
Stochastic geometric series - efavdb
http://efavdb.com/stochastic-geometric-series/
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thanatropism
This is nice. So, follow-up question.

In economics, there's "input-output models" which amount to the summation of
adjacency matrices for inter-sectoral demand linkages such that

S = I + A + A^2 + A^3+... = (I-A)^(-1)

We know this to be true because matrix A is markovian, so it's a bounded
operator, so we have a Neumann series. What happens if we have probability
distributions for the elements of A?

This is the context where it arises: we define a vector of x interindustrial
demands that must satisfy itself through production coefficients A (the
weights in the graph of cross-sector dependencies), such that

x = A _x

but then we give this balanced system an additive shock ("consumer demands") b
so that

x' = x+e = A_x+b

but we still want to satisfy Ax' = x'. So:

A _(x+e) = A_ x+b => A*e = b => e = (I-A)^(-1) b

This is the shock to the interindustrial demands provoked by b. We have to
know that (I-A) is invertible, which by the Perron-Frobenius theorem is true
if A happens to be markovian.

This matters because the interindustrial linkage graph A is very noisy (it's
hammered into place at the national accounts offices from disperse
information), and much basic demand-shock appraisal (in a multi-sector context
at least) still uses this basic ("Leontieff") model; and more sophisticated
models apply variants of this too; but it's very hard to get anything said
about confidence intervals -- other than by means of simulation.

So -- maybe it's an interesting, applied problem to work on!

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efavdb
Awesome idea!

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thanatropism
Can I maybe email you if this is (in your estimation) a conversation worth
continuing? I have a feeling it gets fast lost in an aging HN thread.

At any rate: probably the most important problem to be posed by such a "random
input-output theory" is stochastic divergence. Random matrix theory, of which
I can make little sense, does present some knowledge about the distribution of
eigenvalues given some stochastic structure, which numerically gives us
probability estimates to whether we're breaking Perron-Frobenius and posing an
explosive system. But the relevant problem is in reverse: what is the set of
acceptable stochastic conditions such that we can guarantee that S =
I+A+Aˆ2+... = some finite value?

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efavdb
Hi, I'm very happy to talk some more about this. My email is jslandy@gmail.com
(will delete this post after you email).

