Homework 1. Due Mar 7, 2017
Q1 Let Xn be a submartingale and let Sn = max1≤i≤n Xi . By mimicking the proof of Doob’s
Lp maximal inequality and by using the elementary inequality a log b ≤ a log a + b/e for
a, b > 0, show that
1 + E[Xn+ log+ (Xn+ )]
E[Sn+ ] ≤
,
1 − e−1
where log+ x = max{0, log x}.
Pn
Q2. Let Xn =
i=1 ξi be a random walk where ξi are i.i.d., and let τ be an integrable
stopping time w.r.t. the canonical filtration generated by (Xn )n∈N .
(i) Show that if E|ξ1 | < ∞, then Wald’s first identity E[Xτ ] = E[ξ1 ]E[τ ] holds. (Hint: first
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show that Xτ = ∞
i=1 ξi 1{τ ≥i} ).
(ii) If E[ξ1 ] = 0 and E[ξ12 ] < ∞, then show that Wald’s second identity E[Xτ2 ] = E[ξ12 ]E[τ ]
holds. (Hint: use the fact that Xn2 − nE[ξ12 ] is a martingale and show that Xn∧τ is a
Cauchy sequence in L2 ).
Q3. Let (Xn )n∈N be an L1 -bounded martingale, i.e., supn∈N E|Xn | = K < ∞. Prove that
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almost surely, n≥2 (Xn − Xn−1 )2 < ∞. (Hint: for L > 0, let τL := inf{n ∈ N : |Xn | ≥ L}.
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Then show that E[ ni=2 (Xn − Xn−1 )2 1{τL >n} ] is uniformly bounded in n. )
Q4. Let Z := f (X1 , X2 , . . . , Xn ) be a function of n independent random variables X1 , . . . , Xn .
Let Xi0 be an independent copy of Xi , 1 ≤ i ≤ n. Use martingale decomposition to prove that
Var(Z) ≤
n
2 1X E f (X1 , . . . , Xi−1 , Xi0 , Xi+1 , . . . , Xn ) − f (X1 , . . . , Xn ) .
2
i=1
This inequality gives a way to bound the variance of Z. (Hint: First prove that Var(Y ) =
1
0 2
0
2 E[(Y − Y ) ] for any random variable Y , with Y being an independent copy of Y .)
Q5. Let Sn be the number of individuals in the n-th generation of a branching process (also
called Galton-Watson process). Assume S0 = 1. Each of the Sn individuals in generation n
independently produces a random number of offsprings with common distribution µ for the
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(n + 1)-st generation. Let m := ∞
k=0 kµ(k) be the mean number of off-springs. Prove that
the branching process dies out almost surely, i.e., almost surely Sn = 0 for all n sufficiently
large, if m ≤ 1 and µ(0) > 0. (Hint: Sn /mn is a martingale.)
Q6. Let ξ1 ξ2 · · · ξn · · · be a sequence of i.i.d. Bernoulli random variables with P(ξ1 = 0) =
P(ξ1 = 1) = 21 . Let τ := min{n ≥ 5 : ξn−4 ξn−3 ξn−2 ξn−1 ξn = 10101}. Find E[τ ] by constructing a suitable martingale. (Hint: Make use of the following game. At each time n ∈ N, a new
player enters the game and bets $1. If ξn = 1, then his money doubles to $2 and he stays in
the game; otherwise he loses his money. If ξn+1 = 0, then his money doubles again and he
stays in the game; otherwise he loses all his money. If ξn+2 = 1, then his money doubles yet
again; otherwise he loses all his money. This continues until the player either sees 10101, in
which case he leaves the game with $32; or otherwise he leaves the game earlier with nothing.)
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