Exercise 2 in Random Walks - Martingales
0. (Small exercise. Choose 1 clause, no need to write them all ): If Xn is a
martingale sequence then |Xn|, max(Xn,a), (Xn+c)^2 and actually and
convex function of Xn are submartingale sequences. (3)
1. The Maximal Inequality for Non-negative sub-Martingales.(9)
If Xn is a non-negative sub-martingale and K is a nonnegative
constant, then
Pr[Max0≤i≤n Xi ≥ K ] <= E[Xn]/K
(Note this is an extension of Markov's inequality – simply by taking
Xn==X0)
2. Let Xn be a martingale with E[X0]=0 and E[Xn2]<infinity
Show that P(Max0≤i≤n Xi >r ) ≤ E[Xn2]/(E[Xn2]+r2) (9)
Hints:
a. The Cauchy–Schwarz inequality states that for all vectors x and y
of an inner product space it is true that |<x,y>| <= <x,x> *<y,y>
This is perhaps the most useful inequality in mathematics.
In terms of expectations of random variables it tells us that
𝐸 2 [𝑋𝑌] ≤ 𝐸[𝑋 2 ]𝐸[𝑌 2 ]
If you have not seen a proof, please look for one in a
textbook/Wikipedia or do it yourselves.
b. Use the C-S inequality to prove the following useful inequality on
non-negative random variables:
If X is a non-negative random variable then
𝐸 2 [𝑋]
𝑃(𝑋 > 0) ≥
𝐸[𝑋 2 ]
Note that this is a special case of the more general PaleyZygmund inequality , saying that for a non-negative random variable X, and
0≤a<1
𝐸 2 [𝑋]
𝑃(𝑋 > 𝑎𝐸[𝑋]) ≥ (1 − 𝑎)
𝐸[𝑋 2 ]
2
c.
Now prove an inequality similar to b. for the martingale sequence:
Use C-S with the functions X=(r-Xn) and Y=the indicator function
for {Max0≤i≤nXi <r} .
3. Let 𝑆𝑛 = ∑𝑛𝑖=1 𝑋𝑖 be a p-biased random walk on the integers. (9)
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. (preferably using martingales) :
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)
4. 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). (7)
5. In the primary elections in the "Israel" party, candidate A got a votes
and overcame candidate B who got only b votes. Assume the ballots
are counted in a random order one at a time. Show that the probability
that candidate A lead over candidate b throughout the whole count is
(a-b)/(a+b). (7)
Hint: Let Sk be (#A’s votes) − (#B’s votes) after k votes are
counted; Let n=a+b and look at the sequence Xk =(Sn-k)/(n-k).