Ergo dre Theorem-

We will provethat the tree weeps on eghel to spaceN

averages ,i-e .tEf(xn) = ⇐ Tcs) fcs) if T is the equi-hmmmNu> I

distribution.

Note that LAS depends on trajectory, Rete cot-- one of manifestation of ergodiary as

"forgetfulness ? .

Start with the strong LLN

Thin If Y, . . . . . Yn .. ..We did r

-u.

, Yu> o , f- Yu --µ c es , thenI CE Ei Tu →a) -- t .(Note the difference of usual LLN : IEC# EYu -up e)→ o)

for ay E-

Now,to leather Chairs

The Assume P is irreducible,Xu is Rohr (d. P) process . Then

#( FEI→ Ia ) - 2 ( here VIN) - ÷k(Kik) , umber. of visits to k). . I here = E,I

"

If , in addition ,D is pos . recurrent ,

then, for ay

bounded f :S→R , K¥4, fcxu ) ⇒F) =L ,

where I -- Efes) UTD .SES

as

RI : If P B trainwut,then each TICK.- k) is finite a. s .

,

VINand tenth -so = 's. .#ilet Si' are the lengths of excursions to

k.

N

Then they are iie, E.sc?=mic ; ¥7's ±Ny , ¥7 I > N

.

-

-

Vem VEN) -11so ÷,÷sIe÷assu÷as÷

By Shay LLN, LL, AS → me with B. =L,

as P is recurrent,VIN)→ as

.

So,RC Vance → ⇒ =L ( 1st pot

Now assume 17kt.

Then

I Etch - El-- LECHIA- a.) the EIE-rultfz.IE-rtNw

. E.e¥a7E⇒¥ a.E¥ ÷¥¥+M.

c-E.EE#..r.seaE..thF-rut.E.iD-----

The set J can be chosen fate, so tht he a § .

Then,each of KNIN) - are→ o fur leaf , so

does their finite sum .

So,a. s .

( & II fix. ) - Flee from some point , D

Lefts apply this to question of reconstruction ofMI long trajectories .

If one observes Xu,then what is the best estimate

for the transition ruahix Ree ?

Consider it as a function of unknown probabilities Pnethen

, given these,the chance to observe given trajectory is

Ae.Pee , n

- - -

- pen, en = Ie.IT Reena

,where

k, EES

Nice = #(of transitions k → e.)We want to take pae maximizing this probability .(called : 14L )Obvious Solin

,Due =L is bad : we have the constraint

E Re =L for all k .

So we wax Tl Dunne (or , rather be of it) very LagrangeIe Nae hey Pee t Ifk Ee Pae

→ ex 'm

→ N÷=µ , Pie - Nff = e÷n [ML estimate]

Does Prue -7 Due as # of observations go to co ?

Yes, if it does ! In transient ease , we'll need to

take care of it restarting the chair (not entirely kosher,but

. . .)

Example Ergodic planning )

g"q⇒pp ,-pFfs) - 750 the)= -p2 > tooo

-- 1000

f = 75op-p72_ → max

P + "2 @ -up) Capt ,) - top -42126Irs =p - TF

p!z.pt#,z.3PzpITP-max3-zp-8pZ-bpt4p2--oFcs) -Tcs) + ftp.u-cpy

4=4p2t4p -4

Xu - inch . ; Xu={ O w - pub. ÷,I w pub net

Eckel > e) = # → a

hot true U2 (Xu → o) -IA- ( Xu-- o for all NZN) -TT ( t

-⇒ = 0

WIN

A

-2 1- =ph=N htt