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I have been trying to integrate and compute moments of LogMultinormalDistributionin Mathematica, but the integrals don't seem to evaluate. Say for the first moment, $langle x rangle $

Integrate[
 PDF[LogMultinormalDistribution[
 Subscript[μ, 1], Subscript[μ, 2],
 
 Subscript[σ, 1]^2, Subscript[σ, 1] Subscript[σ, 2] ρ,
 Subscript[σ, 1] Subscript[σ, 2] ρ, Subscript[σ, 2]^2
 
 ]
 , x, y
 ]*x
, x, 0, Infinity
, y, 0, Infinity
]

I also tried typing out the PDF,

Integrate[(2*Pi*σx*σy Sqrt[1 - ρ^2])^-1*
 Exp[-1/2*(1/(
 1 - ρ^2)) (((Log[x] - μx)/σx)^2 - 
 2*ρ*((Log[x] - μx)/σx)*((
 Log[y] - μy)/σy) + ((
 Log[y] - μy)/σy)^2)]*x, x, 0, Infinity, y, 0, Infinity]

But no luck.

The integrals don't seem to evaluate and either I end up aborting them or the program seems to crash.

In general, I would like compute $langle x^n y^m rangle$ for bivariate log normal distribution in mathematica.

Is there a way to speed up the integrals/evaluate them at all.

I am running Mathematica 11.3 on Mac OS Mojave if that helps.

Life is easier when one can go straight to the answer without going through the integration.

Given a mean vector and covariance matrix

μ = μ1, μ2;
Σ = σ1^2, σ1 σ2 ρ, σ1 σ2 ρ, σ2^2;

the moment generating function for a bivariate normal is

mgf[t1_, t2_] := Exp[t1, t2.μ + t1, t2.Σ.t1, t2/2]

Because the definition of the moment generating function is $E[e^t_1 X_1+t_2 X_2]=E[Y_1^t_1 Y_2^t_2]$, then mgf[t1_, t2_] is the function that gives the expectation of $Y_1^t_1 Y_2^t_2$.

mgf[m, n]
(* E^(m μ1 + n μ2 + 1/2 (m (m σ1^2 + n ρ σ1 σ2) + n (m ρ σ1 σ2 + n σ2^2))) *)

You can see that this easily generalizes to more than two dimensions. Here is a reference with the steps of integration: The Lognormal Random Multivariate

For Mathematica the integral can be very complicated if all general cases are considered for each parameter. One option to speed and simplify things is to analyze more specific cases using Assuming.

You would need to decide if these assumptions are valid, and define your own if not, but these seem quite reasonable to me.

pdf = Assuming[
 And[x > 0, y > 0, ρ^2 < 1, σ1, σ2, μ1, μ2 ∈ Reals, σ2 > 0],
 Simplify@PDF[
 LogMultinormalDistribution[
 μ1, μ2,
 
 σ1^2, ρ σ1 σ2,
 ρ σ1 σ2, σ2^2
 
 ]
 , x, y]
 ]

Notice I have avoided the use of Subscript and used Simplify and indentation to make the code more readable.

Assuming[
 And[x > 0, y > 0, ρ^2 < 1, σ1, σ2, μ1, μ2 ∈ Reals, σ2 > 0],
 Integrate[
 pdf
 , x, 0, Infinity
 , y, 0, Infinity
 ]
 ]

(* 1 *)

So the integral is $1$, as expected for a PDF.

Assuming[
 And[x > 0, y > 0, ρ^2 < 1, σ1, σ2, μ1, μ2 ∈ Reals, σ2 > 0],
 Integrate[
 pdf x
 , x, 0, Infinity
 , y, 0, Infinity
 ]
 ]

E^(μ1 + σ1^2/2)

Using AbsoluteTiming the solution took 123 seconds in my computer (Mathematica 12 i7-4770 3.4GHz)

Format[μ[n_]] := Subscript[μ, n];
Format[σ[n_]] := Subscript[σ, n]

dist = LogMultinormalDistribution[μ[1], μ[2],
 σ[1]^2, ρ σ[1] σ[2], ρ σ[1] σ[2], σ[2]^2];

The Moments are built-in for this distribution

Moment[dist, m, n]