$HistoryLength = 1;

k = 3;

Unprotect[
      Power];      (ρ^(l_ /; 
            l > 3  k + 1)):=0

(ϵ^(l_ /; l > 2  k)) := 0;       

    ns = 5  k + 1;

determinant[A_]:=Module[{n = Length[A], a = A}, 
        Do[a[[j]] = 
            a[[j]] - a[[i]] a[[j, i]]/a[[i, i]], {i, 
            n - 1}, {j, i + 1, n}]; 
        Product[a[[i, i]], {i, n}] // Together]

L[i1_, i2_, i3_, i4_, 
        i5_] := D[
          MGF, {(Subscript[t, 1]), i1}, {(Subscript[t, 2]), i2}, {(Subscript[t, 3]), i3}, {(Subscript[t, 4]), i4}, {(Subscript[t, 5]), 
            i5}]/.{(Subscript[t, i])_ -> 0}

est = ((ρ + ϵ (Subscript[Z, 2]) - (ϵ^2) (Subscript[Z, 4]) (Subscript[Z, 5]))/(1 + ϵ (Subscript[Z, 1]) - (ϵ^2) ((Subscript[Z, 3])^2)));    (* 
      defining the estimator *)

adj = 
        Series[F[est], {ϵ, 0, 2  k}] - 
            F[est /. ϵ -> 0] // Normal;

S = 
        Collect[(Subscript[Z, 1]) (Subscript[Z, 2]) (Subscript[Z, 3]) (Subscript[Z, 4]) (Subscript[Z, 5]) {adj, (adj^2), 
                (adj^3), (adj^4)}, ϵ, Expand] /. ϵ -> 
            1;

\(Do[V0 = 
        Table[Which[i == j, 
              1 + If[i == 1 || i == n, 0, ρ\^2], 
              Abs[i - j] == 1, \(-ρ\), True, 0], {i, n}, {j, 
              n}]/\((1 - ρ\^2)\); 

      V = V0 - \(t\_2\) 
            Table[If[Abs[i - j] == 1, 1, 0], {i, n}, {j, n}] - 
          2 \( t\_1\) Table[If[i == j, 1, 0], {i, n}, {j, n}]; 

      Vio = Table[ρ\^Abs[i - j], {i, n}, {j, n}]; \ \ \  (*\ 
        inverse\ of\ V0\ *) 

      T = Table[
          Which[i == 1, t\_3 + t\_4, i == n, t\_3 + t\_5, True, 
            t\_3 + t\_4 + t\_5], {i, 
            n}]; 
aux = \((V - V0)\) . Vio; 

      MGF = \(\@\(determinant[V0]\/determinant[V]\)\) 
          Exp[Nest[Expand[T . Vio - # . aux] &, T . Vio, 2  k - 2] . T/2 // 
              Expand]; 

      MGF = \((Series[
                  MGF\ Exp[\(-\((n - 1)\)\) ρ\ t\_2 - n\ t\_1] /. 
                    t\_i_ -> λ\ t\_i/\((nn - 
                              If[i == 1 || i == 3, 0, 
                                1])\), {λ, 0, 2  k}] // 
                Normal)\) /. λ -> 1 // Expand; \ \ \ \ \  (*\ 
        moment\ generating\ function\ of\ the\ Z\_i' s; \ 
        nn\ represents\ unevaluated\ n\ *) 

      R[n] = \(S /. \(Z\_1\^i1_. \) \(Z\_2\^i2_. \) \(Z\_3\^i3_. \) \(Z\_4\^i4_. \) Z\_5\^i5_.  -> 
              K[i1 - 1, i2 - 1, i3 - 1, i4 - 1, i5 - 1]\) /. 
          K -> L\ \  (*resulting\ set\ of\ expected\ values*) , {n, ns, 
        ns + k}]\)

int = 
        Inverse[Table[Table[(i^j), {j, 0, k}], {i, ns, ns + k}]];

res = 
        Table[(nn^j), {j, 0, k}] . (int . 
              Table[R[ns + j], {j, 0, k}]);

res = Apart[res, nn] ;   res = 
        Collect[res /. 
              nn - 1 -> 1/Sum[(ϵ^(2  j)), {j, 1, k}] /. 
            nn -> 
              1/(ϵ^2), ϵ] ;   (*simplifying*)

F[x_] := x

moment = 
      Collect[{res[[1]] + F[est /. ϵ -> 0], 
            res[[2]] - res[[1]]^2, 
            res[[3]] - 3  res[[2]] res[[1]] + 2  res[[1]]^3, 
            res[[4]] - 4  res[[3]] res[[1]] + 
              6  res[[2]] res[[1]]^2 - 
              3  res[[1]]^4}, ϵ, 
          Simplify] /. (ϵ^l)_ -> 
          1/(n^(l/2 ))

Collect[
          Series[{moment[[3]]/(((moment[[2]] n))^(3/2)), 
                moment[[4]]/(((moment[[2]] n))^2)}, {n, Infinity, 
                k}] // Normal, n, Simplify] {(n^(3/2)), 
          (n^2)} //Expand[#, n] &

F[x_] := ArcTanh[x]

moment = 
      Collect[{res[[1]] + F[est /. ϵ -> 0], 
            res[[2]] - res[[1]]^2, 
            res[[3]] - 3  res[[2]] res[[1]] + 2  res[[1]]^3, 
            res[[4]] - 4  res[[3]] res[[1]] + 
              6  res[[2]] res[[1]]^2 - 
              3  res[[1]]^4}, ϵ, 
          Simplify] /. (ϵ^l)_ -> 1/
            (n^(l/2))

Collect[
          Series[{moment[[3]]/(((moment[[2]] n))^(3/2)), 
                moment[[4]]/(((moment[[2]] n))^2)}, {n, Infinity, 
                k}] // Normal, n, Simplify] {(n^(3/2)), 
          (n^2)}  //Expand[#, n] &

Clear[F]