Martingale Convergence Theorems

Let \(X_{n}, Y_{n}\) be positive, in \(L^{1}\), and measurable with respect to the filtration \(\mathcal {F}_{n}\). Suppose

with \(\sum Y_{n} < \infty \) a.s.

1. Show that \(\prod _{i=1}^{n}(1 + Y_{i})\) converges a.s. to a finite limit.

2. Show

   \[ M_{n} = \frac {X_{n}}{\prod _{i=0}^{n-1}(1 + Y_{i})} \]
   is a super-martingale.
3. Use (a) and (b) to prove that \(X_{n}\) converges a.s. to a finite limit.

Consider a probability space \((\Omega , \mathcal {F}, P)\). Let \(X \in L^{1}\). Suppose \((\mathcal {F}_{n})_{n \in \mathbb {N}}\) is an increasing sequence of \(\sigma \)-algebras such that \(\cup _{n \in \mathbb {N}}\mathcal {F}_{n}\) generates \(\mathcal {F}\). Prove the Doob martingale convergence theorem that states that \(\mathbb {E}[X \mid \mathcal {F}_{n}]\) converges almost surely to \(X\).

Let \(Y_{k}\) be i.i.d., \(Y_{k} \ge 0\), and \(\mathbb {E}[Y_{k}] \le 1\) for all \(k\). Prove the existence of the almost sure limit \(M_{\infty } = \lim _{n \to \infty } \prod _{i=1}^{n} Y_{i}\) and describe this random variable explicitly. Do not forget the degenerate cases.

Let \(Z_n\) be a martingale difference with respect to a filtration \(\mathcal {F}_n\), i.e. \(\mathbb {E}[Z_{n+1}\mid \mathcal {F}_n]=0\) for all \(n\geq 0\). Assume the following hold:

1. \(n^{-1/2}\max _{m\leq n}|Z_m|\to 0\) in probability.
2. \(n^{-1}\sum _{m=1}^{n}|Z_m|^2\to 1\) in probability.
3. \(\sum _{m=1}^{n}|Z_m|^2\leq 2n\) almost surely, for all \(n\).

Prove that \(M_n=n^{-1/2}\sum _{m=1}^{n}Z_m\) converges in distribution to a standard normal.

Hint: Prove that \(|1+ix|\leq e^{x^2}\) for all \(x\in \mathbb {R}\) and \(e^{ix}=(1+ix)e^{-x^2/2+r(x)}\) with \(|r(x)|\leq x^3\) for all \(x\in \mathbb {R}\). Let \[ T_n=\prod _{m=1}^{n}\left (1+itn^{-1/2}Z_m\right ) \] and \[ U_n= e^{-\frac {t^2}{2n}\sum _{m=1}^{n}Z_m^2 +\sum _{m=1}^{n}r(tn^{-1/2}Z_m)}. \] Compute \(\mathbb {E}[T_n]\) then prove that \(\mathbb {E}[T_nU_n]\to e^{-t^2/2}\) and conclude.

Let \(X_1,X_2,\ldots \) be independent random variables with \[ X_n= \begin {cases} 1, & \text {with probability }1/(2n),\\ 0, & \text {with probability }1-1/n,\\ -1, & \text {with probability }1/(2n). \end {cases} \] Let \(Y_1=X_1\) and for \(n\geq 2\) define \[ Y_n= \begin {cases} X_n, & \text {if }Y_{n-1}=0,\\ nY_{n-1}|X_n|, & \text {if }Y_{n-1}\ne 0. \end {cases} \] Show that \(Y_n\) is a martingale with respect to \(\mathcal {F}_n=\sigma (Y_1,\ldots ,Y_n)\). Show that \(Y_n\) does not converge almost surely. Why does the martingale convergence theorem not apply?

(Pólya’s Urn) An urn initially contains \(r\) red and \(b\) blue balls. A ball is chosen uniformly at random (i.e. with probability \(1/(r+b)\) each). If If it comes up red (resp. blue), then it is returned and another red (resp. blue) ball is added to the urn. The process is repeated indefinitely. Let \(R_n\) be the number of red balls in the urn after \(n\) draws.

a) Prove that the fraction of red balls \(M_n=R_n/(r+b+n)\) has an almost sure limit \(M_\infty \) as \(n\to \infty \).

b) Suppose now \(r=b=1\). Prove that \(\mathbb {P}(R_n=k)=1/(n+1)\) for \(1\leq k\leq n+1\). Hint: Write down the probability of choosing \(k\) red and \(n-k\) blue balls in some fixed order.

c) Still assuming \(r=b=1\), what is the distribution of \(M_\infty \)? Hint: use part b) to compute \(\mathbb {P}(M_\infty \leq x)\).

Let \(M_n\) be a martingale with \(M_0=0\) and increments \(X_n=M_n-M_{n-1}\) that satisfy \(\mathbb {E}[X_n^2]<\infty \) for all \(n\). Assume that \((b_n)_{n=1}^{\infty }\) is an increasing sequence of positive real numbers such that \(b_n\to \infty \) and \[ \sum _{n=1}^{\infty }\frac {\mathbb {E}[X_n^2]}{b_n^2}<\infty . \] Show that \(M_n/b_n\to 0\) almost surely as \(n\to \infty \). Hint: Since the sum of the \(X_k\) is a martingale so too should be a weighted sum; prove this idea and use it to produce a new martingale out of the \(X_k\) and \(b_k\). Derive convergence properties of this new martingale, and then use Kronecker’s Lemma: if \(a_n\) is a sequence of numbers such that \(\sum _{n=1}^{\infty }a_n\) exists and is finite and \(b_n\) is a sequence of the type above, then \[ \frac {1}{b_n}\sum _{k=1}^{n}a_kb_k\to 0 \qquad \text {as } n\to \infty . \]

Suppose \((\Omega ,\mathcal {F},\mathbb {P})\) is a probability space, and \(\mathcal {F}_1\subset \mathcal {F}_2\subset \cdots \) defines a filtration of sigma-algebras of subsets of \(\mathcal {F}\).

(a) State, without proof, Doob’s martingale convergence theorem.

(b) Prove that \(\mathcal {L}:=\lim _{n\to \infty }\mathbb {E}(Z\mid \mathcal {F}_n)\) exists almost surely and in \(L^1(\mathbb {P})\) for all \(Z\in L^1(\mathbb {P})\) [this is Lévy’s martingale convergence theorem]. Identify the limit \(\mathcal {L}\).

Let \(X_1,X_2,\ldots \) be i.i.d. uniform\((0,2)\), and put \(M_n:=X_1X_2\cdots X_n\) for each \(n\geq 1\). Show that \(\{M_n\}_{n=1}^{\infty }\) is a martingale. What does the martingale convergence theorem tell us about \(\lim _{n\to \infty }M_n\)? (In particular, evaluate this limit.)

Let \(X_1,X_2,\ldots \) be independent random variables such that \[ X_n= \begin {cases} 1 & \text {with probability }(2n)^{-1},\\ 0 & \text {with probability }1-n^{-1},\\ -1 & \text {with probability }(2n)^{-1}. \end {cases} \] Let \(Y_1:=X_1\) and for \(n\geq 2\) \[ Y_n:= \begin {cases} X_n & \text {if }Y_{n-1}=0,\\ nY_{n-1}|X_n| & \text {if }Y_{n-1}\neq 0. \end {cases} \] Show that \(Y_n\) is a martingale with respect to \(\mathcal {F}_n:=\sigma (Y_1,Y_2,\ldots ,Y_n)\). Show that \(Y_n\) does not converge almost surely. Does \(Y_n\) converge in any way? Why does the martingale convergence theorem not apply?