Harris chain

Harris chain

In the mathematical study of stochastic processes, a Harris chain is a Markov chain satisfying an additional property.

Definition

A Markov chain {"Xn"} on state space Ω with kernel "K" is a "Harris chain" [R. Durrett. "Probability: Theory and Examples". Thomson, 2005. ISBN 0-534-42441-4.] if there exist "A", "B" ⊆ Ω, ϵ > 0, and probability measure ρ with ρ("B") = 1 such that
# If τ"A" := inf {"n" ≥ 0 : "Xn" ∈ "A"}, then P(τ"A" < ∞|"X"0 = "x") > 0 for all "x" ∈ Ω.
# If "x" ∈ "A" and "C" ⊆ "B" then "K"("x", "C") ≥ "&epsilon;ρ"("C").

In essence, this technical definition can be rephrased as follows: given two points "x"1 and "x"2 in "A", then there is at least an ϵ chance that they can be moved together to the same point at the next time step.

Another way to say it is that suppose that "x" and "y" are in "A". Then at the next time step I first flip a Bernoulli with parameter ϵ. If it comes up one, I move the points to a point chosen using ρ. If it comes up zero, the points move independently, with "x" moving according to P("X""n"+1 ∈ C|"Xn" = "x") = "K"("x", "C") − "&epsilon;ρ"("C") and "y" moving according to P("Y""n"+1 ∈ "C"|"Y""n" = "y") = "K"("y", "C") − "&epsilon;ρ"("C").

Examples

Example 1: Countable state space

Given a countable set "S" and a pair ("A"′, "B"′ ) satisfying (1) and (2) in the above definition, we can without loss of generality take "B"′ to be a single point "b". Upon setting "A" = {"b"}, pick "c" such that "K"("b", "c") > 0 and set "B" = {"c"}. Then, (1) and (2) hold with "A" and "B" as singletons.

Example 2: Chains with continuous densities

Let {"X""n"}, "X""n"R"d" be a Markov Chain with a kernel that is absolutely continuous with respect to Lebesgue measure:

: "K"("x", "dy") = "K"("x", "y") "dy"

such that "K"("x", "y") is a continuous function.

Pick ("x"0, "y"0) such that "K"("x"0, "y"0 ) > 0, and let "A" and "B" be open sets containing "x"0 and "y"0 respectively that are sufficiently small so that "K"("x", "y") ≥ "&epsilon;" > 0 on "A" × "B". Letting "ρ"("C") = |"B" ∩ "C"|/|"B"| where |"B"| is the Lebesgue measure of "B", we have that (2) in the above definition holds. If (1) holds, then {"X""n"} is a Harris chain.

Reducibility and periodicity

In the following, "R" := inf {"n" ≥ 1 : "Xn" ∈ "A"}; i.e. "R" is the first time after time 0 that the process enters region "A".

Definition: If for all "L"("X"0), "P"("R" < ∞|"X"0 ∈ "A") = 1, then the Harris chain is called "recurrent."

Definition: A recurrent Harris chain "Xn" is "aperiodic" if ∃"N", such that ∀"n" ≥ "N", ∀"L"("X"0), P("Xn" ∈ "A"|"X"0 ∈ "A") > 0.

Theorem: Let "Xn" be an aperiodic recurrent Harris chain with stationary distribution π. If P("R" < ∞|"X"0 = "x") then as "n" → ∞, distTV ("L"("Xn"|"X"0 = "x"), π) → 0.

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Harris Department Store — Infobox Company company name = The Harris Company company > company type = Department store foundation = 1927 location = San Bernardino, California industry = Retail products = Clothing, footwear, bedding, furniture, jewelry, beauty… …   Wikipedia

  • Harris (disambiguation) — Harris is the southern part of the largest island of Scotland s Outer Hebrides.Harris may also refer to:People*Harris (surname), a family name *Harris (rapper) (born 1976), a German rapper *Harris ( Porridge ), a character from the UK sitcom… …   Wikipedia

  • Harris Farm Markets — is an Australian grocery chain with 18 different locations and 800 employees. The supermarkets specialise in fruit, vegetables, delicatessen, and other grocery items.HistoryHarris Farm Markets was established in 1971, with a single shop in… …   Wikipedia

  • Harris Scarfe — Infobox Company name = Harris Scarfe logo = type = Department Store genre = foundation = 1850 founder = location city = Adelaide location country = South Australia location = origins = key people = area served = industry = Retail products =… …   Wikipedia

  • Harris Teeter — Infobox Company company name = Harris Teeter company company type = Subsidiary company slogan = foundation = 1936, Charlotte, North Carolina industry = Grocery store location = flagicon|North Carolina Matthews, North Carolina key people = Fred… …   Wikipedia

  • Harris Ranch — A Harris Ranch sign in Coalinga, California, displaying the Harris Ranch logo similar to all Harris Ranch packaging …   Wikipedia

  • Chain Home Low — Radar Coverage 1939 1940 Chain Home Low (CHL) was the name of a British radar early warning system, detecting enemy aircraft movement at lower altitudes than and summarily used with the fixed Chain Home system which was operated by the RAF during …   Wikipedia

  • Harris-Ketten — Eine Markow Kette (engl. Markov chain, auch Markow Prozess, nach Andrei Andrejewitsch Markow, andere Schreibweisen: Markov Kette, Markoff Kette) ist eine spezielle Klasse von stochastischen Prozessen. Man unterscheidet eine Markow Kette in… …   Deutsch Wikipedia

  • Markov chain — A simple two state Markov chain. A Markov chain, named for Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized …   Wikipedia

  • Ted Harris (mathematician) — Ted Harris (11 January 1919 ndash; 3 November 2006) was an American mathematician known for his research on stochastic processes, including such areas as general state space Markov chains (oftennow called Harris chains), the theory of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”