Properties Of Random Walk
Song Zhai
Texas A&M University, Math Department
Beihang University, Math Department
5/13/2015
Song Zhai
Properties Of Random Walk
What is random walk?
Definition. A random walk is a path that is created by some
stochastic process.
Note. A random walk cares about not only the random
phenomenon, but also what will happen as time or space changes.
Song Zhai
Properties Of Random Walk
What is random walk?
A simple but interesting example.
Song Zhai
Properties Of Random Walk
What is random walk?
Consider a person standing on the integer line who flips a coin and
moves one unit to the right if it lands on heads, and one unit to
the left if it lands on tails. The path that is created by the random
movements of the walker is a random walk.
Song Zhai
Properties Of Random Walk
What is random walk?
Question. Suppose this person stands at zero point and he wants
to go to x6=0 point. Can he get to x in finite steps? Or what is the
probability that he can arrive at x in finite steps?
Song Zhai
Properties Of Random Walk
Statement of probability space
1.(<,B,µ) B = Borel σ-field µ = Probability measure
2.×∞
i=1 < = {ω = (ω1 , ω2 , · · ·) : ωi ∈ <}
3.⊗∞
i=1 B = B ⊗ B ⊗ · · ·
4.P = µ × µ × · · ·
∞
Thus, our probability space will be (×∞
i=1 <, ⊗i=1 B, P).
We can then define X1 , X2 , · · · i.i.d, taking values in ×∞
i=1 <,
Xi (ω) = ωi .
Song Zhai
Properties Of Random Walk
Stopping time
I would like to introduce some knowledge of stopping time as my
background.
Definition. Fn = σ(X1 , X2 , · · ·, Xn ).
Definition. A random variable N taking values in
{1, 2, · · ·} ∪ {∞} is said to be a stopping time or an optional
random variable if for every n< ∞, {N = n} ∈ Fn .
Definition. Associated with each stopping time N is a σ-field
FN = {A : A ∩ {N = n} ∈ Fn for all n < ∞}.
Song Zhai
Properties Of Random Walk
Random walk Sn (with n fixed)
Definition. Let X1 , X2 , · · · be i.i.d, taking values in ×∞
i=1 < and let
Sn = X1 + X2 + · · · + Xn , Sn is a random walk.
Song Zhai
Properties Of Random Walk
Random walk Sn (with n fixed)
Definition.
1.A finite permutation π is a map from N onto N ,thus,
(πω)i = ωπ(i)
2.An event A is permutable if π −1 A ≡ {ω : πω ∈ A} = A for any
finite permutation π
Song Zhai
Properties Of Random Walk
Random walk Sn (with n fixed)
Hewitt-Savage 0-1 Law
If X1 , X2 , · · · are i.i.d and A is permutable event, then P(A)∈ {0, 1}
Song Zhai
Properties Of Random Walk
Random walk Sn (with n fixed)
Theorem
For a random walk on <, there are only four possibilities,one of
which has probability one.
(a)Sn =0 for all n
(b)Sn → ∞, n → ∞
(c)Sn → −∞, n → ∞
(d)−∞ = liminfSn < limsupSn = ∞
Song Zhai
Properties Of Random Walk
Random walk Sn (with n fixed)
Sketch of proof
By checking that {ω : limsupSn (ω) = c} is a permutable event for
all c∈ [−∞, ∞], we can apply Hewitt-Savage 0-1 law. For any
fixed c0 , P({ω : limsupSn (ω) = c0 }) ∈ {0, 1}, which implies limsup
Sn ≡ c.
Song Zhai
Properties Of Random Walk
Random walk Sn (with n fixed)
Let Sn0 = Sn+1 − X1 , since Sn0 has the same distribution as Sn , by
taking limsup on both sides, we have c=c-X1 . Similarly, we have
liminfSn ≡ b, and b=b-X1 .
Song Zhai
Properties Of Random Walk
Random walk Sn (with n fixed)
My results follows from some simple combinations.
(1) c is finite. (2) b = −∞, c = −∞ (3) b = ∞, c = ∞ (4)
b = −∞, c = ∞
Song Zhai
Properties Of Random Walk
Random walk SN (where N is stopping time)
Before I talk about random walk SN , I would like to show you a
fact.
Fact. Let α(ω) = inf {n : ω1 + · · · + ωn > 0}, where inf = ∞,
and set α(4) = ∞.
then, we have, for arbitrary ω ∈ Ω, Sα (ω) = Sα(ω) (ω) > 0.
Song Zhai
Properties Of Random Walk
Random walk SN (where N is stopping time)
Notice that it’s not ture that for arbitrary ω ∈ Ω, Sn (ω) > 0.
In that case, we say Sα (where α is stopping time) has some more
beautiful properties than Sn . That’s the motivation why I want to
talk about random walk SN (where N is stopping time).
Song Zhai
Properties Of Random Walk
Random walk SN (where N is stopping time)
Wald’s equation
Let X1 , X2 , · · · be i.i.d with E |Xi | < ∞. If N is a stopping time
with EN < ∞, then, ESN = EX1 EN.
Song Zhai
Properties Of Random Walk
Random walk SN (where N is stopping time)
Wald’s second equation
Let X1 , X2 , · · · be i.i.d. with EX1 = 0 and EX12 = σ 2 < ∞. If T is
a stopping time with ET < ∞, then, EST2 = σ 2 ET .
Hint: First show that EST2 ∧n = E (T ∧ n)σ 2 . Then take the limit
on both sides.
Song Zhai
Properties Of Random Walk
Simple random walk
A mathematical model of the question I gave at the beginning.
Simple random walk
Let X1 , X2 , · · · be i.i.d with P(Xi = 1) =
1
2
and P(Xi = −1) = 12 ,
let Tx = inf {n : Sn = x}, ∀x ∈ Z.
(1) P(Tx < ∞) = 1 for all x ∈ Z, which means simple random
walk can arrive at any integer in finite steps with probability 1.
(2) ETx = ∞ for x 6= 0.
Song Zhai
Properties Of Random Walk
Simple random walk
Sketch of proof
Let a < 0 < b be integers, and let N = inf {n : Sn ∈
/ (a, b)}. By
checking EN < ∞, we can apply Wald’s equation and
ESN = EX1 EN = 0.
Song Zhai
Properties Of Random Walk
Simple random walk
By the definition, SN (ω)= a or b. So
bP(SN = b) + aP(SN = a) = ESN = 0, since
P(SN = b) + P(SN = a) = 1, solving this system of equation, we
have
(
P(SN = a) =
P(SN = b) =
Song Zhai
b
b−a
−a
b−a
Properties Of Random Walk
Simple random walk
Let Ta = inf {n : Sn = a}, Tb = inf {n : Sn = b},
P(Ta < Tb ) = P(SN = a) =
b
b−a
for all a < 0 < b
Letting b → ∞ gives P(Ta < ∞) = 1 for all a < 0, because of the
symmetry (and the fact T0 = 0), it follows that P(Tx < ∞) = 1
for all x ∈ Z, where Tx = inf {n : Sn = x}.
Song Zhai
Properties Of Random Walk
Acknowledge
Thanks Doctor Joel Zinn and Doctor Ursula Mueller-Harknett for
helpful discussions.
Song Zhai
Properties Of Random Walk
The end
The End.
Song Zhai
Properties Of Random Walk