Martingales. Discrete and Continuous Time

Let a random variable $\xi \in L^{2}(\Omega , \mathscr {F}, \mathbb {P})$ and a $\sigma$-algebra $\mathscr {A} \subset \mathscr {F}$ be given. Denote by $\operatorname {Pr}_{H}$ the operator of orthogonal projection onto the subspace $H = L^{2}(\Omega , \mathscr {A}, \mathbb {P}) \subset L^{2}(\Omega , \mathscr {F}, \mathbb {P})$. Show that $\mathbb {E}\left[\xi \mid \mathscr {A}\right] = \operatorname {Pr}_{H} \xi$.

Note that since the space $L^{2}(\Omega , \mathscr {F}, \mathbb {P})$ is dense in the space $L^{1}(\Omega , \mathscr {F}, \mathbb {P})$, the operator $\operatorname {Pr}_{H}$ can be extended by continuity to $L^{1}(\Omega , \mathscr {F}, \mathbb {P})$.

Let $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ be a filtration. Prove that $\tau$ is an optional time with respect to the filtration $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ if and only if $\tau$ is a Markov time with respect to the filtration $(\mathscr {F}_{t+})_{t \geqslant 0}$, where $\mathscr {F}_{t+} = \bigcap_{s > t} \mathscr {F}_{s}$, $t \geqslant 0$.

Let $\left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ be a filtration and $\tau$ an optional time with respect to it. Introduce the $\sigma$-algebra

Can one assert that in relation (51) the $\sigma$-algebra $\mathscr {G} = \mathscr {F}_{\sigma +}$?

Relation (51) arises in the proof of Lemma 7. Lemma 7: let $X = \left(X_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a martingale with a.s. right-continuous sample paths, and let $\sigma$ and $\tau$ be optional times (with respect to the filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$) such that $\sigma \leqslant \tau \leqslant c$ a.s., where $c$ is a positive constant; then $\mathbb {E}\left[X_{\tau }\right] = \mathbb {E}\left[X_{\sigma }\right]$. In the proof one sets $\sigma (n) = 2^{-n}\left[2^{n} \sigma +1\right]$, $n \in \mathbb {N}$, where $[\cdot ]$ denotes the integer part of a number, and

where $\overline{\mathscr {A}} = \sigma \left\{ \mathscr {A}, \mathscr {N}\right\}$ for a $\sigma$-algebra $\mathscr {A}$, and $\mathscr {N}$ is the class of $\mathbb {P}$-null sets.

Let $\left\{ \xi_{n}\right\}_{n \geqslant 1}$ be a sequence of integrable random variables, $\mathscr {F}_{n} = \sigma \left\{ \xi_{1}, \ldots , \xi_{n}\right\} , X_{n} = \sum_{k = 1}^{n} \xi_{k}, n \in \mathbb {N}$. Prove that $\left(X_{n}, \mathscr {F}_{n}\right)_{n \geqslant 1}$ is a martingale if and only if

for every bounded Borel function $f_{n}: \mathbb {R}^{n} \rightarrow \mathbb {R}$ and all $n \geqslant 1$.

Prove that if $\left(\xi_{n}, \mathscr {F}_{n}\right)_{n \geqslant 1}$ is a sequence of independent centered random variables and $\mathscr {F}_{n} = \sigma \left\{ \xi_{1}, \ldots , \xi_{n}\right\} , n \geqslant 1$, then the process $\left(S_{n, m}, \mathscr {F}_{n}\right)_{n \geqslant m}$ of the form

is a martingale with zero mean for each $m \geqslant 1$.

Let an urn contain $a$ white and $b$ black balls. Balls are drawn from the urn successively at random, one at a time; each drawn ball is returned to the urn together with $d$ additional balls of the same color as the drawn one. Denote by $X_{n}(n \geqslant 1)$ the ratio of the number of white balls to the total number of balls in the urn after the $n$-th drawing of a ball and its return to the urn together with $d$ balls of the corresponding color $\left(X_{0}: = a /(a+b)\right)$. Prove that $X = \left\{ X_{n}, n \geqslant 0\right\}$ is a martingale.

Let $(\Omega , \mathscr {F}, \mathbb {P}) = ([0,1], \mathscr {B}[0,1], \mathrm{mes})$, where mes is the Lebesgue measure. Let $\left\{ T_{n}, n \geqslant 1\right\}$ be a sequence of refining partitions of the segment $[0,1]$, i.e., $T_{n}$ consists of the points $0 = t_{n, 0} < \cdots < t_{n, m_{n}} = 1$ and $T_{n} \subset T_{n+1}, n \geqslant 1, m_{n} \geqslant 2$. Introduce the $\sigma$-algebras $\mathscr {F}_{n}$ on $[0,1]$ generated by the sets $\Delta_{n, 1} = \left[0, t_{n, 0}\right]$, $\Delta_{n, k} = \left(t_{n, k-1}, t_{n, k}\right], 2 \leqslant k \leqslant m_{n}, n \geqslant 1$. For a function $f:[0,1] \rightarrow \mathbb {R}$ set

Prove that $\left(X_{n}, \mathscr {F}_{n}\right)_{n \geqslant 1}$ is a martingale.

Under the conditions of the previous exercise, let the function $f$ satisfy the Lipschitz condition, i.e., $\left|f(x)-f(y)\right| \leqslant L\left|x-y\right|$ for $x, y \in [0,1]$ and some constant $L > 0$. Let the partitions $T_{n}$ be such that $\max_{0 \leqslant k \leqslant m_{n}}\left(t_{n, k}-t_{n, k-1}\right) \rightarrow 0$, $n \rightarrow \infty$. Prove that $\bigvee_{n = 1}^{\infty } \mathscr {F}_{n}\left(: = \sigma \left\{ \mathscr {F}_{n}, n \in \mathbb {N}\right\} \right) = \mathscr {B}([0,1])$ and the family $\left\{ X_{n}, n \geqslant 1\right\}$ is uniformly integrable. Then by Theorem 7, $X_{n} \rightarrow X_{\infty }$ a.s. and in $L^{1}$ as $n \rightarrow \infty$, where $X_{\infty } \in \mathscr {B}([0,1]) \mid \mathscr {B}(\mathbb {R})$.

Theorem 7. Let $X = \left(X_{n}, \mathscr {F}_{n}\right)_{n \geqslant 1}$ be a martingale. The following conditions are equivalent:

1. $X_{n} = \mathbb {E}\left[\xi \mid \mathscr {F}_{n}\right]$, $n \geqslant 1$, where $\xi$ is some integrable random variable;

2. the family $\left\{ X_{n}, n \geqslant 1\right\}$ is uniformly integrable;

3. there exists an integrable random variable $X_{\infty }$ such that convergence in $L^{1}$ holds:

   $$\mathbb {E}\left[\left|X_{\infty }-X_{n}\right|\right] \rightarrow 0 \quad \text{as} \quad n \rightarrow \infty ;$$

4. $\sup_{n} \mathbb {E}\left[\left|X_{n}\right|\right] < \infty$ and $X_{k} = \mathbb {E}\left[X_{\infty } \mid \mathscr {F}_{k}\right]$ with $X_{\infty } = \lim_{n \rightarrow \infty } X_{n}$ (the limit exists a.s. and is integrable by Theorem 6), $k \geqslant 1$.

Consequently, for any $[a, b] \subset [0,1]$

Since also $\int_{a}^{b} X_{n} d t \rightarrow f(b)-f(a)$ as $n \rightarrow \infty$ (explain why), together with (63) we conclude from this that the function $f$ is absolutely continuous and its density (with respect to Lebesgue measure) can be found as the limit of the variables $X_{n}, n \rightarrow \infty$.

Let $\left\{ X_{n}\right\}_{n \geqslant 1}$ be a sequence of real random variables defined on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$, and let $f_{n}^{0}\left(x_{1}, \ldots , x_{n}\right)$ and $f_{n}^{1}\left(x_{1}, \ldots , x_{n}\right)$ be some probability densities with respect to Lebesgue measure on $\left(\mathbb {R}^{n}, \mathscr {B}\left(\mathbb {R}^{n}\right)\right), n \geqslant 1$. Consider the likelihood ratio

assuming that $f_{n}^{1}(z) = 0$ if $f_{n}^{0}(z) = 0$ (in which case $L_{n}(z): = 0$ ), $z \in \mathbb {R}^{n}$. Show that $\left(L_{n}, \mathscr {F}_{n}\right)_{n \geqslant 1}$ is a martingale, where $\mathscr {F}_{n} = \sigma \left\{ X_{1}, \ldots , X_{n}\right\} , n \geqslant 1$.

Let $T \subset \mathbb {R}_{+}$ and let $\left(X_{t}, \mathscr {F}_{t}\right)_{t \in T}$ be a complex-valued martingale, i.e., $((\operatorname {Re} X_{t}, \operatorname {Im} X_{t}), \mathscr {F}_{t})_{t \in T}$ is a martingale with values in $\mathbb {R}^{2}$, each component of which is (by definition) a martingale. Then $\mathbb {E}\left[X_{t} \mid \mathscr {F}_{s}\right] = X_{s}$ a.s. for $s \leqslant t$, $s, t \in T$ (here $\mathbb {E}\left[Y+i Z \mid \mathscr {A}\right]: = \mathbb {E}\left[Y \mid \mathscr {A}\right]+i \mathbb {E}\left[Z \mid \mathscr {A}\right]$ for integrable $Y, Z$ and a $\sigma$-algebra $\mathscr {A} \subset \mathscr {F})$. Let $\mathbb {E}\left[\left|X_{t}\right|^{2}\right] < \infty$ for $t \in T$. Prove that $X = \left\{ X_{t}, t \in T\right\}$ is a process with uncorrelated increments, i.e., $\mathbb {E}\left[(X_{t}-X_{s})\left(\overline{X_{u}-X_{v}}\right)\right] = 0$ for $v < u \leqslant s < t, v, u, s, t \in T$. (Thus, square-integrable martingales occupy an intermediate position between processes with independent increments having constant mean and processes with uncorrelated increments.)

Let $\left\{ \xi_{n}\right\}_{n \geqslant 1}$ be a sequence of independent real random variables. Set $S_{n} = \sum_{k = 1}^{n} \xi_{k}, n \geqslant 1$. Using martingale methods, prove that the following statements about the series $\sum_{k = 1}^{\infty } \xi_{k}$ are equivalent:

1. the series converges a.s.,

2. the series converges in probability,

3. the series converges in distribution.

Note that by Kolmogorov's zero-one law (Theorem 6 of Chapter III) this series converges or diverges with probability one.

Theorem 6 of Chapter III (zero-one law). Let a family $\left\{ X_{t}, t \in T\right\}$, $T \subset \mathbb {R}$, of independent random elements with values in measurable spaces $(\mathrm{S}_{t}, \mathscr {B}_{t})$, i.e. $X_{t}: \Omega \rightarrow \mathrm{S}_{t}$, $X_{t} \in \mathscr {F} \mid \mathscr {B}_{t}$, $t \in T$, be given on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$. Denote

Then the "tail" $\sigma$-algebra, i.e. the $\sigma$-algebra $\mathscr {G} = \bigcap_{t \in T} \mathscr {F}_{\geqslant t}^{X}$, is degenerate. In other words, for every $A \in \mathscr {G}$ the probability $\mathbb {P}\left(A\right)$ equals zero or one (if $T \cap [t, \infty ) = \varnothing$, then $\mathscr {F}_{\geqslant t}^{X}: = \varnothing$).

Let $Y = \left\{ Y_{t}, t \in T\right\} , T \subset \mathbb {R}$, be a (complex-valued) process with independent increments such that $\mathbb {E}\left[\left|Y_{t}\right|^{2}\right] < \infty , t \in T$. Prove that there exists a nonrandom function $h: T \rightarrow \mathbb {R}$ such that $X_{t} = \left|Y_{t}\right|^{2}-h(t)$ is a martingale with respect to the filtration $\mathscr {F}_{t} = \sigma \left\{ Y_{s}, s \leqslant t, s \in T\right\} = \sigma \left\{ \operatorname {Re} Y_{s}, \operatorname {Im} Y_{s}, s \leqslant t, s \in T\right\} , t \in T$, if and only if

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Brownian motion with the natural filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$.

Find all $\alpha , \beta \in \mathbb {R}$ for which the process

is a martingale.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Brownian motion with the natural filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$.

Let $\tau$ be a stopping time (with respect to the filtration $\mathbb {F}$ ). Prove that $\left(W_{t \wedge \tau }, \mathscr {F}_{t}\right)_{t \geqslant 0}$ is a martingale.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Brownian motion with the natural filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$.

For $\alpha , x \in \mathbb {R}$ and $t \geqslant 0$ define the function

Set

Prove that $\left(h_{k}\left(W_{t}, t\right), \mathscr {F}_{t}\right)_{t \geqslant 0}$ is a martingale for each $k \geqslant 1$ (note also that, by Exercise 13, the process $\left\{ h\left(\alpha , W_{t}, t\right), t \geqslant 0\right\}$ is a martingale for each $\alpha \in \mathbb {R}$ ). Thus the processes $W_{t}, W_{t}^{2}-t, W_{t}^{3}-3 t W_{t}$, $W_{t}^{4}-6 t W_{t}^{2}+3 t^{2}, \ldots$ are martingales $(t \geqslant 0)$.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Brownian motion with the natural filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$.

Prove that $\mathbb {E}\left[\tau_{a}^{2}\right] = 5 a^{4} / 3$ for $\tau_{a} = \inf \left\{ t \geqslant 0:\left|W_{t}\right| = a\right\} , a > 0$.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Brownian motion with the natural filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$.

Let $X_{t} = \gamma t+\sigma W_{t}, t \geqslant 0$, and $R_{a} = \inf \left\{ t: X_{t} = a\right\}$. Prove that if $\gamma > 0, \sigma \neq 0$ and $a < 0$, then

Does condition (9) hold if

(see [12; p. 166]). Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be an $m$-dimensional Brownian motion and $f(x)$ a continuous superharmonic function in $\mathbb {R}^{m}$, i.e., $f(x)$, for any $x \in \mathbb {R}^{m}$, is not less than the integral of this function over any sphere centered at $x$. Prove that the process $X = \left\{ X_{t} = f\left(W_{t}\right), t \geqslant 0\right\}$ is a supermartingale with respect to the natural filtration of the Brownian motion if $\mathbb {E}\left[\left|X_{t}\right|\right] < \infty$ for all $t \geqslant 0$.

(Krickeberg decomposition). A martingale $X = \left(X_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ has the property $\sup \mathbb {E}\left[\left|X_{n}\right|\right] < \infty$ if and only if there exist nonnegative martingales $Y = \left(Y_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ and $Z = \left(Z_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ such that $X_{n} = Y_{n}-Z_{n}, n \geqslant 0$.

(Riesz decomposition). Let $X = \left(X_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ be a uniformly integrable supermartingale. Then there exists a representation of the form $X_{n} = M_{n}+R_{n}$, where $M = \left(M_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ is a uniformly integrable martingale and $R = \left(R_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ is a potential, i.e., a uniformly integrable nonnegative supermartingale such that $R_{n} \rightarrow 0$ a.s. as $n \rightarrow \infty$.

Let $h: \mathbb {R} \rightarrow \mathbb {R}_{+}$ be a nondecreasing convex function and $\left(X_{n}, \mathscr {F}_{n}\right)_{1 \leqslant n \leqslant N}$ a submartingale. Show that then for any $u \in \mathbb {R}$ and $t > 0$

Let $\left(X_{n}, \mathscr {F}_{n}\right)_{1 \leqslant n \leqslant N}$ be a martingale or a nonnegative submartingale. Prove that

where $\log^{+} x = \log x$ for $x \geqslant 1$ and $\log^{+} x = 0$ for $x < 1$.

Construct an example (see [85; vol. 2, p. 678]) showing that for $p = 1$ the double inequality (68) need not hold.

Theorem 11 (Burkholder). Let $X = \left(X_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ be a martingale with $X_{0} = 0$. Then for each $p > 1$ there exist constants $A_{p}$ and $B_{p}$, not depending on $X$, such that

where $[X]_{n}$ is the quadratic variation of $X$ (see (22)).

Definition 5. The quadratic variation of a process $X = \left\{ X_{n}, n \geqslant 0\right\}$ is the process $[X] = \left([X]_{n}\right)_{n \geqslant 0}$ given by the formula

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be an $m$-dimensional Brownian motion. Prove that $\left(\left\| W_{t}\right\| , \mathscr {F}_{t}\right)_{t \geqslant 0}$ is a submartingale (with a.s. continuous sample paths), where $\left\| \cdot \right\|$ is the Euclidean norm in $\mathbb {R}^{m}$. Using Exercise 22 (with the function $h(x) = e^{t x}$ and a suitable value $t > 0$ ), prove that for every $s > 0$ and all $x > \sqrt{m s}$ the following inequality holds

(The process $\left(\left\| W_{t}\right\| , \mathscr {F}_{t}\right)_{t \geqslant 0}$ is called the Bessel process of dimension $m$. )

(see [12; p. 182]). Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be an $m$-dimensional Brownian motion, where $m \geqslant 3$. Prove that $\left\| W_{t}\right\| \rightarrow \infty$ a.s. as $t \rightarrow \infty$. Prove that with probability 1 the process $W$ does not return to the point 0 in any finite time. Explain why for $m = 1$ these statements are false. What does the corresponding result look like in the case $m = 2$ ?

Definition 7. Let $\left(\mathscr {G}_{t}\right)_{t \in T}$ be a decreasing family of $\sigma$-algebras in $\mathscr {F}$, i.e., $\mathscr {G}_{t} \subset \mathscr {G}_{s} \subset \mathscr {F}$ for $s < t ; s, t \in T \subset \mathbb {R}$. Let $X = \left\{ X_{t}, t \in T\right\}$ be a real-valued stochastic process adapted to the filtration $\left(\mathscr {G}_{t}\right)_{t \in T}$. The process $\overleftarrow {X} = \left(X_{t}, \mathscr {G}_{t}\right)_{t \in T}$ is called a reversed martingale (reversed submartingale, reversed supermartingale) if the (reversed) process $\left(X_{u}, \mathscr {G}_{u}\right)_{u \in U}$ with $U = -T = \left\{ -t, t \in T\right\}$ is a martingale (submartingale, supermartingale).

In particular, a stochastic sequence $\overleftarrow {X} = \left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 0}$ is a reversed martingale if $\mathbb {E}\left[X_{n} \mid \mathscr {G}_{n+1}\right] = X_{n+1}, n \geqslant 0$. Replacing " = " in the last equality by the sign " $\geqslant$ ", we obtain the definition of a reversed submartingale (with " $\leqslant$ ", that of a reversed supermartingale). The notion of a reversed martingale also extends to processes with values in $\mathbb {R}^{m}(m \geqslant 1)$.

Let $\xi_{1}, \ldots , \xi_{N}$ be independent identically distributed vectors in $\mathbb {R}^{m} (m \geqslant 1)$ with $\mathbb {E}\left[\left\| \xi_{1}\right\| \right] < \infty$, where $\left\| \cdot \right\|$ is the Euclidean norm. Set $X_{n} = (1 / n) \sum_{k = 1}^{n} \xi_{k}, \mathscr {G}_{n, N} = \sigma \left\{ X_{n}, \ldots , X_{N}\right\}$. Prove that $\left(X_{n}, \mathscr {G}_{n, N}\right)_{1 \leqslant n \leqslant N}$ is a reversed martingale.

Definition 7. Let $\left(\mathscr {G}_{t}\right)_{t \in T}$ be a decreasing family of $\sigma$-algebras in $\mathscr {F}$, i.e., $\mathscr {G}_{t} \subset \mathscr {G}_{s} \subset \mathscr {F}$ for $s < t ; s, t \in T \subset \mathbb {R}$. Let $X = \left\{ X_{t}, t \in T\right\}$ be a real-valued stochastic process adapted to the filtration $\left(\mathscr {G}_{t}\right)_{t \in T}$. The process $\overleftarrow {X} = \left(X_{t}, \mathscr {G}_{t}\right)_{t \in T}$ is called a reversed martingale (reversed submartingale, reversed supermartingale) if the (reversed) process $\left(X_{u}, \mathscr {G}_{u}\right)_{u \in U}$ with $U = -T = \left\{ -t, t \in T\right\}$ is a martingale (submartingale, supermartingale).

In particular, a stochastic sequence $\overleftarrow {X} = \left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 0}$ is a reversed martingale if $\mathbb {E}\left[X_{n} \mid \mathscr {G}_{n+1}\right] = X_{n+1}, n \geqslant 0$. Replacing " = " in the last equality by the sign " $\geqslant$ ", we obtain the definition of a reversed submartingale (with " $\leqslant$ ", that of a reversed supermartingale). The notion of a reversed martingale also extends to processes with values in $\mathbb {R}^{m}(m \geqslant 1)$.

Definition 8. A sequence of real random variables $\left(\xi_{n}\right)_{n \geqslant 1}$ is called exchangeable if $\left(\xi_{i_{1}}, \ldots , \xi_{i_{N}}\right) \stackrel{\mathscr {D}}{ = }\left(\xi_{1}, \ldots , \xi_{N}\right)$ for every $N \geqslant 1$ and every permutation $\left(i_{1}, \ldots , i_{N}\right)$ of the set $\left\{ 1, \ldots , N\right\}$.

Let $m \geqslant 1$ and let $g: \mathbb {R}^{m} \rightarrow \mathbb {R}$ be a Borel function. Consider the sequence of $U$-statistics of the form

where $C_{n}^{m} = n ! /(m !(n-m) !)$. Set $\mathscr {G}_{n, m} = \sigma \left\{ U_{k, m}, k \geqslant n\right\} , n \geqslant m$.

Prove that for the sequence of $U$-statistics introduced above, $\left(U_{n, m}, \mathscr {G}_{n, m}\right)_{n \geqslant m}$ is a reversed martingale.

Let $\mathbb {E}\left[\left|X\right|\right] < \infty$ and let $\left(\mathscr {G}_{n}\right)_{n \geqslant 1}$ be a nonincreasing sequence of $\sigma$-algebras in $(\Omega , \mathscr {F}, \mathbb {P})$. Set $\mathscr {G}_{\infty } = \bigcap_{n = 1}^{\infty } \mathscr {G}_{n}$. Then as $n \rightarrow \infty$

Let $m \geqslant 1$ and let $g: \mathbb {R}^{m} \rightarrow \mathbb {R}$ be a Borel function. Consider the sequence of $U$-statistics of the form

where $C_{n}^{m} = n ! /(m !(n-m) !)$. Set $\mathscr {G}_{n, m} = \sigma \left\{ U_{k, m}, k \geqslant n\right\} , n \geqslant m$.

(continuation of Exercise 28). By Theorem 14 we obtain that

Definition 9. A one-to-one mapping $\pi : \mathbb {N} \rightarrow \mathbb {N}$ is called a finite permutation (we write $\pi \in \Pi (\mathbb {N})$ ) if $\pi (n) \neq n$ for at most finitely many (depending on $\pi$ ) values $n \in \mathbb {N}$. For a sequence of real random variables $\left\{ \xi_{k}\right\}_{k \geqslant 1}$ defined on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ (i.e., for a random element $\xi = \left(\xi_{1}, \xi_{2}, \ldots \right) \in \mathscr {F} \mid \mathscr {B}\left(\mathbb {R}^{\infty }\right)$ ), set $\pi \xi = \left(\xi_{\pi_{1}}, \xi_{\pi_{2}}, \ldots \right)$, where $\pi = \left(\pi_{1}, \pi_{2}, \ldots \right)$ is a permutation of the set $\mathbb {N}$, and define in $\mathscr {F}$ the $\sigma$-algebra of exchangeable events

Theorem 14 (zero-one law; Hewitt-Savage). Let $\left\{ \xi_{k}\right\}_{k \geqslant 1}$ be a sequence of independent identically distributed random variables. Then the $\sigma$-algebra $\mathscr {G}$ of exchangeable events is degenerate, i.e., it contains only events of probability 0 and 1.

Using Theorem 13, prove that if $\left\{ \xi_{k}\right\}_{k \geqslant 1}$ is a sequence of independent identically distributed random variables, then $U_{\infty , m} = \mathbb {E}\left[g\left(\xi_{1}, \ldots , \xi_{m}\right)\right]$.

Theorem 13. Let $\left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 0}$ be a reversed submartingale. Then a.s. there exists $\lim_{n \rightarrow \infty } X_{n} = X_{\infty } \in [-\infty , \infty )$. If, in addition, there exists

then $X_{\infty } \in (-\infty , \infty )$ and $X_{n} \rightarrow X_{\infty }$ in the space $L^{1}(\Omega , \mathscr {F}, \mathbb {P})$ as $n \rightarrow \infty$. If $X_{n} \geqslant 0$ a.s. for $n \geqslant 0$ and $\mathbb {E}\left[X_{0}^{p}\right] < \infty$ for some $p \in (1, \infty )$, then $X_{n} \rightarrow X_{\infty }$ in $L^{p}(\Omega , \mathscr {F}, \mathbb {P})$ as $n \rightarrow \infty$.

Definition 7. Let $\left(\mathscr {G}_{t}\right)_{t \in T}$ be a decreasing family of $\sigma$-algebras in $\mathscr {F}$, i.e., $\mathscr {G}_{t} \subset \mathscr {G}_{s} \subset \mathscr {F}$ for $s < t ; s, t \in T \subset \mathbb {R}$. Let $X = \left\{ X_{t}, t \in T\right\}$ be a real-valued stochastic process adapted to the filtration $\left(\mathscr {G}_{t}\right)_{t \in T}$. The process $\overleftarrow {X} = \left(X_{t}, \mathscr {G}_{t}\right)_{t \in T}$ is called a reversed martingale (reversed submartingale, reversed supermartingale) if the (reversed) process $\left(X_{u}, \mathscr {G}_{u}\right)_{u \in U}$ with $U = -T = \left\{ -t, t \in T\right\}$ is a martingale (submartingale, supermartingale).

In particular, a stochastic sequence $\overleftarrow {X} = \left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 0}$ is a reversed martingale if $\mathbb {E}\left[X_{n} \mid \mathscr {G}_{n+1}\right] = X_{n+1}, n \geqslant 0$. Replacing " = " in the last equality by the sign " $\geqslant$ ", we obtain the definition of a reversed submartingale (with " $\leqslant$ ", that of a reversed supermartingale). The notion of a reversed martingale also extends to processes with values in $\mathbb {R}^{m}(m \geqslant 1)$.

Definition 7. Let $\left(\mathscr {G}_{t}\right)_{t \in T}$ be a decreasing family of $\sigma$-algebras in $\mathscr {F}$, i.e., $\mathscr {G}_{t} \subset \mathscr {G}_{s} \subset \mathscr {F}$ for $s < t ; s, t \in T \subset \mathbb {R}$. Let $X = \left\{ X_{t}, t \in T\right\}$ be a real-valued stochastic process adapted to the filtration $\left(\mathscr {G}_{t}\right)_{t \in T}$. The process $\overleftarrow {X} = \left(X_{t}, \mathscr {G}_{t}\right)_{t \in T}$ is called a reversed martingale (reversed submartingale, reversed supermartingale) if the (reversed) process $\left(X_{u}, \mathscr {G}_{u}\right)_{u \in U}$ with $U = -T = \left\{ -t, t \in T\right\}$ is a martingale (submartingale, supermartingale).

In particular, a stochastic sequence $\overleftarrow {X} = \left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 0}$ is a reversed martingale if $\mathbb {E}\left[X_{n} \mid \mathscr {G}_{n+1}\right] = X_{n+1}, n \geqslant 0$. Replacing " = " in the last equality by the sign " $\geqslant$ ", we obtain the definition of a reversed submartingale (with " $\leqslant$ ", that of a reversed supermartingale). The notion of a reversed martingale also extends to processes with values in $\mathbb {R}^{m}(m \geqslant 1)$.

Let $\left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 1}$ be a reversed submartingale, and suppose condition (71) holds. Prove that $X_{\infty } \leqslant \mathbb {E}\left[X_{m} \mid \mathscr {G}_{\infty }\right], m \geqslant 1$, where $\mathscr {G}_{\infty } = \bigcap_{n = 1}^{\infty } \mathscr {G}_{n}$ (recall that $X_{\infty } = \lim_{n \rightarrow \infty } X_{n}$ exists a.s. and in $L^{1}(\Omega , \mathscr {F}, \mathbb {P})$ by Theorem 13). In other words, the "closed" process $\left(X_{n}, \mathscr {G}_{n}\right)_{1 \leqslant n \leqslant \infty }$ is a reversed submartingale. Show that if $\left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 1}$ is a reversed martingale, then $\left(X_{n}, \mathscr {G}_{n}\right)_{1 \leqslant n \leqslant \infty }$ is also a reversed martingale.

Theorem 13. Let $\left(X_{n}, \mathscr {G}_{n}\right)_{n \geqslant 0}$ be a reversed submartingale. Then a.s. there exists $\lim_{n \rightarrow \infty } X_{n} = X_{\infty } \in [-\infty , \infty )$. If, in addition, there exists

then $X_{\infty } \in (-\infty , \infty )$ and $X_{n} \rightarrow X_{\infty }$ in the space $L^{1}(\Omega , \mathscr {F}, \mathbb {P})$ as $n \rightarrow \infty$. If $X_{n} \geqslant 0$ a.s. for $n \geqslant 0$ and $\mathbb {E}\left[X_{0}^{p}\right] < \infty$ for some $p \in (1, \infty )$, then $X_{n} \rightarrow X_{\infty }$ in $L^{p}(\Omega , \mathscr {F}, \mathbb {P})$ as $n \rightarrow \infty$.