two-layer MLP with ReLU activation as $Proj(\cdot)$ in this work.

In the evaluation stage, we initialize the parameters of the encoder network
$Enc(\cdot)$ with the model obtained from contrastive learning and add an MLP
head $Head(\cdot)$ to the top. For each patient $i$, we have $M$ feature
vectors $\boldsymbol{z}_{i}^{m}\in\mathbb{R}^{K}$. The $M$ vectors are then
fused to get the joint representation
$\boldsymbol{\tilde{z}}_{i}\in\mathbb{R}^{K\times M}$ and passed to
$Head(\cdot)$. One can have different fusion methods for aggregating
information among the $M$ vectors, e. g. using an LSTM cell [11], averaging,
or concatenating.

Figure 3: Schema of the contrastive learning framework with training and
evaluation stages. The training stage exploits the contrastive loss to learn a
representation leveraging the unlabelled images. The evaluation stage exploits
these learned representations in a supervised manner to predict EF.

##### Contrastive Loss for M-mode Images

To account for (dis)similarities, we design two loss functions for learning
both patient- and structure-awareness.

(a) Patient-aware loss: The goal is to attract the representations from the
same patient to be similar while pushing apart representations from different
patients (see Figure 2 (a)). This enforces two M-mode images to be considered
similar if they are from the same patient and dissimilar if they are from
different patients. The patient-aware loss is given as:

$L^{PA}=-\frac{1}{M-1}\sum_{i=1}^{N}\sum_{m=1}^{M}\sum_{l\neq
m}\log\frac{\exp(\boldsymbol{p}_{i}^{m}\cdot\boldsymbol{p}_{i}^{l}/\tau)}{\sum_{j,k}\exp(\boldsymbol{p}_{i}^{m}\cdot\boldsymbol{p}_{j}^{k}/\tau)-\exp(\boldsymbol{p}_{i}^{m}\cdot\boldsymbol{p}_{i}^{m}/\tau)}$
(1)

where $N$ is the number of patients in one batch, $M$ is the number of
original M-mode images used for each patient, and $\tau$ is the temperature
scaling parameter. The term $\boldsymbol{p}_{i}^{m}$ represents the output of
$Proj(\cdot)$.

Inspired by [30], we tried defining a neighborhood function to limit the
similarity of M-mode images from the same patient. However, incorporating
neighbourhood to patient-awareness did not further improve the results; thus,
we used all M-mode images per patient to define the patient-aware loss.

(b) Structure-aware loss: If we only use patient-aware loss $L^{PA}$, there
exists a risk that all images from the same patient collapse to a single point
[30]. So we propose the structure-aware loss to introduce some diversity (see
Figure 2 (b)). To incorporate this into the learned representations, we
construct positive pairs from each M-mode image with its augmentation and
consider other combinations as negative pairs. It is then defined as:

$L^{SA}=-\sum_{i=1}^{N}\sum_{m=1}^{2M}\log\frac{\exp(\boldsymbol{p}_{i}^{m}\cdot\boldsymbol{p}_{i}^{v(m)}/\tau)}{\sum_{l\neq
m}\exp(\boldsymbol{p}_{i}^{m}\cdot\boldsymbol{p}_{i}^{l}/\tau)}$ (2)

If image $m$ is an original image, then $v(m)$ represents its augmented view;
if image $m$ is an augmented image, then $v(m)$ represents the original image.
Minimizing $L^{SA}$ drives the representation pairs from the augmented images
in the numerator close while pushing the representations in the denominator
far away, where the denominator contains M-mode images from the same patient.

Finally, we combine the two losses to get structure-aware and patient-aware
contrastive loss for M-mode images using the hyperparameter $\alpha$ to
control the trade-off between the awareness terms:

$L^{CL}=\alpha L^{PA}+(1-\alpha)L^{SA}.$ (3)

##  3 Experiments and Results

###  3.1 Dataset

We use the publicly available EchoNet-Dynamic dataset [20]. It contains
$10^{\prime}030$ apical-4-chamber echocardiography videos from individuals who
underwent imaging between 2016-2018 as part of routine clinical care at
Stanford University Hospital. Each B-mode video was cropped and masked to
remove information outside the scanning sector and downsampled into
standardized $112\times 112$ pixel videos. For simplicity, we used videos with
at least 112 frames. We use the official splits with $7465$ training, $1289$
validation, and $1282$ test set samples.

###  3.2 Experimental Setup

We evaluate the models’ performance using classification accuracy for five
random seeds and report the mean performance and standard deviation. During
training, all supervised models optimize the estimation of EF as a regression
task. For testing, we use a constant threshold $\tau$ for classifying
cardiomyopathy. In all experiments, we set $\tau=0.5$. Hence, an estimation of
$\hat{\tau}<0.5$ results in classifying a sample as cardiomyopathic.

We evaluate all models using the area under the receiver operating
characteristic (AUROC) and the area under the precision-recall curve (AUPRC)
with respect to whether a patient is correctly classified as healthy or
cardiomyopathic. Additionally, we report the mean absolute error (MAE) and the
root mean squared error (RMSE) of the predicted EF with respect to the true EF
in the Supplementary Material. We report the mean performance, including
standard deviations over five random seeds for all results.

We use the training set from EchoNet for pre-training (SSL), and apply a
linear learning rate scheduler during the first 30 epochs as warm-up. For the
supervised fine-tuning, we select different proportions of the training set in
the limited labeled data scenario. All M-mode models are trained for $100$
epochs using Adam optimizer [12] with an initial learning rate of $0.001$ and
a batch size of $64$. For image augmentation, we apply random horizontal flip
and Gaussian noise. For the fusion method of the the M-mode representations we
used concatenation. For the EchoNet model, we use the same model and
parameters as in [21]. The model is trained for $45$ epochs with a learning
rate of $0.0001$ and a batch size of $20$. We do not use test-time
augmentation for any of the models. We report the full set of hyperparameters
used in our experiments in Table 1.

Table 1: List the hyperparameters used in our experiments. We use the same hyper-parameters for E2E setup and the fine-tuning stage of SSL setup (denoted as "_sup" in Table 1). "_cl" denotes the hyper-parameters used in the SSL pre-training stage. Parameter | Value | Description
---|---|---
lr_sup | $0.001$ | learning rate for supervised training
lr_cl | $1.0$ | learning rate for SSL training
opt | Adam | optimizer for SSL and supervised training
bsz_sup | $64$ | batch size for supervised training
bsz_cl | $256$ | batch size for SSL training
epoch_sup | $100$ | epochs for supervised training
epoch_cl | $300$ | epochs for SSL training
epoch_warm | $30$ | warm-up epochs for SSL training
$\alpha$ | $0.8$ | loss trade-off
$\tau$ | $0.01$ | temperature scaling
Dim_e | $512$ |  $Enc(\cdot)$ output dimension
Dim_ph | $2048$ |  $Proj(\cdot)$ hidden layer dimension
Dim_po | $128$ |  $Proj(\cdot)$ output dimension
Dim_lstm | $256$ | LSTM output dimension

###  3.3 Results and Discussion

####  3.3.1 Evaluating M-mode Images in Supervised Setting

We train and evaluate models with different numbers of M-modes for
$M\in\\{1,2,5,10,20,50\\}$. We use the complete training set, including
labels, as we are interested in the performance of the models depending on the
number of available M-modes. Figure 4 shows the results for different numbers
of M-modes. We see that late fusion models benefit from an increasing number
of modes, whereas the early fusion method overfits quickly and never achieves
a comparable performance.

(a) AUPRC

(b) AUROC

(c) RMSE

(d) MAE

(e) $R^{2}$

Figure 4: Performance for different numbers of M-mode images using early and
late-fusion methods. In (a), we evaluate the classification performance with
respect to AUPRC and AUROC in (b), the regression performance with respect to
RMSE in (c), MAE in (d), and $R^{2}$-score in (e).

####  3.3.2 Evaluating Limited Data Regime

We evaluate the accuracy of the different models introduced in Section 2 for
different amount of labeled training samples. As most medical datasets do not
have the size of EchoNet-Dynamic [13], methods for medical machine learning
should perform best in the limited labeled data regime. We use _E2E_ for the
supervised and _CL_ for the self-supervised setting.

Additionally, we introduce _E2E+_ and _CL+_ , which, inspired by EchoNet [21],
uses random short clips for each training epoch. Both models use M-mode images
of 32 frames with a sampling period of 2. We train and evaluate models using
$p\%$ of the full training set for $p\in\\{1,2,3,5,10,20,30,$ $50,75,100\\}$.
All M-mode methods are trained with $M=10$.

Figure 5 shows the limited labeled data experiment results. Although we are
not able to reach the performance of the EchoNet model for any number of modes
(see Figure 4(b)) if the number of labeled training samples is high (see
Figure 5(a)), both supervised and self-supervised learning methods using
M-mode instead of B-mode can outperform the EchoNet model in the low labeled
data regime ($p<5\%$, Figure 5(b)). Also, we observe that using shorter clips
is useful for the self-supervised learning methods, with _CL+_ being able to
achieve an AUROC over $0.85$ with only around $200$ labeled samples.

2

(a) 10% - 100%

(b) 1% - 10%

Figure 5:  Results for different training set sizes using the proposed end-to-
end supervised (E2E) and contrastive learning (CL) approaches. In (a), we
train and evaluate the models on 10%-100% labeled training samples, in (b)
only on 1%-10% of the samples. E2E and CL models are trained using a fixed
long clip with length 112; E2E+ and CL+ are trained using random short clips
with length 32. CL freeze and CL+ freeze are fine-tuned with the encoder
parameters frozen.

####  3.3.3 Computational Cost

Furthermore, we compare the number of parameters and computational costs for
different models in Table 2, where we used a multi-GPU setup with four NVIDIA
GeForce RTX 2080 Ti GPUs. We report the computation time in seconds per batch
(sec/B) and milliseconds per sample (msec/sample), and the memory requirements
in gigabytes per batch (GB/B).

Our proposed M-mode image based models require around six times less time and
ten times less memory to train and run inference per sample. Given the used
memory per batch, we could increase the batch size for the M-mode methods,
lowering the computation time per sample even further, whereas the baseline
model is already at the limit due to its architecture.

Table 2: Computational costs. We evaluate the EchoNet and the proposed M-mode methods with respect to the number of parameters, the computation time, and the memory requirements. All M-mode models are evaluated using $M=10$. E2E defines the end-to-end supervised and CL the contrastive learning approach.  |  |  | Time (sec/B) | Time (msec/sample) | Memory (GB/B)
---|---|---|---|---|---
Model | BS | #Params (Mio.) | Train | Test | Train | Test | Train | Test
EchoNet | 20 | 31.5 | 2.898 | 2.474 | 144.9 | 123.7 | 5.294 | 1.187
E2E & CL | 64 | 11.7 | 1.568 | 1.330 | 24.5 | 21.1 | 1.013 | 0.120

##  4 Discussion and Conclusion

In this work, we propose to generate M-mode images from readily available
B-mode echocardiography videos and fuse these to estimate EF and, thus,
cardiac dysfunction. Our results show that M-mode-based prediction methods are
comparable to the baseline method while avoiding its complex training routine
and reducing the computational cost and the need for expensive expert input.

Conventional M-mode images have a very high sampling rate, which results in a
high temporal resolution so that even very rapid motion can be recorded. The
generated M-mode images have significantly less temporal resolution than the
conventional M-mode images from US machines. However, our results indicate
that exploiting generated M-mode images does not limit the performance for EF
estimation. As we do not use the M-mode images collected directly from the US
machines, there is no need for an additional data collection step.

Additionally, we show the potential of pre-trained methods. In scenarios where
expensive expert labels are not readily available, pre-training using
unlabeled M-mode images outperforms more complicated pipelines highlighting
the potential of M-Mode based pipelines for clinical use cases. In our future
work, we want to investigate the use cases for M-mode on different diseases
and further improve the performance of the proposed pre-training pipeline.

##  5 Acknowledgements

EO was supported by the SNSF grant P500PT-206746 and TS by the grant 2021-911
of the Strategic Focal Area “Personalized Health and Related Technologies
(PHRT)” of the ETH Domain (Swiss Federal Institutes of Technology).

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# Flexible Sampling of Discrete Scale Invariant Markov Processes: Covariance
and Spectrum

N. Modarresi and S. Rezakhah  Faculty of Mathematics and Computer Science,
Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran.
E-mail<EMAIL_ADDRESS>(N. Modarresi<EMAIL_ADDRESS>(S. Rezakhah).

###### Abstract

In this paper we consider some flexible discrete sampling of a discrete scale
invariant process $\\{X(t),t\in{\bf R^{+}}\\}$ with scale $l>1$. By this
method we plan to have $q$ samples at arbitrary points ${\bf s}_{0},{\bf
s}_{1},\ldots,{\bf s}_{q-1}$ in interval $[1,l)$ and proceed our sampling in
the intervals $[l^{n},l^{n+1})$ at points $l^{n}{\bf s}_{0},l^{n}{\bf
s}_{1},\ldots,l^{n}{\bf s}_{q-1}$, $n\in{\bf Z}$. Thus we have a discrete time
scale invariant (DT-SI) process and introduce an embedded DT-SI process as
$W(nq+k)=X(l^{n}{\bf s}_{k})$, $q\in{\bf N}$, $k=0,\ldots,q-1$. We also
consider $V(n)=\big{(}V^{0}(n),\ldots,V^{q-1}(n)\big{)}$ where
$V^{k}(n)=W(nq+k)$, as an embedded $q$-dimensional discrete time self-similar
(DT-SS) process. By introducing quasi Lamperti transformation, we find
spectral representation of such process and its spectral density matrix is
given. Finally by imposing wide sense Markov property for $W(\cdot)$ and
$V(\cdot)$, we show that the spectral density matrix of $V(\cdot)$ and
spectral density function of $W(\cdot)$ can be characterized by
$\\{R_{j}(1),R_{j}(0),j=0,\ldots,q-1\\}$ where $R_{j}(k)=E[W(j+k)W(j)]$.

AMS 2000 Subject Classification: 60G18, 62M15.

Keywords: Discrete scale invariance; Wide sense Markov; Multi-dimensional
self-similar processes.

##  1 Introduction

The concept of stationarity and self-similarity are used as a fundamental
property to handle many natural phenomena. Lamperti transformation defines a
one to one correspondence between stationary and self-similar processes.
Discrete scale invariance (DSI) process can be defined as the Lamperti
transform of periodically correlated (PC) process. Many critical systems, like
statistical physics, textures in geophysics, network traffic and image
processing can be interpreted by these processes [1]. Fourier transform is
known as a suited representation for stationarity, but not for self-
similarity. A harmonic like representation of self-similar process is
introduced by using Mellin transform [4].

A process which is Markov and self-similar, is called self-similar Markov
process. These processes are involved in various parts of probability theory,
such as branching processes and fragmentation theory [2].

Current authors considered DSI processes in the wide sense with some scale
$l>1$. They proposed to have some fixed number of samples, say $T$, in each
scale at points $\alpha^{k}$, $k\in{\bf Z}$ where $l=\alpha^{T}$, $T\in{\bf
N}$. By such sampling they provided a discrete time scale invariant process in
the wide sense and found a closed formula for its covariance function [6]. In
this paper we consider $X(\cdot)$ as DSI process with scale $l>1$, and
sampling at arbitrary points $1\leqslant{\bf s}_{0}<{\bf s}_{1}\ldots<{\bf
s}_{q-1}<l$ in the interval $[1,l)$. We also take our samples at points
$l^{n}{\bf s}_{0},l^{n}{\bf s}_{1},\ldots,l^{n}{\bf s}_{q-1}$, $n\in{\bf Z}$
in the intervals $[l^{n},l^{n+1})$. Then we introduce some discrete time
embedded scale invariant (DT-ESI) process $W(nq+k)=X(l^{n}{\bf s}_{k})$,
$q\in{\bf N}$, $k=0,\ldots,q-1$ and corresponding multi-dimensional discrete
time embedded self-similar (DT-ESS) process as
$V(n)=\big{(}V^{0}(n),\ldots,V^{q-1}(n)\big{)}$ where $V^{k}(n)=W(nq+k)$. We
investigate properties of these processes when they are Markov in the wide
sense.

This paper is organized as follows. In section 2, we present a review of
multi-dimensional stationary, periodically correlated, self-similar and
discrete scale invariant processes. Then we define discrete time self-similar
(DT-SS) and scale invariant (DT-SI) processes. We also introduce quasi
Lamperti transformation in this section. Section 3 is devoted to the structure
of the multi-dimensional DT-SS process resulting from the above method of
sampling. We define DT-ESI process and corresponding multi-dimensional DT-ESS
process and characterize the spectral density matrix of it in this section.
Finally covariance function and spectral density matrix of the discrete time
embedded scale invariant Markov (DT-ESIM) processes and corresponding multi-
dimensional discrete time embedded self-similar Markov (DT-ESSM) are obtained
in section 4.

##  2 Theoretical framework

This section is organized in tree subsections. First we review the structure
of the covariance function and spectral distribution matrix of multi-
dimensional stationary processes. We present definitions of DT-SS, DT-SI, wide
sense self-similar and scale invariant processes in subsection 2.2. We define
quasi Lamperti transformation and present its properties which provide a one
to one correspondence between DT-SS and discrete time stationary processes and
also between DT-SI and DT-PC processes.

###  2.1 Stationary and multi-dimensional stationary processes

###### Definition 2.1

A process $\\{Y(t),t\in{\bf R}\\}$ is said to be stationary, if for any
$t,\tau\in{\bf R}$

$\\{Y(t+\tau)\\}\stackrel{{\scriptstyle d}}{{=}}\\{Y(t)\\}$ (2.1)

where $\stackrel{{\scriptstyle d}}{{=}}$ is the equality of all finite-
dimensional distributions.

If $(2.1)$ holds for some $\tau\in{\bf R}$, the process is said to be
periodically correlated. The smallest of such $\tau$ is called period of the
process.

By Rozanov [8], if $Y(t)=\\{Y^{k}(t)\\}_{k=1,\ldots,n}$ be an $n$-dimensional
stationary process, then

$Y(t)=\int e^{i\lambda t}\phi(d\lambda)$ (2.2)

is its spectral representation, where $\phi=\\{\varphi_{k}\\}_{k=1,\ldots,n}$
and $\varphi_{k}$ is the random spectral measure associated with the $k$th
component $Y^{k}$ of the $n$-dimensional process $Y$. Let

$B_{kr}(\tau)=E[Y^{k}(\tau+t)\overline{Y^{r}(t)}],\hskip
19.91692ptk,r=1,\ldots,n$

and $B(\tau)=[B_{kr}(\tau)]_{k,r=1,\ldots,n}$ be the correlation matrix of
$Y$. The components of the correlation matrix of the process $Y$ can be
represented as

$B_{kr}(\tau)=\int e^{i\lambda\tau}F_{kr}(d\lambda),\hskip
19.91692ptk,r=1,\ldots,n$ (2.3)

where for any Borel set $\Delta$,
$F_{kr}(\Delta)=E[\varphi_{k}(\Delta)\overline{\varphi_{r}(\Delta)}]$ are the
complex valued set functions which are $\sigma$-additive and have bounded
variation. For any $k,r=1,\ldots,n$, if the sets $\Delta$ and
$\Delta^{\prime}$ do not intersect,
$E[\varphi_{k}(\Delta)\overline{\varphi_{r}(\Delta^{\prime})}]=0$. For any
interval $\Delta=(\lambda_{1},\lambda_{2})$ when
$F_{kr}(\\{\lambda_{1}\\})=F_{kr}(\\{\lambda_{2}\\})=0$ the following relation
holds

$F_{kr}(\Delta)=\frac{1}{2\pi}\int_{\Delta}\sum_{\tau=-\infty}^{\infty}B_{kr}(\tau)e^{-i\lambda\tau}d\lambda$
(2.4)
$=\frac{1}{2\pi}B_{kr}(0)[\lambda_{2}-\lambda_{1}]+\lim_{T\rightarrow\infty}\frac{1}{2\pi}\sum_{0<|\tau|\leqslant
T}B_{kr}(\tau)\frac{e^{-i\lambda_{2}\tau}-e^{-i\lambda_{1}\tau}}{-i\tau}$

in the discrete parameter case, and

$F_{kr}(\Delta)=\lim_{a\rightarrow\infty}\frac{1}{2\pi}\int_{-a}^{a}\frac{e^{-i\lambda_{2}\tau}-e^{-i\lambda_{1}\tau}}{-i\tau}B_{kr}(\tau)d\tau$

in the continuous parameter case.

###  2.2 Discrete time scale invariant processes

###### Definition 2.2

A process $\\{X(t),t\in{\bf R^{+}}\\}$ is said to be self-similar of index
$H>0$, if for any ${\lambda}>0$

$\\{\lambda^{-H}X(\lambda t)\\}\stackrel{{\scriptstyle d}}{{=}}\\{X(t)\\}.$
(2.5)

The process is said to be DSI of index $H$ and scaling factor
${\lambda}_{0}>0$ or (H,${\lambda}_{0}$)-DSI, if $(2.5)$ holds for
$\lambda=\lambda_{0}$.

As an intuition, self-similarity refers to an invariance with respect to any
dilation factor. However, this may be a too strong requirement for capturing
in situations that scaling properties are only observed for some preferred
dilation factors.

###### Definition 2.3

A process $\\{X(k),k\in{\bf\check{T}}\\}$ is called discrete time self-similar
(DT-SS) process with parameter space $\check{T}$, where $\check{T}$ is any
subset of countable distinct points of positive real numbers, if for any
$k_{1},k_{2}\in\check{T}$

$\\{X(k_{2})\\}\stackrel{{\scriptstyle
d}}{{=}}(\frac{k_{2}}{k_{1}})^{H}\\{X(k_{1})\\}.$ (2.6)

The process $X(\cdot)$ is called discrete time scale invariance (DT-SI) with
scale $l>0$ and parameter space $\check{T}$, if for any
$k_{1},k_{2}=lk_{1}\in\check{T}$, $(2.6)$ holds.

###### Remark 2.1

If the process $\\{X(t),t\in{\bf R^{+}}\\}$ is DSI with scale $l=\alpha^{T}$
for fixed $T\in{\bf N}$ and $\alpha>1$, then by sampling of the process at
points $\alpha^{k},k\in{\bf Z}$, we have $X(\cdot)$ as a DT-SI process with
parameter space $\check{T}=\\{\alpha^{k},k\in{\bf Z}\\}$ and scale
$l=\alpha^{T}$. If we consider sampling of $X(\cdot)$ at points
$\alpha^{nT+k},n\in{\bf Z},\mbox{for fixed}\ k=0,1,\ldots,T-1$, then
$X(\cdot)$ is a DT-SS process with parameter space
$\check{T}=\\{\alpha^{nT+k},n\in{\bf Z}\\}$.

Yazici et.al. [9] introduced wide sense self-similar processes as the
following definition, which can be obtained by applying the Lamperti
transformation ${\cal L}_{H}$ to the class of wide-sense stationary processes.
This class encompasses all strictly self-similar processes with finite
variance, including Gaussian processes such as fractional Brownian motion but
no other alpha-stable processes.

###### Definition 2.4

A random process $\\{X(t),t\in{\bf R^{+}}\\}$ is said to be wide sense self-
similar with index H, for some $H>0$ if the following properties are satisfied
for each $c>0$, $t,t_{1},t_{2}>0$

$(i)\,\,\ E[X^{2}(t)]<\infty$,

$(ii)\,\,E[X(ct)]=c^{H}E[X(t)]$,

$(iii)\,\,E[X(ct_{1})X(ct_{2})]=c^{2H}E[X(t_{1})X(t_{2})]$.

This process is called wide sense DSI of index $H$ and scaling factor
$c_{0}>0$, if the above conditions hold for some $c=c_{0}$.

###### Definition 2.5

A random process $\\{X(k),k\in\check{T}\\}$ is called DT-SS in the wide sense
with index $H>0$ and with parameter space $\check{T}$, where $\check{T}$ is
any subset of distinct countable points of positive real numbers, if for all
$k,k_{1}\in\check{T}$ and all $c>0$, where $ck,ck_{1}\in\check{T}:$

$(i)\,\,\ E[X^{2}(k)]<\infty$,

$(ii)\,\,E[X(ck)]=c^{H}E[X(k)]$,

$(iii)\,\,E[X(ck)X(ck_{1})]=c^{2H}E[X(k)X(k_{1})]$.

If the above conditions hold for some fixed $c=c_{0}$, then the process is
called DT-SI in the wide sense with scale $c_{0}$.

###### Remark 2.2

Let $\\{X(t),t\in{\bf R^{+}}\\}$ in Remark $2.1$ be DSI in the wide sense,
with the same scale $l=\alpha^{T}$. Then $X(\cdot)$ with parameter space
$\check{T}=\\{\alpha^{k},k\in{\bf Z}\\}$ for $\alpha>1$ is DT-SI in the wide
sense and $X(\cdot)$ with parameter space $\check{T}=\\{\alpha^{nT+k},n\in{\bf
Z}\\}$ for fixed $T\in{\bf N}$, $\alpha>1$ is DT-SS for $k=0,\ldots,T-1$ in
the wide sense.

Through this paper we are dealt with wide sense self-similar and wide sense
scale invariant process, and for simplicity we omit the term ”in the wide
sense” hereafter.

###  2.3 Quasi Lamperti transformation

We introduce the quasi Lamperti transformation and its properties by
followings.

###### Definition 2.6

The quasi Lamperti transform with positive index $H$ and $\alpha>1$, denoted
by ${\cal L}_{H,\alpha}$ operates on a random process $\\{Y(t),t\in{\bf R}\\}$
as

${\cal L}_{H,\alpha}Y(t)=t^{H}Y(\log_{\alpha}t)$ (2.7)

and the corresponding inverse quasi Lamperti transform ${\cal
L}^{-1}_{H,\alpha}$ on process $\\{X(t),t\in{\bf R^{+}}\\}$ acts as

${\cal L}^{-1}_{H,\alpha}X(t)={\alpha}^{-tH}X(\alpha^{t}).$ (2.8)

###### Corollary 2.1

If $\\{Y(t),t\in{\bf R}\\}$ is stationary process, its quasi Lamperti
transform $\\{{\cal L}_{H,\alpha}Y(t),t\in{\bf R^{+}}\\}$ is self-similar.
Conversely if $\\{X(t),t\in{\bf R^{+}}\\}$ is self-similar process, its
inverse quasi Lamperti transform $\\{{\cal L}^{-1}_{H,\alpha}X(t),t\in{\bf
R}\\}$ is stationary.

###### Corollary 2.2

If $\\{X(t),t\in{\bf R^{+}}\\}$ is $(H,{\alpha}^{T}$)-DSI then ${\cal
L}^{-1}_{H,\alpha}X(t)=Y(t)$ is PC with period $T>0$. Conversely if
$\\{Y(t),t\in{\bf R}\\}$ is PC with period $T$ then ${\cal
L}_{H,\alpha}Y(t)=X(t)$ is $(H,{\alpha}^{T}$)-DSI.

###### Remark 2.3

If $X(\cdot)$ is a DT-SS process with parameter space
$\check{T}=\\{l^{n},n\in{\bf Z}\\}$, then its stationary counterpart
$Y(\cdot)$ has parameter space $\check{T}=\\{nT,n\in{\bf Z}\\}$

$X(l^{n})={\cal
L}_{H,\alpha}Y(l^{n})=l^{nH}Y(\log_{\alpha}{\alpha^{nT}})=\alpha^{nTH}Y(nT).$

Also it is clear by the following relation that if $X(\cdot)$ is a DT-SI
process with scale $l=\alpha^{T}$, $T\in{\bf N}$ and parameter space
$\check{T}=\\{\alpha^{n},n\in{\bf Z}\\}$, then $Y(\cdot)$ is a discrete time
periodically correlated (DT-PC) process with period $T$ and parameter space
$\check{T}=\\{n,n\in{\bf Z}\\}$

$Y(n)={\cal L}^{-1}_{H,\alpha}X(n)=\alpha^{-nH}X(\alpha^{n}).$

##  3 Structure of the process

In this section we define a multi-dimensional DT-SS process in the wide sense.
We also introduce a new method for sampling of a DSI process with scale $l>1$,
which provide sampling at arbitrary points in the interval $[1,l)$ and at
multiple $l^{n}$ of such points in the intervals $[l^{n},l^{n+1})$, $n\in{\bf
N}$. We introduce DT-ESI process corresponding to the multi-dimensional DT-ESS
process. Finally in Theorem 3.1 we find harmonic like representation and
spectral density matrix of the multi-dimensional DT-ESS process.

###### Definition 3.1

The process $U(t)=(U^{0}(t),U^{1}(t),\ldots,U^{q-1}(t))$ with parameter space
$\check{T}=\\{l^{n},n\in{\bf Z}\\}$, $l=\alpha^{T}$, $\alpha>1$ and $T\in{\bf
N}$ is a q-dimensional discrete time self similar process in the wide sense,
where

$\bf(a)$ $\\{U^{j}(\cdot)\\}$ for all $j=0,1,\cdots,q-1$ is DT-SS process with
parameter space

$\check{T}^{j}=\\{l^{n},n\in{\bf Z}\\}$.

$\bf(b)$ For every $n,\tau\in{\bf Z},\,\ j,k=0,1,\cdots,q-1$

$\mathrm{Cov}\big{(}U^{j}(l^{n+\tau}),U^{k}(l^{n})\big{)}=l^{2nH}\mathrm{Cov}\big{(}U^{j}(l^{\tau}),U^{k}(1)\big{)}.$

Our method of sampling is to provide enough flexibility to choose arbitrary
sample points of a discrete time scale invariant process. So, one could decide
to have $T$ partitions in each scale interval $I_{n}=[l^{n},l^{n+1})$,
$n\in{\bf Z}$ of a continuous time DSI process $X(\cdot)$ with scale $l>1$ and
find $\alpha$ by $l=\alpha^{T}$. Then our partitions in scale interval $I_{n}$
are

$[\alpha^{nT},\alpha^{nT+1}),[\alpha^{nT+1},\alpha^{nT+2}),\ldots,[\alpha^{nT+T-1},\alpha^{(n+1)T}).$

So we consider to have $n_{k}$ samples in partition
$[\alpha^{nT+k},\alpha^{nT+k+1})$ at points

$\alpha^{nT+k}s_{k_{1}},\alpha^{nT+k}s_{k_{2}},\ldots,\alpha^{nT+k}s_{k_{n_{k}}}$

where $1\leqslant s_{k_{1}}<s_{k_{2}}<\ldots<s_{k_{n_{k}}}<\alpha$,
$k=0,\ldots,T-1$ and $q=\sum_{i=0}^{T-1}n_{i}$. Now we can state the following
remark.

###### Remark 3.1

Let $U^{k}(l^{n})=X(l^{n}{\bf s}_{u})$ in Definition $3.1$, where ${\bf
s}_{u}=\alpha^{k}s_{k_{x}}$ in which $\sum_{i=-1}^{k-1}n_{i}\leqslant
u<\sum_{i=-1}^{k}n_{i}$, $n_{-1}=0$ and $u=x+\sum_{i=0}^{k-1}n_{i}$,
$x=1,\ldots,n_{k}$. Thus $X(l^{n}{\bf s}_{u})$ for $u=0,\ldots,q-1$ is a DT-SS
process and $U(l^{n})=\big{(}X(l^{n}{\bf s}_{0}),\ldots,X(l^{n}{\bf
s}_{q-1})\big{)}$ is a $q$-dimensional DT-SS process.

By such method of sampling at discrete points we provide a $q$-dimensional DT-
ESS process $V(n)$ as

$V(n)=\big{(}V^{0}(n),V^{1}(n),\ldots,V^{q-1}(n)\big{)},\hskip
28.45274ptn\in{\bf Z}$

where $q=\sum_{i=0}^{T-1}n_{i}$ and

$V^{u}(n):=X(\alpha^{nT}{\bf s}_{u})$ (3.1)

$\sum_{i=-1}^{k-1}n_{i}\leqslant u<\sum_{i=-1}^{k}n_{i}$, $n_{-1}=0$, ${\bf
s}_{u}=\alpha^{k}s_{k_{x}}$ and $x=u-\sum_{i=0}^{k-1}n_{i}$, $u=0,\ldots,q-1$.

###### Remark 3.2

Corresponding to the $q$-dimensional DT-ESS process $V(n)$ there exist a DT-
ESI process $W(\kappa)$ with scale $l=\alpha^{T}$ as

$W(\kappa):=V^{u}(n)=X(\alpha^{nT}{\bf s}_{u})\hskip 28.45274pt\kappa\in{\bf
Z}$ (3.2)

where $u=\kappa-q[\frac{\kappa}{q}]$, $n=[\frac{\kappa}{q}]$ and
$\kappa=nq+u$, since by $(3.1)$ and $(3.2)$

$W(\kappa+q)=X(\alpha^{(n+1)T}{\bf s}_{u})\stackrel{{\scriptstyle
d}}{{=}}\alpha^{TH}X(\alpha^{nT}{\bf s}_{u})=l^{H}W(\kappa).$

By the following theorem, the spectral density matrix of the $q$-dimensional
DT-ESS process and harmonic like representation of each column is obtained.

###### Theorem 3.1

Let $X(\cdot)$ be a DSI process with scale $l=\alpha^{T}$ and $1\leqslant{\bf
s}_{0}<{\bf s}_{1}<\ldots<{\bf s}_{q-1}<\alpha^{T}$, then
$V(n)=\big{(}V^{0}(n),\ldots,V^{q-1}(n)\big{)}$, where
$V^{u}(n)=X(\alpha^{nT}{\bf s}_{u})$, $n\in{\bf Z}$ and $u=0,\ldots,q-1$ is a
$q$-dimensional DT-ESS process and

(i) The harmonic like representation of $V^{u}(n)$ is

$V^{u}(n)=(\alpha^{nT}{\bf s}_{u})^{H}\int_{0}^{2\pi}e^{i\omega
n}d\phi_{u}(\omega)$ (3.3)

where $\phi_{u}(\omega)$ is an orthogonal spectral measure, that is
$E[d\phi_{u}(\omega)\overline{d\phi_{\nu}(\omega^{\prime})}]=0$,
$u,\nu=0,\ldots,q-1$ when $\omega\neq\omega^{\prime}$.

(ii) The corresponding spectral density matrix of $V(n)$ is
$g^{H}(\omega)=[g_{u,\nu}^{H}(\omega)]_{u,\nu=0,\ldots,q-1}$, where

$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}e^{-i\omega\tau}Q^{H}_{u,\nu}(\tau)$
(3.4)

$\tau\in{\bf N}$ and $Q^{H}_{u,\nu}(\tau)$ is the covariance function of
$V^{u}(\tau)$ and $V^{\nu}(0)$.

Proof of (i): Remark $2.3$ implies that

$V^{u}(n)=X(\alpha^{nT}{\bf s}_{u})={\cal L}_{H,\alpha}Y(\alpha^{nT}{\bf
s}_{u})=(\alpha^{nT}{\bf s}_{u})^{H}\eta^{u}(n)$

where $\eta^{u}(n)=Y(nT+\log_{\alpha}{\bf s}_{u})$. Thus $V^{u}(n)$ for every
$u=0,1,\ldots,q-1$ is a DT-ESS process in $n$, where its discrete time
stationary counterpart $\eta^{u}(n)$ for fixed $u=0,1,\ldots,q-1$ has spectral
representation $\eta^{u}(n)=\int_{0}^{2\pi}e^{i\omega n}d\phi_{u}(\omega)$.

Proof of (ii): The covariance matrix of $V(n)$ is denoted by
$Q^{H}(n,\tau)=[Q^{H}_{u,\nu}(n,\tau)]_{u,\nu=0,\ldots,q-1}$ where

$Q^{H}_{u,\nu}(n,\tau)=E[V^{u}(n+\tau)V^{\nu}(n)]=E[X(\alpha^{(n+\tau)T}{\bf
s}_{u})X(\alpha^{nT}{\bf s}_{\nu})]$

By the scale invariant property of the process $X(\cdot)$ we have that

$Q^{H}_{u,\nu}(n,\tau)=\alpha^{2nTH}E[X(\alpha^{\tau T}{\bf s}_{u})X({\bf
s}_{\nu})]=\alpha^{2nTH}Q^{H}_{u,\nu}(\tau)$ (3.5)

where $Q^{H}_{u,\nu}(\tau)=Q^{H}_{u,\nu}(0,\tau)=E[V^{u}(\tau)V^{\nu}(0)]$,
then by (3.3)

$Q^{H}_{u,\nu}(\tau)=E[(\alpha^{\tau T}{\bf s}_{u})^{H}({\bf
s}_{\nu})^{H}\int_{0}^{2\pi}e^{i\omega\tau}d\phi_{u}(\omega)\int_{0}^{2\pi}\overline{d\phi_{v}(\omega^{\prime})}]$
$=\alpha^{\tau TH}({\bf s}_{u}{\bf
s}_{\nu})^{H}\int_{0}^{2\pi}e^{i\omega\tau}dG^{H}_{u,\nu}(\omega)$ (3.6)

where
$E[d\phi_{u}(\omega)\overline{d\phi_{\nu}(\omega^{\prime})}]=dG^{H}_{u,\nu}(\omega)$
when $\omega=\omega^{\prime}$ and is $0$ when $\omega\neq\omega^{\prime}$.

On the other hand, by the definition of $\eta^{u}(n)$ in the proof of part
$(i)$

$Q^{H}_{u,\nu}(\tau)=E[X(\alpha^{\tau T}{\bf s}_{u})X({\bf s}_{\nu})]=E[{\cal
L}_{H,\alpha}Y(\alpha^{\tau T}{\bf s}_{u}){\cal L}_{H,\alpha}Y({\bf
s}_{\nu})]$ $=(\alpha^{\tau T}{\bf s}_{u}{\bf s}_{\nu})^{H}E[Y(\tau
T+\log_{\alpha}{\bf s}_{u})Y(\log_{\alpha}{\bf s}_{\nu})]$ $=(\alpha^{\tau
T}{\bf s}_{u}{\bf s}_{\nu})^{H}E[\eta^{u}(\tau)\eta^{\nu}(0)]=(\alpha^{\tau
T}{\bf s}_{u}{\bf s}_{\nu})^{H}B_{u,\nu}(\tau).$

Then by (3.6)

$B_{u,\nu}(\tau)=\int_{0}^{2\pi}e^{i\omega\tau}dG^{H}_{u,\nu}(\omega),\hskip
14.22636ptu,\nu=0,\ldots,q-1$

Now by (2.3) and (2.4) for $u,\nu=0,\ldots,q-1$ we have that

$G^{H}_{u,\nu}(A)=\frac{1}{2\pi}\int_{A}\sum_{\tau=-\infty}^{\infty}B_{u,\nu}(\tau)e^{-i\lambda\tau}d\lambda.$

By substituting $B_{u,\nu}(\tau)=(\alpha^{\tau T}{\bf s}_{u}{\bf
s}_{\nu})^{-H}Q^{H}_{u,\nu}(\tau)$, the elements of the spectral distribution
function, $G^{H}_{u,\nu}(\cdot)$ has the following representation

$G^{H}_{u,\nu}(A)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\int_{A}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}e^{-i\lambda\tau}Q^{H}_{u,\nu}(\tau)d\lambda.$
(3.7)

Let $A=(\omega,\omega+d\omega]$, then the elements of the spectral density
matrix, $g_{u,\nu}^{H}(\omega)$ are

$g_{u,\nu}^{H}(\omega):=\frac{G^{H}_{u,\nu}(d\omega)}{d\omega}=\frac{({\bf
s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}\big{(}\frac{1}{d\omega}\int_{\omega}^{\omega+d\omega}e^{-i\lambda\tau}d\lambda\big{)}Q^{H}_{u,\nu}(\tau)$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}\big{(}\frac{1}{-i\tau}\lim_{d\omega\rightarrow
0}\frac{e^{-i{(\omega+d\omega)\tau}}-e^{-i\omega\tau}}{d\omega}\big{)}Q^{H}_{u,\nu}(\tau)$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=-\infty}^{\infty}\alpha^{-TH\tau}\big{(}(\frac{1}{-i\tau})(-i\tau)e^{-i\omega\tau}\big{)}Q^{H}_{u,\nu}(\tau).$

Thus we get to the assertion of part (ii) of the theorem.$\square$

##  4 Multi-dimensional DT-ESSM process

Using our method of sampling in section 3, we find the covariance function of
the DT-ESI process $W(\cdot)$, which is defined in (3.2) and its corresponding
multi-dimensional DT-ESS process $V(\cdot)$, defined in (3.1) for the case
that they are Markov in the wide sense as well, which we call them DT-ESIM and
DT-ESSM respectively in subsection 4.1. We find the spectral density matrix of
these processes in subsection 4.2.

###  4.1 Covariance function of DT-ESIM

Here we characterize the covariance function of the DT-ESIM process
$\\{W(\kappa),\kappa\in{\bf Z}\\}$ in Theorem 4.1 and the covariance function
of the associated $q$-dimensional DT-ESSM process in Theorem 4.2.

###### Theorem 4.1

Let $\\{W(\kappa),\kappa\in{\bf Z}\\}$, defined in $(3.2)$, be DT-ESI and
Markov in the wide sense DT-ESIM, with scale $\alpha^{T}$. Then for
$\tau\in{\bf W}=\\{0,1,\ldots\\}$, $\kappa=nq+\nu$, $\kappa+\tau=mq+u$,
$u,\nu=0,\ldots,q-1$ and $n,m\in{\bf Z}$, the covariance function

$R_{\kappa}(\tau):=E[W(\kappa+\tau)W(\kappa)]=E[X(\alpha^{mT}{\bf
s}_{u})X(\alpha^{nT}{\bf s}_{\nu})]$ (4.1)

can be characterized as

$R_{\kappa}(tq+s)=[\tilde{f}(q-1)]^{t}\tilde{f}(\kappa+s-1)[\tilde{f}(\kappa-1)]^{-1}R_{\kappa}(0)$
(4.2) $R_{\kappa}(-tq+s)=\alpha^{-2tqH}R_{\kappa+s}((t-1)q+q-s)$

where $1\leqslant{\bf s}_{0}<{\bf s}_{1}<\ldots<{\bf s}_{q-1}<\alpha^{T}$,
$t\in{\bf Z}$, $s=0,\ldots,q-1$

$\tilde{f}(r)=\prod_{j=0}^{r}f(j)=\prod_{j=0}^{r}R_{j}(1)/R_{j}(0),\hskip
19.91692ptr\in{\bf Z}$ (4.3)

and $\tilde{f}(-1)=1$.

Before proceeding to the proof of the theorem we present the main property of
covariance function of the wide sense Markov process.

Let $\\{X(n),n\in{\bf Z}\\}$ be a second order process of centered random
variables, $E[X(n)]=0$ and $E[|X(n)|^{2}]<\infty$, $n\in{\bf Z}$. Following
Doob [3], the real valued second order process $X(\cdot)$ is Markov in the
wide sense if

$R(n_{1},n_{2})=G\big{(}\min(n_{1},n_{2})\big{)}H\big{(}\max(n_{1},n_{2})\big{)}$
(4.4)

where $R(n_{1},n_{2}):=E[X(n_{1})X(n_{2})]$ is the covariance function of
$X(\cdot)$ and $G$ and $H$ are defined uniquely up to a constant multiple and
the ratio $G/H$ is a positive nondecreasing function.

Proof of the theorem: As $\\{W(\kappa),\kappa\in{\bf Z}\\}$ is DT-ESI with
scale $\alpha^{T}$, this theorem fully characterize the covariance function of
the DT-ESIM process. From the Markov property (4.4), $R_{\kappa}(\tau)$
defined in (4.1), satisfies

$R_{\kappa}(\tau)=G(\alpha^{nT}{\bf s}_{\nu})H(\alpha^{mT}{\bf s}_{u}),\hskip
28.45274pt\tau\in{\bf Z},\alpha>1$ (4.5)

By substituting $\tau=0$ in the above relation we have $m=n$, then

$G(\alpha^{nT}{\bf s}_{\nu})=\frac{R_{\kappa}(0)}{H(\alpha^{nT}{\bf
s}_{\nu})}.$

Therefore

$R_{\kappa}(\tau)=\frac{H(\alpha^{mT}{\bf s}_{u})}{H(\alpha^{nT}{\bf
s}_{\nu})}R_{\kappa}(0),\hskip 28.45274pt\tau\in{\bf Z}$ (4.6)

Thus

$H(\alpha^{mT}{\bf
s}_{u})=\frac{R_{\kappa}(\tau)}{R_{\kappa}(0)}H(\alpha^{nT}{\bf s}_{\nu}).$

So

${\bf H}(\kappa+\tau)=\frac{R_{\kappa}(\tau)}{R_{\kappa}(0)}{\bf H}(\kappa)$

where ${\bf H}(\kappa)=H(\alpha^{nT}{\bf s}_{\nu})$ and ${\bf
H}(\kappa+\tau)=H(\alpha^{mT}{\bf s}_{u})$. Therefore

$R_{\kappa}(\tau)=\frac{{\bf H}(\kappa+\tau)}{{\bf H}(\kappa)}R_{\kappa}(0).$
(4.7)

For $\tau=1$, we have

${\bf H}(\kappa+1)=\frac{R_{\kappa}(1)}{R_{\kappa}(0)}{\bf H}(\kappa).$

By the recursive relation, it follows that

${\bf
H}(\kappa+1)=\frac{R_{\kappa}(1)}{R_{\kappa}(0)}\frac{R_{\kappa-1}(1)}{R_{\kappa-1}(0)}\ldots\frac{R_{0}(1)}{R_{0}(0)}{\bf
H}(0)={\bf H}(0)\prod_{j=0}^{\kappa}f(j)$

and

${\bf H}(\kappa)={\bf H}(0)\prod_{j=0}^{\kappa-1}f(j)$

where $f(j)=R_{j}(1)/R_{j}(0)$. By the assumptions $n=[\frac{\kappa}{q}]$,
$\nu=\kappa-q[\frac{\kappa}{q}]$ we have $\kappa=nq+\nu$, then

${\bf H}(nq+\nu)={\bf H}(0)\prod_{j=0}^{nq+\nu-1}f(j).$ (4.8)

As mentioned in Remark 3.2, $\\{W(\kappa),\kappa\in{\bf Z}\\}$ is DT-ESI with
scale $l$, then

$f(\kappa+q)=\frac{R_{\kappa+q}(1)}{R_{\kappa+q}(0)}=\frac{E[W(\kappa+q+1)W(\kappa+q)]}{E[W(\kappa+q)W(\kappa+q)]}$
$=\frac{\alpha^{2TH}E[W(\kappa+1)W(\kappa)]}{\alpha^{2TH}E[W(\kappa)W(\kappa)]}=\frac{R_{\kappa}(1)}{R_{\kappa}(0)}=f(\kappa).$

Hence by (4.8)

${\bf H}(nq+\nu)={\bf
H}(0)\big{[}\prod_{j=0}^{q-1}f(j)\big{]}^{n}\prod_{j=0}^{\nu-1}f(j),\hskip
19.91692pt\nu\geqslant 1$

By the definition of $\tilde{f}$ in (4.3)

${\bf H}(nq+\nu)={\bf H}(0)[\tilde{f}(q-1)]^{n}\tilde{f}(\nu-1).$ (4.9)

By a similar method one can verify that

${\bf H}(-nq+\nu)={\bf H}(0)[\tilde{f}(q-1)]^{-n}\tilde{f}(\nu-1).$

Let $\tau=tq+s$ in (4.7), $t\in{\bf W}$ and $s=0,1,\ldots,q-1$, then it
follows from (4.9) that

$R_{\kappa}(tq+s)=\frac{{\bf H}(\kappa+tq+s)}{{\bf
H}(\kappa)}R_{\kappa}(0)=\frac{{\bf
H}(0)[\tilde{f}(q-1)]^{t}\tilde{f}(\kappa+s-1)}{{\bf
H}(0)\tilde{f}(\kappa-1)}R_{\kappa}(0).$

For $\tau=-tq+s$ we have that

$R_{\kappa}(-tq+s)=E[X(\alpha^{-tq+\kappa+s})X(\alpha^{\kappa})]=\alpha^{-2tqH}E[X(\alpha^{\kappa+s})X(\alpha^{tq+\kappa})]$
$=\alpha^{-2tqH}R_{\kappa+s}(tq-s)=\alpha^{-2tqH}R_{\kappa+s}((t-1)q+q-s).\square$

Now we can use this theorem to prove the next result as follows, for
$q$-dimensional DT-ESSM process.

###### Theorem 4.2

Let $\\{W(\kappa),\kappa\in{\bf Z}\\}$ be a DT-ESIM process, and
$\\{V(n),n\in{\bf Z}\\}$ be its associated $q$-dimensional DT-ESSM process
with covariance matrix $Q^{H}(n,\tau)$ which is defined by $(3.5)$. Then

$Q^{H}(n,\tau)=\alpha^{2nTH}[\tilde{f}(q-1)]^{\tau}CR,\hskip
19.91692pt\tau\in{\bf Z}$ (4.10)

where $\tilde{f}(\cdot)$ is defined in $(4.3)$ and the matrices $C$ and $R$
are given by $C=[C_{u,\nu}]_{u,\nu=0,\ldots,q-1}$, where
$C_{u,\nu}=\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}$, and $R$ is a diagonal
matrix with diagonal elements $R_{\nu}(0)$, $\nu=0,1,\ldots,q-1$, which is
defined in $(4.1)$.

Proof: As $W(\cdot)$ is DT-ESI with scale $l$, (3.2) and (3.5) indicate that
$Q^{H}_{u,\nu}(n,\tau)=\alpha^{2nTH}Q^{H}_{u,\nu}(\tau)$. Now by the
assumption $\kappa=nq+\nu$ and $\kappa+\tau=mq+u$ where $m,n\in{\bf Z}$,
$\tau\in{\bf W}$, we have $\tau=(m-n)q+u-\nu$ and therefore

$R_{\kappa}(\tau)=R_{nq+\nu}((m-n)q+u-\nu)=E[W(mq+u)W(nq+\nu)]$
$=E[X(\alpha^{mT}{\bf s}_{u})X(\alpha^{nT}{\bf s}_{\nu})].$

Hence

$Q^{H}_{u,\nu}(\tau)=E[X(\alpha^{\tau T}{\bf s}_{u})X({\bf
s}_{\nu})]=R_{\nu}(\tau q+u-\nu)$ (4.11)

and by the Markov property of $W(\cdot)$ from (4.2) we have

$R_{\nu}(\tau
q+u-\nu)=[\tilde{f}(q-1)]^{\tau}\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}R_{\nu}(0)$

for $u,\nu=0,\ldots,q-1$. Let
$C_{u,\nu}=\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}$, so

$Q^{H}_{u,\nu}(\tau)=[\tilde{f}(q-1)]^{\tau}C_{u,\nu}R_{\nu}(0).$ (4.12)

Thus we can represent the elements of the covariance matrix of $q$-dimensional
DT-ESSM process as

$Q^{H}_{u,\nu}(n,\tau)=\alpha^{2nTH}[\tilde{f}(q-1)]^{\tau}C_{u,\nu}R_{\nu}(0).\square$

###  4.2 Spectral representation of the process

The spectral density matrix of the $q$-dimensional DT-ESSM process is
characterized by the following lemma which is proved in [7].

###### Lemma 4.1

The spectral density matrix
$g^{H}(\omega)=[g^{H}_{u,\nu}(\omega)]_{u,\nu=0,\ldots,q-1}$ of the
$q$-dimensional DT-ESSM process $V(n)$ is specified by

$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\left[\frac{\tilde{f}(u-1)R_{\nu}(0)}{\tilde{f}(\nu-1)\big{(}1-e^{-i\omega}\alpha^{-HT}\tilde{f}(q-1)\big{)}}-\frac{\tilde{f}(\nu-1)R_{u}(0)}{\tilde{f}(u-1)\big{(}1-e^{-i\omega}\alpha^{HT}\tilde{f}^{-1}(q-1)\big{)}}\right]$

where $R_{k}(0)$ is the variance of $W(k)$ and $\tilde{f}(\cdot)$ is defined
by $(4.3)$.

Proof: By applying (3.4) and (4.12), the spectral density matrix of the
process $\\{V(n),n\in{\bf Z}\\}$ which is denoted by
$g^{H}(\omega)=[g^{H}_{u,\nu}(\omega)]_{u,\nu=0,\ldots,q-1}$ can be written as

$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\Big{[}\sum_{\tau=0}^{\infty}\alpha^{-TH\tau}e^{-i\omega\tau}Q^{H}_{u,\nu}(\tau)$
$+\sum_{\tau=-\infty}^{-1}\alpha^{-TH\tau}e^{-i\omega\tau}Q^{H}_{u,\nu}(\tau)\Big{]}=g_{u,\nu,1}^{H}(\omega)+g_{u,\nu,2}^{H}(\omega)$

where

$g_{u,\nu,1}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\sum_{\tau=0}^{\infty}\alpha^{-TH\tau}e^{-i\omega\tau}[\tilde{f}(q-1)]^{\tau}\tilde{f}(u-1)[\tilde{f}(\nu-1)]^{-1}R_{\nu}(0)$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}\tilde{f}(u-1)R_{\nu}(0)}{2\pi\tilde{f}(\nu-1)}\sum_{\tau=0}^{\infty}\big{(}\alpha^{-TH}e^{-i\omega}\tilde{f}(q-1)\big{)}^{\tau}.$
(4.13)

By Remark 3.2, the scale invariant property of $W(\kappa)$ and the assumption,
that at least one of the $\text{Corr}[W(j)W(j+1)]$ be smaller than one, we
have that $|\tilde{f}(q-1)|<\alpha^{TH}$ for $j=0,\ldots,q-1$. Thus

$|e^{-i\omega}\alpha^{-TH}\tilde{f}(q-1)|=|\alpha^{-TH}\tilde{f}(q-1)|<1,$

and (4.13) for $\tau\in{\bf W}$ is convergent. By the equality

$Q_{u,\nu}(-\tau)=E[X(\alpha^{-\tau T}{\bf s}_{u})X({\bf
s}_{\nu})]=\alpha^{-2\tau TH}E[X(\alpha^{\tau T}{\bf s}_{\nu})X({\bf
s}_{u})]=\alpha^{-2\tau TH}Q_{\nu,u}(\tau),$

convergence of $g_{u,\nu,2}^{H}(\omega)$ follows by a similar method.
Therefore

$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\Big{[}\frac{R_{\nu}(0)\tilde{f}(u-1)}{\tilde{f}(\nu-1)}\sum_{\tau=0}^{\infty}\big{(}\alpha^{-TH}e^{-i\omega}\tilde{f}(q-1)\big{)}^{\tau}$
$+\frac{R_{u}(0)\tilde{f}(\nu-1)}{\tilde{f}(u-1)}\sum_{\tau=1}^{\infty}\big{(}\alpha^{-TH}e^{i\omega}\tilde{f}(q-1)\big{)}^{\tau}\Big{]}$
$=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}}{2\pi}\Big{[}\frac{R_{\nu}(0)\tilde{f}(u-1)}{\tilde{f}(\nu-1)\big{(}1-\alpha^{-TH}e^{-i\omega}\tilde{f}(q-1)\big{)}}+\frac{R_{u}(0)\tilde{f}(\nu-1)\alpha^{-TH}e^{i\omega}\tilde{f}(q-1)}{\tilde{f}(u-1)\big{(}1-\alpha^{-TH}e^{i\omega}\tilde{f}(q-1)\big{)}}\Big{]},$

so we arrive at the assertion of the lemma.$\square$

###### Example 4.1

Let

$X(t)=\sum_{n=1}^{\infty}\lambda^{n(H-\frac{1}{2})}I_{[\lambda^{n-1},\lambda^{n})}(t)B(t)$

where $B(\cdot)$ is the standard Brownian motion, $I(\cdot)$ indicator
function, $H>0$ and $\lambda>1$. We call this process Simple Brownian Motion.
We showed in [7] that $\\{X(t),t\in R^{+}\\}$ is DSI and Markov with Hurst
index $H$ and scale $\lambda$. By sampling of this process at points
$\alpha^{nT}{\bf s}_{u}$, $n\in{\bf W}$, where $1\leq{\bf s}_{0}\leq{\bf
s}_{1},\cdots,{\bf s}_{q-1}<\alpha^{T}$, and by assuming $\lambda=\alpha^{T}$,

$W(\kappa):=X(\alpha^{nT}{\bf s}_{u}),$

is a DT-ESIM process, and $V(n)=\big{(}V^{0}(n),\ldots,V_{q-1}(n)\big{)}$
where $V^{u}(n)=W(\kappa)$ is the associated $q$-dimensional DT-ESSM process
where $u=\kappa-q[\frac{\kappa}{q}]$, $n=[\frac{\kappa}{q}]$. By (4.1) we have
that $R_{j}^{H}(0)=R_{j}^{H}(1)=\alpha^{2TH^{\prime}}{\bf s}_{j}$ for
$j=0,\cdots,q-2$ and
$R_{q-1}^{H}(1)=\alpha^{TH^{\prime}}R_{q-1}^{H}(0)=\alpha^{3TH^{\prime}}{\bf
s}_{q-1}$, where $H^{\prime}=H-\frac{1}{2}$. So
$R_{u}(0)=\alpha^{2TH^{\prime}}{\bf s}_{u}$,
$\;R_{\nu}(0)=\alpha^{2TH^{\prime}}{\bf s}_{\nu}$. Also (4.3) implies that
$\tilde{f}(u-1)=\tilde{f}(\nu-1)=1$, $\tilde{f}(q-1)=\alpha^{TH^{\prime}}$.
Thus By Lemma 4.1, the spectral density matrix of $V(n)$ is

$g_{u,\nu}^{H}(\omega)=\frac{({\bf s}_{u}{\bf
s}_{\nu})^{-H}\alpha^{2TH^{\prime}}}{2\pi}\left[\frac{{\bf
s}_{\nu}}{1-e^{-i\omega}\alpha^{-T/2}}-\frac{{\bf
s}_{u}}{1-e^{-i\omega}\alpha^{T/2}}\right].$

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