
but satisfying the remaining criteria. 
 \item[\textsc{\bf LGO3}] has 804 candidates obtained by relaxing the density contrast condition 
 \ref{item:d} only. 
\item[\textsc{\bf LGO4}] of 1561 candidates with the conditions \ref{item:v} \& \ref{item:d}
relaxed simultaneously. 
\item[\textsc{\bf LGO5}] has 1197 observers without imposing the constraint on the host halo 
mass but with all the other criteria fulfilled. 
\item[\textsc{\bf LGO--NOV}] contains 772543 candidate observers satisfying all conditions except the 
proximity to a Virgo-like cluster. 
\item[\textsc{\bf RNDO}] is a list of observers with randomly selected 
positions in the simulation box. This set is used as a benchmark for comparison. 
\end{description} 

Based on the number of candidate observers in each set, we conclude that 
the proximity to a Virgo-like cluster is the strongest discriminator among
all the conditions.
Moreover, positions of observers in each of the 5 sets {\textsc{\bf LGO1}}-{\textsc{\bf LGO5}} 
are highly correlated, as they are constrained to reside 
in the same vicinities of Virgo like objects. 
Therefore, in order to speed up the calculations, we consider only a 
sub-sample of the list of observes, not reducing however the statistical significance of the results. 
 This is done by laying a uniform coarse $8^3$ grid in the box and selecting, 
for each set of observers, one random observer per grid cell, should the cell 
contain any observers. This 
gives an average number of $60$ observers for each of the 5 sets. To match
the sample variance we also keep only 64 observers in the
\textsc{\bf LGO--NOV}{} and \textsc{\bf RNDO}{} sets. 
As we have already pointed out, currently available 
peculiar velocity catalogues are relatively shallow due to 
the difficulty in measuring distances especially for distant galaxies. 
Furthermore, additional distance cuts and trimming of the data 
are usually imposed on velocity catalogues in order to avoid 
very large errors and uncontrolled observational systematics.
To get closer to a realistic catalogue, 
we implement two simple data weighting schemes.
The first scheme mimics simple radial selection cuts that one
can always implement for a given peculiar velocity catalogue.
It is defined by a single ``depth" parameter, $r_w$.
Here a halo at a distance $r$ from the observer is assigned a weight, $w_h$, given by
\begin{equation}
\label{eqn:data_weighting}
w_{h}=
\begin{cases}
1, & \text{if}\ r\leqr_w \\
0, & \text{otherwise}\,.
\end{cases}
\end{equation}
The second scheme aims at mimicking a sample with a flattened radial distribution
of galaxies, similar to the one describing the CosmicFlows-3 catalogue \citep{Tully2016}.
Here, the weighting is characterised by a power-law and, in addition to the depth parameter $r_W$,
is also a function of the ''steepness`` parameter $m$. The corresponding formula
for $w_h$ is
\begin{equation}
\label{eqn:data_weighting2}
w_{h}=
\begin{cases}
1, & \text{if}\ r\leqr_w \\
(r/r_w)^{-m}, & \text{otherwise}\,.
\end{cases}
\end{equation}
Here we consider $r_w=20h^{-1}\,{\rm Mpc}$ and $m=2,3$ and dub the corresponding
catalogues {\it CF3-like m=2} and {\it CF3-like m=3} accordingly.
We will use these data weighting schemes to further investigate
how the velocity statistics depend on the catalogue depth.

\label{sec:LG_observers}
\begin{figure*}
 \includegraphics[angle=-90,width=170mm]{fig/observers_bias_v12_p1_p2.pdf}
 \caption{The effects of \textsc{\bf LGO}{}-like observer location and various selection functions on
 the velocity statistics. Error-bars and filled regions mark $1\sigma$ observer-to-observer scatter
 around ensemble mean. From to to bottom panels show results for $v_{12}$, $\psi_1$ and $\psi_2$ respectively.
 The left column illustrate the effects for different set of observers, but with the same imposed
 radial selection cut of $r_w=80h^{-1}\,{\rm Mpc}$. The right column of panels focuses on our main Local Group (\textsc{\bf LGO1}{})
 observers sample and the comparison of various selection functions and data weights.}
\label{fig:observers_bias}
\end{figure*}
\begin{figure}
 \includegraphics[angle=-90,width=85mm]{fig/rw_bias.pdf}
 \caption{The biasing of the LG observer estimators with respect to a Copernican observer 
 as a function of the velocity catalogue depth parameter, $r_w$.
 The ratios for each estimator are taken at two pair separation scales: $20h^{-1}\,{\rm Mpc}$ and $50h^{-1}\,{\rm Mpc}$.}
\label{fig:rw_bias}
\end{figure}
 
The three panels to the left in Fig.~\ref{fig:observers_bias} show the statistics derived for 
all sets of observers, with $r_w=80h^{-1}\,{\rm Mpc}$ and the first weighting scheme applied. 
The curves are (ensemble) averages over all observers in each set 
(as indicated in the figure) and the attached error-bars and filled regions represent 
the corresponding $1\sigma$ scatter. 
The error-bars in the \textsc{\bf LGO}{} series are similar and for clarity they are attached only to \textsc{\bf LGO1}. 
Since we do not include observational errors, this scatter is entirely due to 
cosmic variance among the observers in each set. 
Plotted also are results for the ``Copernican'' observer,
computed from the full catalogue for the \textsc{\bf RNDO}{} observer set. 
The small error-bars here 
reflect the fact that different observers see different (radial) velocity components of the same galaxies.
The \textsc{\bf LGO}{} curves in all the panels differ systematically from the Copernican \textsc{\bf RNDO}{} result. However, the
\textsc{\bf LGO--NOV}{} and \textsc{\bf RNDO}{} curves are almost indistinguishable up to pair separations of $R\sim55h^{-1}\,{\rm Mpc}$,
meaning that the proximity to Virgo is the only significant criterion
in the selection of the LG candidate.
The average streaming velocity, $v_{12}$, defined in Eqn.~(\ref{eqn:v12_obs_estim}), 
in the top-left panel is significantly affected
by the LG selection criteria at pair separations $R\stackrel{>}{{}_\sim} 40h^{-1}\,{\rm Mpc}$. 
At those scales, LG-observers are deviating 
from the ``Copernican'' curve by more than $1\sigma$ getting values
lower then the cosmic mean observer.
However, the observer-to-observer induced variance is large. So even for
smaller scales, where the both averages agree within the scatter, the amplitude
of the difference is large and can typically take from 50 to 100 km/s.
This is already a $100\%$ level effect
at $R=40h^{-1}\,{\rm Mpc}$, but it quickly grows, reaching
$200\%$ magnitude difference already at separations of $\sim 60h^{-1}\,{\rm Mpc}$.
At large separations closer to $r_w$, fewer galaxy/halo pairs
are found which explains the rapid increase of the error-bars for LG-observers.

In the middle and bottom-left panels of Fig.~\ref{fig:observers_bias} we consider the correlation 
functions, $\psi_1$ and $\psi_2$.
For all \textsc{\bf LGO}{} LG-analogues the amplitude of $\psi_1$
is systematically larger than the black curve corresponding to 
the Copernican observer, up-to separations of $\sim75h^{-1}\,{\rm Mpc}$.
At larger separation the sign of the effect is flipped and all \textsc{\bf LGO}{} $\psi_1$'s
take smaller amplitudes than a random observer measurement.
This is a clear sign of the imposed catalogue depth, with our radial cut of $r_w=80h^{-1}\,{\rm Mpc}$. 
Here again the observer induced scatter is large making the \textsc{\bf LGO}{}
curves to ``agree'' within $1\sigma$ with the Copernican observer, even though 