Smart Art diagram

Question

I want to update the Smart Art diagram down below:

to the following one: .

My source code for the first one is:

\documentclass[12pt,a4paper,landscape]{article}
\usepackage{tikz}
\usepackage{geometry}
\usetikzlibrary{positioning,calc}
\geometry{left=0.75in,right=0.75in,top=0.75in,bottom=0.75in}

\begin{document}
\begin{center}
  \begin{tikzpicture}[scale=.9,transform shape]
    \tikzset{
      every node/.style={draw,rounded corners,font=\fontsize{14}{16}\selectfont\bfseries,align=center,on grid,inner sep=8pt},
      node distance=2cm and 9cm,
      every path/.style={line width=2pt,line cap=round}
    }
    \node (c1)[]{
      Almost uniform
      convergence
    };
    \node (c2) [left =of c1,] {Complete convergence};
    \node (c3) [below = of c2,] {Strong convergence};
    \node (c4) [below = of c1,] {Almost everywhere convergence};
    \node (c5) [right =of c4,] {Asymptotically uniformly \\ convergence};
    \node (c6) [below left =6cm and 9cm of c4,] {Convergence in probability};
    \node (c7) [below right =3cm and 9cm of c4,] {
      Convergence in \\
      $r$th mean, $r>0$
    };
    \node (c12) [below=3cm of c7,] {Convergence in $L^q,0<q<r$};
    \node (c8) [below =5cm of c6,] {Convergence in distribution};
    \node (c9) [right=of c8,] {Convergence ($P_{X_n}\rightarrow P_X$)};
    \node (c10) [right =of c9,] {
      Convergence in characteristic\\
      function
    };


    \draw[bend left=45,black,latex-,line width=2pt] (c3.west) to (c2.west);
    \draw[bend left=45,blue,latex-,line width=2pt] (c2.east) to (c3.east);
    \draw[black,latex-,->,line width=2pt] (c4.west) to (c3.east);
    \draw[black,latex-,->,line width=2pt] (c1.south) to (c4.north);
    \draw[black,latex-,<->,line width=2pt] (c5) to (c4);
    \draw[bend left=25,black,latex-,line width=2pt] (c6.north) to (c4.south west);
    \draw[bend left=25,red,latex-,line width=2pt] (c4.south) to (c6.east);
    \draw[bend left=10,red,latex-,line width=2pt] (c4.south east) to (c7.north);
    \draw[bend left=25,blue,latex-,line width=2pt] (c7.west) to (c4.south);
    \draw[black,latex-,->,line width=2pt] (c7) to (c12);
    \draw[bend left=45,black,latex-,line width=2pt] (c8.west) to (c6.west);
    \draw[bend left=45,blue,latex-,line width=2pt] (c6.east) to (c8.east);
    \draw[black,latex-,<->,line width=2pt] (c8) to (c9);
    \draw[black,latex-,<->,line width=2pt] (c9) to (c10);
    \draw[bend left=35,blue,latex-,line width=2pt] (c4.south) to (c8.east);

    \coordinate(sep aux)at($(c6)!.5!(c8)$);
    \coordinate(sep)at($(current bounding box.west|-sep aux)+(-2cm,0)$);
    \coordinate(T1)at($(current bounding box.north west-|sep)!.5!(sep)$);
    \coordinate(T2)at($(current bounding box.south west-|sep)!.5!(sep)$);

    \draw(sep)--(sep-|current bounding box.east);
    \node[anchor=west,draw=none]at(T1){\Large Type I};
    \node[anchor=west,draw=none]at(T2){\Large Type II};
  \end{tikzpicture}

\break

\end{center}
\end{document} 

My problems in constructing the new details are:

1. I don't find a manner how include a label above the arrow.
2. Then the construction of the blue left arrow (labeled Skorohod), I have tried a lot of combinations for the curvature but never seen like I am waiting for.
3. Moving the Type X on up-right.
4. My professor recomands me to use another type of arrow, dot arrow, double arrow or another decoration arrow, but the simple commands have no effects.

Answer 1

Here is an approach done by tikz-cd. You can take most of the commands I am using in normal TikZ as well.

% arara: pdflatex

\documentclass{standalone}
\usepackage{tikz-cd}
\newcommand{\textCD}[1]{\begin{tabular}{@{}c@{}}#1\end{tabular}}

\begin{document}
    \tikzset{%
        ,every path/.style={%
            ,line width=2pt
            ,line cap=round
        }
    }
    \begin{tikzcd}[%
        ,column sep=4em,row sep=5ex
        ,cells={nodes={draw, rounded corners, align=center,font=\fontsize{14}{16}\selectfont\bfseries, inner sep=8pt}}
        ,every arrow/.append style={line width=2pt,-latex}
        ,every label/.append style={black,font=\Large,align=center}
        ,every matrix/.append style={name=m}
        ,execute at end picture={%
            \node [above=3ex, font=\Large\bfseries] at (X.north) {Type 1};
            \node [below=3ex, font=\Large\bfseries] at (Y.south) {Type 2};
            \draw (current bounding box.190) -- (current bounding box.350);
            }
        ]
        Complete convergence \dar[start anchor=west, end anchor=west, bend right=45]& 
        Almost uniform convergence \dar&\\
        Strong convergence \ar[blue, start anchor=east, end anchor= east, bend right=45]{u}& 
        Almost everywhere convergence\ar[start anchor=190, bend right=20]{ddl}\lar\ar[latex-latex]{r}\ar[blue, bend right=10, end anchor=west]{dr}[swap]{Lebesgue}& 
        |[alias=X]|\textCD{Asymptotically uniformly\\convergence}\\
        & & \textCD{Convergence in\\$r$th mean, $r>0$}\ar[red, start anchor=north, end anchor=south east, out=150, in=320]{ul}\\
        Convergence in probability\ar[xshift=-5em]{dd}\ar[red,start anchor={[yshift=1em]east}, bend right]{uur}\ar[start anchor=east]{urr}\ar[blue,bend right=10, start anchor={[yshift=-1em]east}, end anchor=south west]{urr}[swap]{Lebesgue} & &
        Convergence in $L^q,0<q<r$\\
        &&&\\
        Convergence in distribution\ar[latex-latex]{r}{Levy}\ar[blue,xshift=5em]{uu}\ar[blue, start anchor=north west, out=140, end anchor=south west, in=190]{uuuur}{Scorohod} & 
        Convergence $(P_{X_n}\rightarrow P_X)$\ar[latex-latex]{r} &
        |[alias=Y]|\textCD{Convergence in characteristic\\function}\\
    \end{tikzcd}
\end{document} 

---

You may try arrow options such as dotted, dashed, dash, Rightarrow, equal...