# TELESIM: A Modular and Plug-and-Play Framework for Robotic Arm Teleoperation
using a Digital Twin

Florent P Audonnet1∗, Jonathan Grizou2∗, Andrew Hamilton3 and Gerardo Aragon-
Camarasa4∗ This research has been supported by EPSRC DTA No. 2605103 and EPSRC
Grant No. EP/S019472/1∗School of Computing Science, University of Glasgow, G12
8QQ, Scotland, United
Kingdom1f.audonnet.1@research.gla.ac.uk2jonathan.grizou@glasgow.ac.uk3andrew.w.hamilton<EMAIL_ADDRESS>
###### Abstract

We present TELESIM, a modular and plug-and-play framework for direct
teleoperation of a robotic arm using a digital twin as the interface between
the user and the robotic system. We tested TELESIM by performing a user survey
with 37 participants on two different robots using two different control
modalities: a virtual reality controller and a finger mapping hardware
controller using different grasping systems. Users were asked to teleoperate
the robot to pick and place 3 cubes in a tower and to repeat this task as many
times as possible in 10 minutes, with only 5 minutes of training beforehand.
Our experimental results show that most users were able to succeed by building
at least a tower of 3 cubes regardless of the control modality or robot used,
demonstrating the user-friendliness of TELESIM.

##  I Introduction

Robot teleoperation is difficult for non-experts [1, 2]. Recently, the ANA
Avatar XPRIZE Challenge [3] set a series of challenging tasks to test the
limits of teleoperation. The best systems that completed the challenges were
rewarded with a prize pool of $10 million. At its core, the challenge involves
the direct teleoperation of a robot with minimal latency and the capacity to
experience the environment from the robot’s perspective. However, direct
teleoperation still places a heavy physical and mental strain on the user, as
Pettinger et al. [4] reported that a user performing a pick and place task was
faster and had fewer errors while reporting the task was more accessible when
shared autonomy systems were enabled. Hence, researchers from HCI, medicine,
robotics, and others, have explored different means of control for
teleoperation to address these limitations. While most research efforts focus
on a physical control device such as a Virtual Reality Controller [5, 6], a
Joystick [7, 8], or phone [9], others decided to use cameras to track the
whole body [10, 11], or just the gaze [12]. There has yet to be an overall
consensus on the most appropriate type of control for direct teleoperation
with specific applications requiring specific implementations.

In this paper, we develop a modular and plug-and-play direct teleoperation
framework called TELESIM that non-experts can use without specialised training
using off-the-shelf Virtual Reality (VR) technologies. Specifically, TELESIM
objective is to allow for the direct teleoperation of any robotic arm using a
digital twin as the interface between the user and the robotic system. We then
demonstrate TELESIM’s user-friendliness using a user study and the users’
success rate at completing the task using two different types of control and
grasping systems. Specifically, we use a virtual reality controller and a
finger mapping hardware controller mounted on two robotic manipulators using
different grasping systems. We compare their performance to study whether
additional degrees of freedom in the control scheme enhance performance while
performing a simple task. Our contributions are:

  * •

A modular and plug-and-play framework for teleoperation for any robotic arm
using a digital twin.

  * •

An experimental validation for testing the framework’s performance through a
simple non-expert task.

  * •

A rigorous evaluation involving 37 participants demonstrating the user-
friendliness of TELESIM.

Figure 1: Our modular and plug-and-play TELESIM framework is being used to
control a UR3 Robot (top-left) and a Baxter Robot (top-right) and its digital
twin (bottom-right). The robot’s digital twins can be seen underneath their
respective real robots Figure 2: Overview of the experimental setup. The Steam
Index VR Headset [13] is marked as (1) on the far left, which acts as the
world’s origin. The Baxter robot on the left (2) is controlled by the Steam
Index controller (5). In front of it, the UR3 is on the right (3), with the
Yale OpenHand T42 gripper [14], controlled by the Senseglove and HTC Vive
tracker (4) on the left side of the brown table. Additionally, in the upper
right corner (7), a bird eye view of the task, which consists of 3 cubes in a
triangle pattern (described in Section IV), while on the brown table, the
cubes are arranged in the goal configuration (6)

##  II Background

Direct teleoperation is considered a stepping stone for shared autonomy [15].
This is because direct teleoperation causes significant cognitive strain on
the user [4], and the user may not be capable of millimetre-scale adjustment
to the position of the robot end effector. While in medicine, the user’s
movement is scaled down to allow for more precision [16, 6], it may not be
suitable for all types of manipulation tasks as some require significant arm
movements to move an object from one place to another.

Hence, researchers have explored different control methods to reduce the
cognitive strain while giving the highest amount of precision. For instance,
low degree-of-freedom control methods such as a keyboard [17], a joystick [18,
7], a touchscreen [19], or a gamepad[20] have brought an improved level of
control [17] to address the user’s mental strain. However, with the advent of
VR technologies, researchers have investigated whether these technologies are
appropriate for direct teleoperation. For example, they have proposed using a
VR controller such as [21, 5] or a phone [22]. While others have investigated
the use of motion mapping of the user’s body [10, 23] or only gaze control
[12]. However, for the latter, the added mobility generates a higher cognitive
load [4], and mapping motions to robot movements is challenging due to
differences in kinematics chains between robot arms and users [24].

Recently, Gottardi et al. [25] have investigated combining multiple control
systems, such as a VR controller and a tracking band on the upper arm, to
track the user’s movements. Rakita et al.[6] also compared various control
methods; a stylus, a touchscreen, and a VR controller. These were then
integrated into a custom inverse kinematics solver that adjusted the tolerance
level when matching the end-effector pose to that of the user. The authors
showed that users preferred the VR controller as they were more successful at
completing pick-and-place tasks, such as picking up bottles or plates.

To mitigate the limitation of direct teleoperation, researchers have focused
on how much shared autonomy improved the success of a given task. For this,
research works have aimed at comparing direct teleoperation with respect to an
assisted version to analyze the impact of shared autonomy on task success. For
example, Chen et al. [10] created a system in which the operator, using a
joystick, manipulated the robot’s end effector to an object, and then the
robot could either grasp the object autonomously or assist the user in fine-
tuning the robot position for a more optimal grasp. Later, [11, 4, 25] built
on [10] where the user teleoperated the robots directly to a planned position
but allowed the robot to perform the grasp automatically or, in the case of
[4], turn a valve handle. Lin et al. and Gottardi et al. conducted a user
survey and confirmed that users preferred the shared autonomy approach, as it
reduced complexity and mental strain. Furthermore, [25] observed results
similar to [8], who hypothesised that users preferred to give up control if it
meant increasing the task completion rate. However, Javdani et al. [8, 26]
have falsified this hypothesis using a system similar to [11, 4, 25]. The
authors concluded that users preferred to lose control if it meant an increase
in a task’s success rate only for a more complex task, while, for simple
tasks, users still preferred to have more control. These works have focused on
one robotic system and conducted their experimental survey on a small user
base (between 8 and 12 participants). Furthermore, they focused on different
autonomy levels and not on different control methods.

Figure 3: Overview of TELESIM, our direct teleoperation framework. The
controllers (in the black dotted line) can be any system that outputs a 3D
pose, while our framework is depicted in the blue dotted line. Finally,
TELESIM can be plugged into any robotic system via a ROS2 robot controller, as
shown in the red dotted line.

Although this paper focuses on TELESIM as a framework, our evaluation also
addresses two main limitations of previous work: (1) researchers have only
used one robot per study, and (2) most user studies consider a small user
base, which does not represent a statistically significant sample. It also
addresses a gap discussed by Rea & Seo [1], which states that there needs to
be more non-expert evaluation of robotic teleoperation for general tasks such
as picking and placing common objects. Therefore, to advance the state-of-the-
art in robotic teleoperation, we investigate the performance of different
control modalities for direct teleoperation using a VR controller and finger
mapping. Similarly, we evaluate our framework on 2 different robots, a Rethink
Robotics Baxter and a Universal Robotic 3, and ask 37 non-expert participants
to carry out a simple pick-and-place task. Additionally, by using two
different control modalities, our goal is to bridge the gap between VR control
methods[21, 5] and complete body mapping [10, 11] by investigating the
performance of hand and finger tracking through a SenseGlove.

##  III Direct Teleoperation Framework

TELESIM (Fig. 3) consists of 3 components. The controller (1) can be any type
of control system that outputs a 3D pose. In this paper, the controller is
either a VR controller or a SenseGlove with an HTC Vive Tracker. Our framework
(2) calculates a motion plan to the pose given by (1) and sends it through
ROS2 (Robot Operating System) [27] to any type of robotic arm that supports
ROS control (3). In this paper, we use either a Baxter robot or a UR3 robot.
The following subsections detail the design and implementation of each
component.

###  III-A Hardware Setup

The hardware used in this paper consists of a SteamVR [28] tracking a Steam
Index controller and HTC Vive Tracker, as shown in Figs. 2 points 5 and 4,
respectively. The Baxter robot (point 2 in Fig. 2) is controlled using the
Steam Index controller, and the gripper can be closed by pressing the main
trigger on the Steam Index controller. The UR robot (point 3 in Fig. 2) is
controlled using a Senseglove development kit, which allows for mapping of all
individual finger movements, with an HTC Vive Tracker mounted on top and is
linked to a T42 gripper from the Yale OpenHand project [14]. These control
methods are represented as point 1 in Fig. 3

The T42 gripper is an underactuated 2-finger gripper that we modified to have
a 1 degree of freedom per finger. The interface with the Senseglove is thus
restricted to movements of the user’s thumb and index finger, and these are
mapped to a single degree of freedom to result in the opening and closing of
the gripper. We adopted this approach to compare the Steam Index controller
and the Senseglove fairly, but still allowed us to test if the increase in
complexity of adding an extra degree of freedom on the user side has effects
on the success rate.

We also developed a custom controller for the gripper using two separate
Arduino UNOs, as they only have one serial port per board. Joint positions are
converted from degrees to motor commands, which are then sent to the first
Arduino through a serial interface. Then, commands are relayed to the second
Arduino through I2C and transmitted to the two MX-28 Dynamixels[29]. The
second Arduino is also responsible for reading the motors’ current position
and load and transmitting it back to the computer. Additionally, we restricted
the amount of force the user can transmit to the gripper, as without it, it
leads to breakage of the controlling string or exceeding the amount of
resistance allowed by the motor. Since we were interested in developing a
simple task that non-experts can carry out, we decided not to implement haptic
feedback as it would give the users an advantage of sensing whether an object
is grasped. Thus, this will result in an unfair comparison between the VR
controller and the SenseGlove. Therefore, we leave haptic feedback for future
work.

The VR headset (1 in Fig. 2) acts as the origin of both robots, giving the
user an easy reference point for teleoperation. The SteamVR outputs the
controller position in a 3D space with respect to the headset. The origin is
thus transformed into the user’s resting hand position when initiated. This
method of tracking the position is preferred by multiple researchers[4, 30],
as well as many of the participants in the ANA Avatar XPRIZE Challenge [5,
31]. The Senseglove is also used by the winning team [31], but to control a
Schunk robotic hand that replicates a human hand.

###  III-B Digital Twin

The position in the 3D space from the SteamVR is transmitted through ROS2 to a
full digital twin created in NVIDIA Isaac Sim[32]. This flow of information
can be seen in Fig. 3 point 2. Isaac Sim, a recent ray-tracing simulation
software, is used to calculate the robot motion plan using RMPFlow [33], which
is a motion generation based on Riemannian Motion Policies [34] (Fig. 3 point
3). Isaac Sim was chosen as it is the most realistic simulation software
compatible with ROS2 from our simulation benchmark [35].

Isaac Sim takes in a URDF of a robot for visualisation along with a robot
description file, describing the joints that can be actuated by the motion
planner and the robot’s collision as spheres, as Isaac Sim uses them for
collision checking. This file can be created with an included extension (Lula
Robot Description Editor[36]). The ability to add new robots is a significant
part of what makes TELESIM modular, along with ROS2.

We decided to use ROS2 as it is the most used framework for controlling a
variety of robots, making our framework plug-and-play. Views of the UR3 and
the Baxter robot from Isaac Sim are shown in 1. The grey square in between the
gripper (Fig. 3 point 4) is the point that is controlled by the teleoperation
system and indicates where Isaac Sim must find a path.

The fact that Isaac Sim acts as a complete digital twin allows the robot to
avoid collision with the world around it and damage itself. Additionally, both
robots have systems that allow them to work alongside humans. Our system is
capable, with minimal configuration, of handling the restrictions placed by
their need to be safe around humans. Finally, Isaac Sim transmits the position
of each joint to the real robot through ROS2 and passes this information to
the robotic system, as shown in Fig. 3 points 5 and 6.

###  III-C Robot Control

ROS control [37], provides a wrapper that facilitates the interface of
different robots. However, each robot needs to be adapted to work with
specific hardware. For this paper, we implemented the Universal Robot ROS2
Control package and used it to transfer the joint states from Isaac Sim to the
robots (Fig. 3 point 5).

Specifically, for the UR3, the size of the gripper and the safety regulations
of the laboratory reduced the available workspace for the robot. We handled
these limitations by adding safety planes and limitations on the range of
motion of the real robot. For Baxter, we used a ROS1 to ROS2 bidirectional
bridge, as Baxter only works with ROS1 internally. The pipeline is thus
converting Isaac Sim’s joint states into Baxter messages in ROS2 and then they
are sent through the bridge as ROS1 messages to the robot. The robot outputs
its current state, which is converted to ROS2 through the same bridge.

Our framework introduces a slight amount of lag (500 ms) between the user
movement and the robot movement, partly due to the path planning step, the
time taken by the actuator to move the robot arm to the desired position and
mostly due to safety restrictions that caps the maximum speed of our robots.
The faster the user moves from one position to another, the higher the delay
as the arm tries to catch up. This is easily accounted for by the user by
making small and slow movements, as confirmed by Schwarz and Behnkle [38].
Although we acknowledge lag is present in our system, Schwarz and Behnkle [38]
found that minor delays do not impact performance, and none of our users
mentioned lag as an issue in our experiments (see Section V).

##  IV Experiments

###  IV-A Methodology

As described in Section III-A, the user controls a robot by teleoperating it
using either a VR controller or the SenseGlove. For this experiment, we
consider a simple task where the user has to pick up 3 cubes on a table. The
user then needs to bring them individually to the centre of the table to
complete a tower. Once the tower is completed, the cubes are returned to their
original position, and the user is asked to repeat the task as many times as
possible within 10 minutes.

The task definition consists of 3 cubes positioned at each vertex of an
isosceles triangle and at similar distances from the robots’ base on each
robot’s table (Fig. 3 point 7). We have placed markers of the vertices of this
isosceles triangle on each robot’s operating table for repeatability of the
task between attempts and among users. The front cube at the top of the
triangle is positioned such that it is at the maximum reach limit of each
robot’s overhead rotation. This means that the end-effector’s z-axis is
perpendicular to the table, which makes it difficult for the user to pick up
the cube from overhead. Thus, the user has to add some rotation in the x- or
y-axis of the gripper to pick the cube successfully. The right cube is the
furthest away from the user in both robots and adds a degree of difficulty due
to the user’s viewpoint, but still within reach. Finally, the left cube is
placed such that the user has to move their body. That is, for Baxter, the
location where the user needs to pick up the left cube is approximately at the
waist of the user, while for the UR3, the location is on the other side of
(farthest from) the headset. The cube positions were chosen to let users be
spatially aware of their position and its relationship to the robot. Finally,
the location where the users need to stack the three cubes is easily
accessible by the robot, and this position is marked by a red square in red
tape, 2 cm larger than the cube. This position can be seen in Fig. 2 point 6,
with the cubes stacked as required.

In order to teleoperate using the Baxter robot, the users need to stand with
their back to the VR headset, while for teleoperating using the UR3, users
need to stand with the headset on their right and the Baxter robot behind
them. This difference in position is due to the space constraint of the room
in which we ran our experiments. The operating room can be seen in Fig. 2,
with the headset on the left of the picture shown as point 1.

###  IV-B User Survey

In our experiments, we asked 37 participants (29 male and 8 female) from
various backgrounds aged 19 to 51 (mean: 25.32, 1 standard deviation: 6.26) to
teleoperate both robots and stack 3 cubes without a monetary reward.
Participants reported having, on a 5-point Likert scale (going from
”Experienced” with a score of 1 to ”No Experience” with a score of 5), a 3.03
mean experience with Virtual Reality with a standard deviation of 1.2. They
also reported having a mean experience of 3.24 with a robot with a standard
deviation of 1.24

Each participant completed the short questionnaire described above at the
beginning of the experiment. After being asked to position their back to the
VR headset, an explanation was given on how to control the robot, emphasising
that all of their hand movements and rotation will be mapped one-to-one to the
robot. They were instructed to try to grasp a cube from both sides. They had 5
minutes to get used to the control without a specific task objective. Most of
the participants picked up and placed a cube during this time.

After 5 minutes, the participants performed the task of stacking the 3 cubes
in the given location without any restriction on the cubes’ pose and order.
Users were asked to stack cubes as many times as possible in 10 minutes. Once
a tower has been completed, we reset the cubes to their initial configuration.
Users’ actions were recorded, such as the time taken for each tower and for
individual actions for each pick, place, and drop (i.e. failures).

After 10 minutes, users were given the option to take a break while answering
the Single-Ease Question (SEQ) [39]. Then, they were asked to repeat the same
experiment but with the UR3 robot. SEQ was chosen instead of other metrics
such as the System Usability Scale (SUS) [40] as Hodrien and Fernando [39]
have argued that it is a good end-of-task metric.

##  V Evaluation

Figure 4: Average Time Taken and Percentage of Population for each Tower
Completed for Baxter Figure 5: Average Time Taken and Percentage of Population
for each Tower Completed for UR

Fig. 4 shows that $85\%$ of the participants can build at least one tower in
10 minutes using Baxter and the VR controller. However, there is a steady
decline for each of the following towers, with only $5\%$ of the users able to
build 8 towers. This is in direct comparison to the UR3 as shown in Fig. 5,
with slightly less than $50\%$ of the population failing to build one tower
and $5\%$ managing to build 4, half as many towers for Baxter.

The box plots in Fig. 4 and 5 show the average and variance of the time taken
by users to complete the towers. In particular, the first tower for both
robots took most of the task duration because some participants could not
build one tower. This time completion trend shows TELESIM’s user-friendliness
as $60\%$ of users for Baxter managed with minimal training to complete a full
tower, which means 3 different pick-and-place operations in around 2 minutes.
Similarly, this can also be observed for the UR3 as $25\%$ of the users
managed to build a tower in 4 minutes.

TABLE I: Additional Statistics Collected

(a) Baxter  | Min | Mean $\pm$ Std | Max
---|---|---|---
Placing Rate | 25.00% | 77.42% $\pm$ 15.54% | 100.00
Dropping Rate | 3.70% | 23.83% $\pm$ 14.08% | 66.67
Collapse Rate | 5.56% | 18.44% $\pm$ 11.66% | 57.14
Still in Place Rate | 24.31% | 75.21% $\pm$ 15.20% | 95.92

(b) UR | Min | Mean $\pm$ Std | Max
---|---|---|---
Placing Rate | 12.50% | 46.29% $\pm$ 17.97% | 86.67
Dropping Rate | 13.33% | 53.37% $\pm$ 18.63% | 87.50
Collapse Rate | 4.76% | 22.25% $\pm$ 12.46% | 50.00
Still in Place Rate | 14.88% | 46.93% $\pm$ 16.75% | 84.44

Table I shows the additional statistics collected during the experiment, such
as the percentage of times the user dropped a cube that caused the tower to
collapse. Table I(a) indicates that for the Baxter robot, $75\%$ of the
picking actions resulted in a correctly placed cube that did not collapse due
to incorrect placement or the user inadvertently moved the robot in the
tower’s path. Similarly, in Table I(b), $46\%$ of all the picking actions
resulted in a correct place. The difference in the number of towers built,
shown in Fig. 5, can be explained by a greater difficulty in picking the cube.
Specifically, our results indicate that there is no significant difference in
the difficulty of picking a cube ($P>0.05$), nor is there a difference in the
amount of time that the user collapses a tower while placing a cube ($P>0.1$).
This lack of difference shows the stability of the teleoperation, as the
difference in placing rates and the number of towers can be explained by the
difference in control modality and the difference in robots.

The collapse rate in both Table I(a) and Table I(b) is similar and indicates
that the type of robot does not influence the difficulty in safely placing a
cube in a specific spot. However, the difference in drop rate for the UR3 can
be related to the limitations of the gripper described in Section III-A, such
as the limitation of the grip strength of the closed finger, to prevent the
cable from breaking, and the limited range of motions, since we observed that
some users let the cube fall while the gripper was closed. However, successful
users moved slowly to prevent unnecessary movement, thus reducing the risk of
dropping. This limitation is also visible in the placing rate; the placing and
dropping rates are complementary, as these are the only two outcomes after
picking up a cube.

Results of the Single Ease Question asked at the end of each task, in which a
higher score means that TELESIM is easy to use, can be seen in Fig. 6. They
show that the user was able to detect how well they performed and that their
estimate is consistent with the result shown in Fig. 4 and Fig. 5.
Specifically, Baxter obtained a mean of 3.32 with a standard deviation of
1.27, while UR3 obtained a mean of 2.19 with a standard deviation of 1.14.
Furthermore, Fig. 6 shows that no user gave the maximum score for UR3, while
they did for Baxter. Additionally, the UR3 has a sharp decline in score after
a SEQ score of 3, while Baxter’s is more spread out.

Figure 6: Single Ease question violin plot for Baxter (orange) and UR3 (blue)
(Higher number means easier to use)

##  VI Conclusion and Future Works

In this paper, we have investigated the performance of TELESIM by conducting a
medium-scale user survey with 37 participants who were asked to build towers
of 3 cubes by teleoperating robots. We tested TELESIM’s modularity on two
different robots with two different control modalities. Our experimental
results show that TELESIM is modular, plug-and-play and user-friendly, as not
only were we able to deploy it on 2 robots with different modalities, but most
users were able to succeed by building at least once a tower of 3 cubes, with
only 5 minutes of training, regardless of the control modality or robot used.
We thus bridged the gap pointed out by Rea & Seo [1], where they state that
there is a lack of non-expert evaluation of robotic teleoperation for general
tasks such as picking and placing common objects. TELESIM is available on
GitHub111https://github.com/cvas-ug/telesim_pnp, allowing developers to
perform teleoperation on their robots with minimal setup time.

Our underlying motivation for choosing direct teleoperation in this paper is
to establish a baseline for further research on shared autonomy, which could
combine human intuition and a high-level overview of a task while giving
freedom to the robot to perform, for example, accurate picking and placing
objects. Additionally, we plan to remove the constraint of having the VR
headset behind the user and allow them to wear the headset to operate either
in VR in the digital twin view of Isaac Sim or Augmented Reality by allowing
the user to move around the robot and have different viewpoints while
manipulating, thus enhancing the precision of the teleoperation. However, the
choice of control input is fundamental to success. Future work consists of
carrying out a survey using the VR controller and the UR3 to dissociate the
robot and control method; as for our current evaluation, we hypothesise they
are closely linked.

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# On set-convergence statistically modulated

María Pilar Romero de la Rosa  Department of Mathematics University of Cádiz
Avda. de la Universidad s/n 11403-Jerez de la Frontera. Spain.
<EMAIL_ADDRESS>
###### Abstract

We explore some convergence notions for set-convergence coming from modern
summability methods. Specifically we will see the connections between Wijsman
$f$-statistical convergence and Wijsman $f$-strong Cesàro convergence, when
$f$ is a modulus function that induces a density in $\mathbb{N}$.

###### keywords:

Set convergence, Wijsman convergence, statistical convergence
MSC , 40H05 , 40A35

##  1 Introduction

Set-convergences show up naturally in Mathematics. A century ago, it appear
for the first time with the work of Painlevé (1902), that was supported and
disseminate by Hausdorff and Kuratowski in their 1927 and 1933 books,
respectively (see [14, Chapter 4] for historical details). In the twenty
century, the study of set convergence was centered on topological aspects by
considering a set as a point in a topological space. These studies were
collected on Beer’s monograph (see [2]). Currently, an increasing number on
applied variational problems and the constant queries coming from the modern
data science, have renewed the interest in set-convergence from an applied
point of view.

Imagine that we wish to know if a sequence of closed data sets $(A_{n})_{n}$,
obtaining experimentally, converges to a given closed subset $A$. Assume that
every subset is contained in a metric space $(X,d)$, and let us denote by
$CL(X)$ the family of all closed subsets of $X$. A way to see convergence of
sets is to consider for any $x\in X$ the distances $(d(x,A_{n}))_{n}$ and to
check if they converges to $d(x,A)$. In this case the sequence $A_{n}$ is said
to be Wijsman convergent to $A$. However this is a strong condition, because
the data sets were obtained experimentally and probably many data were
collected erroneously. One way to relax this situation is to consider weak
convergence methods.

For instance, a sequence $(x_{n})_{n}$, in a normed space, is said to be
statistically convergent to $L$ if for any $\varepsilon>0$ the subset
$A_{\varepsilon}=\\{k\leq n\,:\,\|x_{n}-L\|>\varepsilon\\}$ has zero density
on $\mathbb{N}$. And with this weaker convergence in hand, Nuray and Rhoades
[13] relaxed the concept of Wijsman convergence by saying that a sequence of
closed subsets $(A_{k})_{k}$ is Wijsman statistically convergent to $A$ if for
any $x\in X$ the distances $(d(x,A_{n}))_{n}$ converges statistically to
$d(x,A)$. They also connect this concept with the concept of Wijsman strong
Cesàro convergence which is a reminiscent of the classical strong Cesàro
convergence. That is, a sequence $(A_{k})_{k}\subset CL(X)$ converges Wijsman
strong Cesàro to $A\in CL(X)$ provided for any $x\in X$ the sequence of real
numbers $(d(x,A_{n}))_{n}$ converges strong Cesàro to $L=d(x,A)$, that is:

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}|d(x,A_{k})-d(x,A)|=0$

However, sometimes statistical convergence is a drastic way to filter data.
And what is needed is to filter the data in a more accurate way. We can obtain
a fine filter data by means of a modulus function. Indeed, a modulus function
will be a function $f\,:\,\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying

  1. 1.

$f(x)=0$ if and only if $x=0$.

  2. 2.

$f(x+y)\leq f(x)+f(y)$ for every $x,y\in\mathbb{R}^{+}$.

  3. 3.

$f$ is increasing.

  4. 4.

$f$ is continuous from the right at $0$.

Attempts to obtain convergence methods by means of modulus functions date back
to the works of Maddox [12] in the nineties. The study of convergences through
modulo functions has a wide applicability, not only for the case of
convergence of sets. For instance, it can be used by F. García-Pacheco and R.
Kama for refinements of the concept of derivative. And it have been used by C.
Belen adn Yildrim to see new points of view for convergence of power series
([1, 3, 8]).

We will suppose that the modulus functions are unbounded to avoid
trivialities. Let us denote by $\\#(A)$ the cardinal of a finite subset $A$.
More recently, these weak convergences are transferred to convergence of sets
in [5]. For instance, a sequence $(A_{n})n\subset CL(X)$ is said to be Wijsman
$f$-statistically convergent to $A$ if for any $x\in X$ the sequence of
distances $(d(x,A_{n}))_{n}$ converges $f$-statistically to $d(x,A)$ that is,
for any $\varepsilon>0$

$\lim_{n\to\infty}\frac{1}{f(n)}f\left(\\#\\{k\leq
n\,:\,|d(x,A_{n})-d(x,A)|>\varepsilon\\}\right)=0.$

Moreover, in [5] there is a great effort initiated by V. K. Bhardwaj and S.
Dhawan, to relate the $f$-statistical Wijsman convergence with the strong
Cesàro convergence modulated by a modulus function, introduced by Maddox [12].
However, as seen in previous studies [10, 11, 15], in order to obtain such a
structure between both convergences, it is necessary to introduce a smooth
change in the notion of strong $f$-Cesàro convergence introduced by Maddox.

If it is used lacunary sequences in the framework of the classic concepts of
statistical convergence and strong Cesàro convergence, it is also possible to
open or close the degree of precision in filtering data sets. Freeman and
Sember, in their 1978 visionary paper and inspired by the classical studies on
Cesàro convergence by Hardy-Littlewood and Fekete, studied the concept of
strong lacunary convergence in [6]. These studies were refined by Fridy and
Orhan [7] by relating them to a new concept: lacunary statistical convergence,
where it underlies new ways of measuring the density of subsets in
$\mathbb{N}$.

Ulusu and Nuray [16] were the first to explore these convergences within the
the framework of convergence of sets, studying the Wijsman lacunary
statistical convergence and the Wijsman lacunary strong Cesàro convergence.
These rich Wijsman lacunary convergences were sharpened by V. K. Bhardwaj, S.
Dhawan [4] using modulus functions.

In this paper we will complete the studies initiated in [5] and [4] by V. K.
Bhardwaj, S. Dhawan and coworkers by fully characterizing the connection
between Wijsman $f$-statistical convergence and Wijsman $f$-strong Cesàro
convergence. We explore the modulus functions that offer set-convergence
methods which are less permissive with the errors obtained experimentally.
Specifically this phenomenon happens when the modulus function $f$ is not
compatible. On the contrary, when the modulus functions are compatible (see
Definition 2.1) then the Wijsman $f$-statistically convergence is exactly the
Wijsman statistically convergence introduced by Nuray and Rhoades [13]. We
analyze also the existing structure between the Wijsman convergence of sets
when it is induced by lacunary $f$-statistical convergences.

The paper is structured as follows. In Section 2 we will explore the
connections between Wijsman $f$-statistical convergence and Wijsman $f$-strong
Cesàro convergence. In Section 3 we fully characterize the relationship
between the Wijsman $f$-statistical convergence and the Wijsman $f$-strong
Cesàro convergence in the lacunary setting.

##  2 Compatible modulus functions versus Wijsman $f$-statistical convergence

In [10] the concept of compatible modulus functions was introduced. Basically,
for the compatible modulus functions it was obtained in [10] that
$f$-statistical convergence and statistical convergence are equivalent.

######  Definition 2.1.

Let us denote by
$\varphi(\varepsilon)=\limsup_{n}\frac{f(n\varepsilon)}{f(n)}$. A modulus
function $f$ is said to be compatible if $\lim_{\varepsilon\to
0}\varphi(\varepsilon)=0$.

######  Remark 2.2.

Examples of compatible modulus functions are $f(x)=x^{p}+x^{q}$, $0<p,q\leq
1$, $f(x)=x^{p}+\log(x+1)$, $f(x)=x+\frac{x}{x+1}$. And $f(x)=\log(x+1)$,
$f(x)=W(x)$ (where $W$ is the $W$-Lambert function restricted to
$\mathbb{R}^{+}$, that is, the inverse of $xe^{x}$) are modulus functions
which are not compatible. By taking inverses, there is a plenty number of non-
compatible modulus functions. For instance, we consider the inverse of the
functions $x^{p}e^{x}$, which are generalized $W$-Lambert functions.

Let us show that $f(x)=x+\log(x+1)$ is compatible.

$\lim_{n\to\infty}\frac{f(n\varepsilon^{\prime})}{f(n)}=\lim_{n\to\infty}\frac{n\varepsilon^{\prime}+\log(1+n\varepsilon^{\prime})}{n+\log(n+1)}=\varepsilon^{\prime}$

On the other hand if $f(x)=\log(x+1)$, since

$\lim_{n\to\infty}\frac{\log(1+n\varepsilon^{\prime})}{\log(1+n)}=1$

we obtain that $f(x)=\log(x+1)$ is not compatible.

To simplify the language, let us denote by
$A_{n}\overset{WS}{\longrightarrow}A$ when $(A_{n})_{n}$ Wijsman statistically
converges to $A$ and by $WS$ the set of all sequences in $CL(X)$ that are
Wijsman statistically convergent. Analogously we denote by
$A_{n}\overset{WS^{f}}{\longrightarrow}A$ if $(A_{n})_{n}$ is Wijsman
$f$-statistially convergent to $A$ and by $WS^{f}$ the set of all sequences in
$CL(X)$ that are Wijsman $f$-statistically convergent.

From the point of view of data filtering, which is our main interest in this
work, the following result shows that the interesting modulus functions are
those that are not compatible.

######  Theorem 2.3.

Let $(X,d)$ be metric space. Assume that $f$ is a compatible modulus function,
and $(A_{k})_{k}\subset CL(X)$, $A\in CL(X)$. The following conditions are
equivalents:

a)

$(A_{k})_{k}$ converges Wijsman $f$-statistically to $A$.

b)

$(A_{k})_{k}$ converges Wijsman statistically to $A$.

###### Proof.

Condition $a)$ implies $b)$ follows without the hypothesis on compatibility of
$f$ and it was proved in [5, Theorem 2.1].

Now assume that $A_{n}\overset{WS}{\longrightarrow}A$. For each $x\in X$ we
have that $(d(x,A_{n}))_{n}$ converges statistically to $d(x,A)$. Since $f$ is
compatible, according to Proposition 2.7 in [10] we get that
$(d(x,A_{n}))_{n}$ converges $f$-statistically to $d(x,A)$ for any $x\in X$,
that is, $A_{n}\overset{WS^{f}}{\longrightarrow}A$ as desired. ∎

In brief, when $f$ is compatible then $WS=WS^{f}$. Moreover when this
phenomenon happens then $f$ must be compatible.

######  Theorem 2.4.

Assume that for some $f$ we have $WS=WS^{f}$ then $f$ is compatible.

###### Proof.

Indeed, if $f$ is not compatible then there exists $c>0$ and a subsequence
$\varepsilon_{k}$ such that
$\varphi(\varepsilon_{k})=\limsup_{n}\frac{f(n\varepsilon_{k})}{f(n)}>c$. In
fact since $\varphi(\cdot)$ is increasing, we have that
$\varphi(\varepsilon)>c$ for any $\varepsilon>0$.

Let us fix a decreasing sequence $\varepsilon_{k}$ converging to $0$. For each
$k$ we can select $m_{k}$ large enough such that $f(m_{k}\varepsilon_{k})\geq
cf(m_{k})$. We can select $m_{k+1}$ inductively satisfying

$1-\varepsilon_{k+1}-\frac{1}{m_{k+1}}>\frac{(1-\varepsilon_{k})m_{k}}{m_{k+1}}$
(1)

Now the idea is to construct a subset with prescribed density. Let us denote
$\lfloor x\rfloor$ the integer part of $x\in\mathbb{R}$. Set $n_{k}=\lfloor
m_{k}\varepsilon_{k}\rfloor+1$. Extracting a subsequence if it is necessary,
we can assume that $n_{1}<n_{2}<\cdots$, $m_{1}<m_{2}<\cdots$. Let us define
$A_{k}=[m_{k+1}-(n_{k+1}-n_{k})]\cap\mathbb{N}$. Condition (1) is to guarantee
that $A_{k}\subset[m_{k},m_{k+1}]$. Let us denote $A=\bigcup_{k}A_{k}$.

Without loss of generality, we can suppose that the metric space is the
complex plane with the euclidean metric. Let us consider the following
sequence of subsets:

$B_{k}=\begin{cases}\\{1\\}&k\in A\\\ \\{0\\}&k\notin A.\end{cases}$

Assume that $B=\\{0\\}$, and let us see that
$B_{k}\overset{WS^{f}}{\nrightarrow}B$. Indeed, if we take $x_{0}=0$ then

$d(0,B_{k})=\begin{cases}1&k\in A\\\ 0&k\notin A\end{cases}$

and $d(0,B)=0$ therefore

$\displaystyle\frac{f(\\#\\{k\leq
m_{k}\,/|d(0,B_{k})-d(0,B)|>1/2\\})}{f(m_{k})}$ $\displaystyle=$
$\displaystyle\frac{f(n_{k})}{f(m_{k})}$ $\displaystyle\geq$
$\displaystyle\frac{f(m_{k}\varepsilon_{k})}{f(m_{k})}\geq c.$

Now, let us show that $B_{k}\overset{WS}{\rightarrow}B$. Indeed, for any $m$,
there exists $k$ such that $m_{k}<m\leq m_{k+1}$. Moreover, we can suppose
without loss that $m\in A$, that is, $m_{k+1}-n_{k+1}+n_{k}\leq m$. Let us
consider $x\in\mathbb{C}$ we can suppose without loss that $|1-x|\neq|x|$.
Thus for any $\varepsilon>0$:

$\displaystyle\frac{\\#\\{l\leq m\,:\,|d(x,B_{l})-d(x,B)|>\varepsilon\\}}{m}$
$\displaystyle\leq$ $\displaystyle\frac{\\#\\{l\leq
m_{k}\,:\,|d(x,B_{k})-d(x,B)|>\varepsilon\\}}{m_{k}}$
$\displaystyle+\frac{n_{k+1}-n_{k}}{m_{k+1}-n_{k+1}+n_{k}}$
$\displaystyle\leq$
$\displaystyle\frac{n_{k}}{m_{k}}+\frac{1}{\frac{m_{k+1}}{n_{k+1}-n_{k}}-1}\to
0$

as $k\to\infty$, which proves the desired result. ∎

Now, we turn our attention to the Wijsman $f$-strong Cesàro convergence. Let
us recall that a sequence $(x_{n})_{n}$ is said to be $f$-strong Cesàro
convergent to $L$ if

$\lim_{n\to\infty}\frac{f\left(\sum_{k=1}^{n}|x_{k}-L|\right)}{f(n)}=0.$

The above definition suggests the following notion.

######  Definition 2.5.

Assume that $(X,d)$ is a metric space, $(A_{k})_{k}\subset CL(X)$ and $A\in
CL(X)$. We say that $(A_{k})_{k}$ is Wijsman $f$-strong Cesàro convergent to
$A$ if for any $x\in X$ the sequence $(d(x,A_{k}))_{k}$ is $f$-strong Cesàro
convergent to $d(x,A)$.

A result similar to Theorem 2.3 is true for the Wijsman $f$-strong Cesàro
convergence.

######  Theorem 2.6.

Let $(X,d)$ be metric space. Assume that $f$ is a compatible modulus function,
and $(A_{k})_{k}\subset CL(X)$, $A\in CL(X)$. The following conditions are
equivalents:

a)

$(A_{k})_{k}$ converges Wijsman $f$-strong Cesàro to $A$.

b)

$(A_{k})_{k}$ converges Wijsman strong Cesàro to $A$.

###### Proof.

According to Proposition 2.3 in [10], since for any modulus function, a
$f$-strong Cesàro convergent sequence is strong Cesàro convergent, we get that
$a)$ implies $b)$ follows directly without using the hypothesis on
compatibility on $f$.

Now, for each $x\in X$ we have that $(d(x,A_{n}))_{n}$ is strong Cesàro
convergent to $d(x,A)$. Since $f$ is compatible, according to Proposition 2.8
in [10] we get that $(d(x,A_{n}))_{n}$ is $f$-strong Cesàro convergent to
$d(x,A)$ for any $x\in X$, that is, $(A_{n})_{n}$ is Wijsman $f$-strong Cesàro
convergent to $A$ as we desired. ∎

Let us denote by $WN$ the class of sequences of sets that are Wijsman strong
Cesàro convergent and by $WN^{f}$ the class of sequences which are Wijsman
$f$-strong Cesàro convergent. When $f$ is compatible then $WN=WN^{f}$. Now let
us shown the converse result.

######  Theorem 2.7.

Assume that for some $f$ we have $WN=WN^{f}$ then $f$ is compatible.

###### Proof.

Indeed, let us consider the sequences
$(\varepsilon_{k})_{k},(m_{k})_{k},(n_{k})_{k}$ and $(A_{k})_{k}$ provided by
Theorem 2.4. Let us denote $A=\bigcup_{k}A_{k}$.

Again without loss, we can suppose that the metric space is the complex plane
with the euclidean metric. Let us consider the following sequence of subsets:

$B_{k}=\begin{cases}\\{1\\}&k\in A\\\ \\{0\\}&k\notin A.\end{cases}$

Assume that $B=\\{0\\}$, and let us see that
$B_{k}\overset{WN^{f}}{\nrightarrow}B$. Indeed, let us consider $x_{0}=0$,
then

$\displaystyle\frac{f\left(\sum_{k=1}^{m_{k}}|d(0,B_{k})-d(0,B)|\right)}{f(m_{k})}$
$\displaystyle=$ $\displaystyle\frac{f(\\#\\{k\leq
m_{k}\,/|d(0,B_{k})-d(0,B)|>1/2\\})}{f(m_{k})}$ $\displaystyle=$
$\displaystyle\frac{f(n_{k})}{f(m_{k})}$ $\displaystyle\geq$
$\displaystyle\frac{f(m_{k}\varepsilon_{k})}{f(m_{k})}\geq c.$

However $B_{k}\overset{WN}{\rightarrow}B$. Indeed, since for any $x\in X$
$\sum_{k=1}^{m}|d(x,B_{k})-d(x,B)|=\\#\\{k\leq
m\,/|d(x,B_{k})-d(x,B)|>\varepsilon\\}$ and since by Theorem 2.4 the sequence
$(B_{k})_{k}$ is Wijsman statistically convergent to $B$, we obtain that
$(B_{k})_{k}$ is Wijsman strong convergent to $B$ as we desired to prove. ∎

Now, to the end of this section, let us explore the relationship between the
concept of Wijsman $f$-statistically convergence and Wijsman $f$-strong Cesàro
convergence.

To establish this connection we need the extra hypothesis introduced in [5]:
Wijsman boundedness of a sequence of subsets. A sequence $(A_{k})_{k}\subset
CL(X)$ is said to be Wijsman bounded if for any $x\in X$ the sequence
$(d(x,A_{n}))_{n}$ is bounded. Let us observe that by the triangular
inequality a sequence $(A_{k})_{k}\subset CL(X)$ is Wijsman bounded if there
exists $x_{0}\in X$ such that $(d(x_{0},A_{n}))_{n}$ is bounded. Let us
observe also that Wijsman boundedness notion does not imply boundedness of the
subsets $(A_{k})$.

In fact, this boundedness condition can be sharpened. Khan and Orhan in [9]
discovered that the classical result that connect statistically convergence
with strong Cesàro convergence can be established for uniform integrable
sequences. Let $Y$ be a normed space. A sequence $(x_{n})_{n}\subset Y$ is
said to be uniformly integrable if

$\lim_{c\to\infty}\sup_{n}\frac{1}{n}\sum_{\begin{subarray}{c}k=1\\\
\|x_{k}\|\geq c\end{subarray}}^{n}\|x_{k}\|=0.$

If a sequence $(x_{n})_{n}$ is uniformly integrable then $(x_{n}-L)_{n}$ is
also uniformly integrable for every $L\in Y$.

Our result pivots on this notion. Let us refine also the notion of Wijsman
boundedness:

######  Definition 2.8.

Let $(X,d)$ be a metric space. A sequence $(A_{k})_{k}\in CL(X)$ is said to be
Wijsman uniform integrable if for any $x\in X$ the sequence $(d(x,A_{n}))_{n}$
is uniform integrable.

Now, we are in position to state our result.

######  Theorem 2.9.

Assume $(X,d)$ is a metric space and $(A_{k})_{k}\subset CL(X)$, $A\in CL(X)$.

a)

If $(A_{k})_{k}$ is Wijsman $f$-strong Cesàro convergent to $A$ then
$(A_{n})_{n}$ is Wijsman $f$-statistically convergent to $A$ and $(A_{k})_{k}$
is Wijsman uniformly integrable.

b)

Assume that $f$ is compatible. If $(A_{k})_{k}$ is Wijsman $f$-statistically
convergent to $A$ and $(A_{k})_{k}$ is Wijsman uniformly bounded then
$(A_{k})_{k}$ is Wijsman $f$-strong Cesàro convergent to $A$.

###### Proof.

To show a), let us observe that for any $x\in X$ the sequence
$(d(x,A_{n}))_{n}$ is $f$-strong Cesàro convergent to $d(x,A)$. Therefore,
according to Theorem 3.2 in [10], the sequence $(d(x,A_{n}))_{n}$ is
$f$-statistically convergent to $d(x,A)$ and it is uniformly integrable. That
is, $(A_{k})_{k}$ is Wijsman $f$-statistically convergent to $A$ and Wijsman
uniformly integrable.

On the other hand, if for any $x\in X$ the sequence $(d(x,A_{n}))_{n}$ is
$f$-statistically convergent to $d(x,A)$ and it is uniformly integrable, since
$f$ is compatible, according to Theorem 3.4 in [10] we get that
$(d(x,A_{n}))_{n}$ is $f$-strong Cesàro convergent to $d(x,A)$. Hence,
$(A_{k})_{k}$ is Wijsman $f$-strong Cesàro convergent to $A$ as we desired. ∎

There is a converse of the above result. Let us denote by $WI$ the set of all
Wijsman uniformly integrable sequence of subsets. We have shown that for any
modulus function $f$, $WN^{f}\subset WS^{f}\cap WI$, and if $f$ is compatible
then $WN^{f}=WS^{f}\cap WI$.

######  Theorem 2.10.

Assume that $WN^{f}=WS^{f}\cap WI$ then $f$ is compatible.

###### Proof.

If $f$ is not compatible. Thus, as in the proof of Theorem 2.4 we can
construct sequences $(\varepsilon_{k})_{k}$, $(m_{k})_{k}$ such that
$f(m_{k}\varepsilon_{k})\geq cf(m_{k})$ for some $c>0$. Moreover, we can
construct $(m_{k})_{k}$ inductively, such that the sequence

$r_{k}=\frac{m_{k+1}\varepsilon_{k+1}-m_{k}\varepsilon_{k}}{m_{k+1}-m_{k}}$

is decreasing and converging to $0$.

Let us consider the following subsets:

$B_{l}=\begin{cases}\\{r_{k}\\}&l\in A_{k}\\\ \\{0\\}&k\notin A.\end{cases}$

Since $(r_{k})_{k}$ is decreasing, clearly $(B_{k})$ is Wijsman bounded and
therefore Wijsman integrable. Moreover since $(r_{k})_{k}$ is decreasing to
$0$, the sequence $(B_{j})_{j}$ is Wijsman statistically convergent to
$B=\\{0\\}$. However if we take $x_{0}=0$ then

$\frac{f\left(\sum_{l=1}^{m_{k}}|d(0,B_{l})-d(0,B)|\right)}{f(m_{k})}=\frac{f(m_{k}\varepsilon_{k})}{f(m_{k})}\geq
0$

which proves that $(B_{k})_{k}$ is not Wijsman $f$-strong Cesàro convergent to
$B=\\{0\\}$ as we desired to prove. ∎

##  3 $\Theta$-compatible modulus functions versus Wijsman lacunary
$f$-statistical convergence

By a lacunary sequence $\theta=\\{k_{r}\\}_{r\geq 0}\subset\mathbb{N}$ with
$k_{0}=0$, we will means an increasing sequence of natural numbers such that
$h_{r}=k_{r}-k_{r-1}\to\infty$ as $r\to\infty$. Assume that $f$ is a modulus
function. Denoting the intervals $I_{r}=(k_{r-1},k_{r}]$, a sequence $(x_{n})$
on a normed space $X$ is said to be $f$ strong lacunary convergent to $L$
provided (see [6])

$\lim_{r\to\infty}\frac{1}{f(h_{r})}f\left(\sum_{k\in
I_{r}}^{\infty}\|x_{k}-L\|\right)=0.$

and is said to be lacunary $f$-statistical convergent to $L$ (see [7]) if for
any $\varepsilon>0$ the subset
$A_{\varepsilon}=\\{k\in\mathbb{N}\,:\,\|x_{k}-L\|>\varepsilon\\}$ has
($\theta,f)$-density equal $0$. That is,

$d_{\theta,f}(A_{\varepsilon})=\lim_{r\to\infty}\frac{1}{f(h_{r})}f\left(\\#\\{k\in
I_{r}\,:\,\|x_{k}-L\|>\varepsilon\\}\right)=0.$

In [4] V. K. Bhardwaj, S. Dhawan applied these new convergence methods in the
study of convergence of sets. We say that a sequence $(A_{k})_{k}\subset
CL(X)$ is Wijsman lacunary $f$-statistical convergent to $A\in CL(X)$ if for
any $x\in X$ the sequence $(d(x,A_{k}))_{k}$ is lacunary $f$-statistical
convergent to $d(x,A)$. Let us denote by $WS_{\theta}^{f}$ the set of all
Wijsman $\theta$-lacunary $f$-statistical convergent sequences. Analogously we
say that $(A_{k})_{k}\subset CL(X)$ is Wijsman lacunary $f$-strong Cesàro
convergent to $A\in CL(X)$ if for any $x\in X$ the sequence $(d(x,A_{k}))_{k}$
is lacunary $f$-strong Cesàro convergent to $d(x,A)$. Let us denote by
$WN_{\theta}^{f}$ the set of all Wijsman $\theta$-lacunary $f$-strong Cesàro
convergent sequences.

Our results pivots on the following notion.

######  Definition 3.1.

Let us denote
$\varphi_{\theta}(\varepsilon)=\limsup_{t\to\infty}\frac{f(h_{t}\varepsilon)}{f(h_{t})}$.
We will say that $f$ is $\theta$-compatible if if $\lim_{\varepsilon\to
0}\varphi_{\theta}(\varepsilon)=0$.

Clearly, if $f$ is compatible then for any lacunary sequence $\theta$, $f$ is
$\theta$-compatible.

######  Theorem 3.2.

Let $(X,d)$ be a metric space and $(A_{k})_{k}\subset CL(X)$, $A\in CL(X)$.
Assume that $\theta=(k_{t})_{t}$ is a lacunary sequence.

a)

For any modulus function $f$, if $(A_{n})_{n}$ is Wijsman lacunary
$f$-statistical convergent to $A$ then $(A_{n})_{n}$ also converges Wijsman
lacunary statistical convergent to $A$.

b)

If $f$ is $\theta$-compatible and $(A_{n})_{n}$ is Wijsman lacunary
statistical convergent to $A$ then $(A_{n})_{n}$ is Wijsman $f$-statistical
convergent to $A$.

###### Proof.

Indeed, part a) is exactly Theorem 3.5 in [4].

Let us shown part b). If $(A_{n})_{n}$ is Wijsman lacunary statistical
convergent to $A$ then for any $x\in X$ the sequence $(d(x,A_{n}))_{n}$ is
$\theta$-lacunary statistical convergent to $d(x,A)$. Hence, since $f$ is
$\theta$-compatible, according to Theorem 3.2 in [15] we get that
$(d(x,A_{n}))_{n}$ is $\theta$-lacunary $f$-statistical convergent to
$d(x,A)$, therefore $(A_{n})$ is Wijsman $\theta$-lacunary $f$-statistical
convergent to $A$ as desired. ∎

######  Theorem 3.3.

Let $(X,d)$ be a metric space and $(A_{k})_{k}\subset CL(X)$, $A\in CL(X)$.
Assume that $\theta=(k_{t})_{t}$ is a lacunary sequence.

a)

For any modulus function $f$, if $(A_{n})_{n}$ is Wijsman $\theta$-lacunary
$f$-strong Cesàro convergent to $A$ then $(A_{n})_{n}$ also converges Wijsman
$\theta$-lacunary statistical convergent to $A$.

b)

If $f$ is $\theta$-compatible and $(A_{n})_{n}$ is Wijsman lacunary strong
Cesàro convergent to $A$ then $(A_{n})_{n}$ is Wijsman $\theta$-lacunary
$f$-strong Cesàro convergent to $A$.

###### Proof.

To show $a)$, assume that $(A_{n})_{n}$ is Wijsman $\theta$-lacunary
$f$-strong Cesàro convergent to $A$, then for any $x\in X$ we know that
$(d(x,A_{n}))_{n}$ is $\theta$-lacunary $f$-strong Cesàro convergent to
$d(x,A)$. Therefore, according to Theorem in [15], $(d(x,A_{n}))_{n}$ is
$\theta$-lacunary strong Cesàro convergent to $d(x,A)$ which yields the
result.

On the other hand if $(A_{n})_{n}$ is Wijsman lacunary strong Cesàro
convergent to $A$ then for any $x\in X$, the sequence $(d(x,A_{n}))_{n}$ is
$\theta$-lacunary strong Cesàro convergent to $d(x,A)$, but since $f$ is
$\theta$-compatible we get that $(d(x,A_{n}))_{n}$ is $\theta$-lacunary
$f$-strong Cesàro convergent to $d(x,A)$ which is what we are looking for. ∎

######  Theorem 3.4.

a)

Assume that $WS_{\theta}^{f}=WS_{\theta}$ then $f$ is $\theta$-compatible.

b)

Assume that $WN_{\theta}^{f}=WN_{\theta}$ then $f$ is $\theta$-compatible.

###### Proof.

To show $a)$, let us assume that $f$ is not a $\theta$-compatible modulus
function. On the metric space $(\mathbb{R},|\cdot|)$ we will construct a
sequence $(B_{n})_{n}\in WS_{\theta}\setminus WS_{\theta}^{f}$. An easy
modification provides a counterexample on any metric space.

Since $\varphi_{\theta}(\varepsilon)$ is an increasing function, if $f$ is not
$\theta$-compatible then there exists $c>0$ such that for any $\varepsilon>0$:
$\varphi_{\theta}(\varepsilon)>c$.

Let us fix $\varepsilon_{k}\to 0$. Thus, for each $k$ there exists $h_{r_{k}}$
such that $f(h_{r_{k}}\varepsilon_{k})\geq cf(h_{r_{k}})$. Moreover, since
$h_{r}$ is increasing we can suppose that

$h_{r_{k}}(1-\varepsilon_{k})-1>0.$ (2)

Let us denote by $\lfloor x\rfloor$ the integer part of $x$. We set
$n_{k}=\lfloor h_{r_{k}}\varepsilon_{k}\rfloor+1$. According to equation (2),
we get that $h_{r_{k}}-n_{k}>0$. Let us define the subset
$A_{k}=[k_{r_{k}}-n_{k},k_{r_{k}}]\cap\mathbb{N}\subset I_{r_{k}}$, and
$A=\bigcup_{k}A_{k}$.

Let us define

$B_{k}=\begin{cases}\\{1\\}&k\in A\\\ \\{0\\}&k\notin A\end{cases}$

We claim that the sequence $(B_{n})_{n}$, is Wijsman $\theta$-lacunary
statistically convergent to $B=\\{0\\}$ but not Wijsman $\theta$-lacunary
$f$-statistically convergent, a contradiction.

Let us observe that for any $x$, $d(x,B)=|x|$ and

$d(x,B_{k})=\begin{cases}|1-x|&k\in A\\\ |x|&k\notin A.\end{cases}$

Therefore, if $r\neq r_{k}$ for any $k$, then

$\frac{\\#\\{l\in
I_{r}\,:\,|d(x,B_{l})-d(x,B)|>\varepsilon\\}}{h_{r}}\leq\frac{0}{h_{r}}=0.$

And for $r=r_{k}$:

$\frac{\\#\\{l\in
I_{r_{k}}\,:\,|d(x,B_{l})-d(x,B)|>\varepsilon\\}}{h_{r_{k}}}=\frac{n_{k}}{h_{r_{k}}}\to
0$

as $k\to\infty$. That is, $(B_{k})_{k}$ is Wijsman $\theta$-lacunary
statistical convergent to $B=\\{0\\}$.

On the other hand,

$\frac{f(\\#\\{l\in
I_{r_{k}}\,:\,|d(x,B_{l})-d(x,B)|>\varepsilon\\})}{f(h_{r_{k}})}=\frac{f(n_{k})}{f(h_{r_{k}})}\geq\frac{f(h_{r_{k}}\varepsilon_{k})}{f(h_{r_{k}})}\geq
c$

which yields the desired result.

For the part $b)$ an easy check show that the sequence $(B_{k})$ constructed
before belong to $WN_{\theta}\setminus WN_{\theta}^{f}$. ∎

Finally we wish to explore the connections between Wijsman lacunary
$f$-statistical convergence and Wijsman lacunary $f$-strong Cesàro
convergence. To this end, let us refine the the concept of sequence Wijsman
Uniform integrable.

Let $\theta=(k_{t})$ be a lacunary sequence. A sequence $(x_{n})$ in a normed
space $Y$ is said to be $\theta$-lacunary uniformly integrable if

$\lim_{M\to\infty}\sup_{t}\sum_{\begin{subarray}{c}k\in I_{t}\\\ \|x_{k}\|\geq
M\end{subarray}}\|x_{k}\|=0.$

######  Definition 3.5.

Let $\theta=(k_{t})$ be a lacunary sequence and $(X,d)$ a metric space. A
sequence $(A_{n})_{n}\subset CL(X)$ is said to be Wijsman $\theta$-lacunary
uniformly integrable if for any $x\in X$ the sequence $(d(x,A_{n}))_{n}$ is
$\theta$-lacunary uniformly integrable. Let us denote by $WI_{\theta}$ the
subsets of all Wijsman $\theta$-lacunary uniformly integrable sequences.

Let us point out that if a sequence $(A_{k})_{k}\subset CL(X)$ is Wijsman
uniformly integrable then it is Wijsman $\theta$-lacunary uniformly
integrable.

######  Theorem 3.6.

a)

Assume that $\theta=(k_{t})$ is a lacunary sequence and $f$ is a modulus
function. Then $WN_{\theta}^{f}\subset WS_{\theta}^{f}$.

b)

Assume that $\theta=(k_{t})$ is a lacunary sequence and $f$ is a
$\theta$-compatible modulus function. Then $WS_{\theta}^{f}\cap
WI_{\theta}\subset WN_{\theta}^{f}$.

###### Proof.

If $(A_{k})_{k}\subset CL(X)$ is Wijsman $\theta$-lacunary $f$-statistical
convergent to $A$ then for any $x\in X$, $(d(x,A_{n}))_{n}$ is
$\theta$-lacunary $f$-strong Cesàro convergent to $d(x,A)$ therefore according
to Theorem 4.1 in [15] we get that $(d(x,A_{n}))_{n}$ is $\theta$-lacunary
$f$-statistical convergent to $d(x,A)$ which means that $(A_{n})_{n}$ is
Wijsman $\theta$-lacunary $f$-statistical convergent to $A$.

To prove $b)$, let $(A_{k})\in WS_{\theta}^{f}\cap WI_{\theta}$, Wijsman
$\theta$-lacunary $f$-statistical convergent to $A$. Then for any $x\in X$ the
sequence $(d(x,A_{n}))_{n}$ is $\theta$-lacunary $f$-statistical convergent to
$d(x,A)$. Additionally $(d(x,A_{n}))_{n}$ is uniformly integrable. Therefore,
since $f$ is $\theta$-compatible according to Theorem 4.3 in [racsman] we get
that $(d(x,A_{n}))_{n}$ is $\theta$-lacunary $f$-strong Cesàro convergent to
$d(x,A)$, that is $(A_{n})_{n}\in WN_{\theta}^{f}$ as desired. ∎

Finally let us shown that $\theta$-compatibility is again necessary for such
phenomena.

######  Theorem 3.7.

Let $\theta=(k_{t})_{t}$ be a lacunary sequence. Assume that for some modulus
function $f$, $WS_{\theta}^{f}\cap WI_{\theta}\subset WN_{\theta}^{f}$ then
$f$ is $\theta$-compatible.

###### Proof.

Let us assume that $f$ is not a $\theta$-compatible modulus function. On the
metric space $(\mathbb{R},|\cdot|)$ we will construct a sequence
$(B_{n})_{n}\in WS_{\theta}^{f}\cap WI_{\theta}\setminus WN_{\theta}^{f}$. An
easy modification provides a counterexample on any metric space.

Since $f$ is not $\theta$-compatible given $(\varepsilon_{k})$ a decreasing
sequence converging to zero, there exists a subsequence $r_{k}$ such that
$f(h_{r_{k}}\varepsilon_{k})\geq cf(h_{r_{k}})$, for some $c>0$.

Let us consider the following subsets

$B_{l}=\begin{cases}\\{\varepsilon_{k}\\}&l\in(k_{r_{k}-1},k_{r_{k}}]\\\
\\{0\\}&\textrm{other case}\end{cases}$

Since $(B_{l})_{l}$ is bounded, $(B_{l})_{l}\in WI_{\theta}$. Since
$(\varepsilon_{k})_{k}$ is decreasing $(B_{l})_{l}\in WS_{\theta}^{f}$.

However, if we take $x=0$ and $B=\\{0\\}$ we get that
$B_{k}\overset{WN_{\theta}^{f}}{\nrightarrow}B$. Indeed

$\frac{1}{f(h_{r_{k}})}f\left(\sum_{n\in
I_{r_{k}}}|d(x,B_{n})|\right)=\frac{f(h_{r_{k}}\varepsilon_{k})}{f(h_{r_{k}})}\geq
c$

which gives the desired result. ∎

## Conclusions

The notion of statistical convergence modulated by an unbounded modulus
function, allow us to define convergences of sets, more restrictive than the
Wijsman statistical convergence defined through the classical statistical
convergence. We conclude that in order to obtain new, more restrictive data
filtering methods, it is necessary to consider non-compatible modulus
functions, because for compatible modulus functions both convergence methods:
Wijsman statistical convergence and Wijsman $f$-statistical convergence, are
equivalents. Such structure remain stable if we consider the lacunary version
of these convergence methods.

## Availability of data and material

Not applicable.

## Acknowledgements

We want to thank to the Vicerretorado de Investigación of University of Cádiz
who supported partially this research.

## Competing interests

The author declare that they have no competing interests.

## Funding

The author was supported by Ministerio de Ciencia, Innovación y Universidades
under grant PGC2018-101514-B-I00.

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