Exercise #4 in random walks - fixes in red
1. Let 𝑆𝑛 = ∑𝑛𝑖=1 𝑋𝑖 be a p-biased random walk on the integers.
i.e. fix some 0<p<1 and let Xi be i.i.d. with
P(Xi=1)=p, P(Xi=-1) = 1-p.
Recall that for a set A we denote by A=min{k≥0: SkA} the (first)
hitting time of A, and for a singleton we denote x={x}. Let a,b be
positive integers.
a. Calculate P({-a,b}={-a})
b. Calculate E{-a,b}
c. Use a. to calculate the chance to never return to 0. i.e.
P(Sk≠0 for all k>0)
2. A gambler plays the following game. In each round, he can pay
any 0 < p < 1 dollars, and win $1 with probability p
(independently). Show that the probability that the gambler's net
gain exceeds h at any of the first n rounds is at most exp(-h2/2n).
3. A general random walk on the line is a sum of i.i.d. random
variables 𝑆𝑛 = ∑𝑛𝑖=1 𝑋𝑖 . (Xi need not even be integer valued)
Assume Xi are bounded, and that P(Xi=0)<1.
a. Prove that if E(X1)=0 then
1
lim 𝑃(𝜏(−∞,−𝑛] < 𝜏[𝑛,∞) ) =
𝑛→∞
2
b. Prove that if E(X1)=0 then almost surely
lim𝑠𝑢𝑝 𝑆𝑛 = ∞ and lim𝑖𝑛𝑓 𝑆𝑛 = −∞
𝑛→∞
𝑛→∞
c. Assume now that X1 is integer valued. *(hard) Does
E(X1)=0 imply that the walk is recurrent?.
d. Find an example of a symmetric integer valued
distribution with E(|X1|)=∞ such that the corresponding
random walk is transient. (the Xi are not bounded for
this clause)
Note that all the Xi have the same distribution as X1.
4. Harmonic functions on the tree – Let T2 be the infinite binary
rooted tree (i.e. there is a root vertex having a degree of 2 - no
"parent" and 2 children, and each other vertex has one parent and
2 children (degree 3). Call one of the neighbors of the root the left
child and the other the right child. The descendants of the left
(right) child are called the left(right) sub-tree. Define the "level" or
"depth" of a vertex v to be the distance between v and the root.
a. Let Xn be a simple random walk on T. Show that Yn=Depth(Xn)
behaves like a biased random walk on the natural numbers.
b. Define 𝑓(𝑣) = 𝑃𝑣 (∃𝑁 ∶ ∀𝑛 > 𝑁 𝑋𝑛 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑠𝑢𝑏𝑡𝑟𝑒𝑒)
(The probability that the random walk is eventually in the left
sub-tree)
Show that f is harmonic and bounded in T2.
Use (a) to show that f is non-constant. Actually show that
𝑓(𝑣) → 0 as v goes deeper into the right sub-tree and 𝑓(𝑣) →
1
as v goes deeper into the left sub-tree.
c. Calculate f(v) explicitly , using (a) and/or symmetry.
d. (hard) Describe the set of all bounded harmonic functions on
T2 . (Alternatively (easier) describe some examples of harmonic
functions on T2)