Processes with Independent Increments. Poisson and Gaussian Processes

Find the covariance function of a Poisson process with constant intensity $\lambda$. Compare the answer obtained with the covariance function of the Wiener process.

Let $r = r(s, t)$ be a nonnegative definite function $(s, t \in T$, where $T$ is some set). Does there exist a non-Gaussian process $X = \left\{ X_{t}, t \in T\right\}$ such that $\operatorname {cov}\left(X_{s}, X_{t}\right) = r(s, t)$ for all $s, t \in T$ ?

Prove that if $P_{n}\left(z_{1}, \ldots , z_{m}\right)$ is a polynomial of degree $n$ (in $m$ variables) with positive coefficients and $r_{k}(s, t)(k = 1, \ldots , m ; s, t \in T)$ are covariance functions, then $P_{n}\left(r_{1}(s, t), \ldots , r_{m}(s, t)\right)$ is also a covariance function.

For an $L^{2}$-process $X = \left\{ X_{t}, t \in T\right\}$ defined on $(\Omega , \mathscr {F}, \mathbb {P})$, denote by $L^{2}[X]$ the closure in the space $L^{2}(\Omega , \mathscr {F}, \mathbb {P})$ of the linear span of all the variables $X_{t}$, $t \in T$. Prove that if $X$ is a Gaussian process, then $L^{2}[X]$ is a Gaussian system, i.e., for any $n \in \mathbb {N}$ and any $Y_{1}, \ldots , Y_{n} \in L^{2}[X]$ the vector $\left(Y_{1}, \ldots , Y_{n}\right)$ has a Gaussian distribution.

(self-similarity property). Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process and $c$ a positive constant. Prove that the process $X = \left\{ X(t) = \frac{1}{\sqrt{c}} W(c t), t \geqslant 0\right\}$ is a Wiener process.

Prove Theorem 8 by verifying the original definition of the Wiener process, i.e., prove that the process given by formula (31) has independent increments. By Doob's theorem (see Appendix 5), this process with continuous sample paths will automatically be Gaussian.

Theorem 8. Let $\left\{ \xi_{k}\right\}_{k \geqslant 1}$ be a sequence of independent random variables having the standard normal distribution $\mathscr {N}\left(0, 1\right)$ and defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$. The Haar functions $H_{k}(t)$, $t \in [0,1]$, $k = 1, 2, \ldots$, are given by the formulas $H_{1}(t) \equiv 1$, $H_{2}(t) = \mathbf{1}_{[0,1 / 2]}(t)-\mathbf{1}_{(1 / 2,1]}(t)$, and for $2^{n} < k \leqslant 2^{n+1}$

where $I_{n, k} = \left[a_{n, k}, a_{n, k}+2^{-n-1}\right]$, $J_{n, k} = \left(a_{n, k}+2^{-n-1}, a_{n, k}+2^{-n}\right]$, $a_{n, k} = 2^{-n}\left(k-2^{n}-1\right)$, $n \in \mathbb {N}$; the Schauder functions are $S_{k}(t) = \int_{0}^{t} H_{k}(y)\, d y$, $t \in [0,1]$, $k \in \mathbb {N}$. Formula (31) is the equality

and Theorem 8 asserts that $W = \left\{ W(t), t \in [0,1]\right\}$ is a Wiener process on $[0,1]$ which, moreover, has continuous sample paths with probability one.

Definition 3. A Wiener process, or Brownian motion, is a stochastic process $W = \left\{ W(t), t \geqslant 0\right\}$ such that

1. $W(0) = 0$ a.s.;

2. the process $W$ has independent increments;

3. the variables $W(t)-W(s) \sim \mathscr {N}\left(0, t-s\right)$ for all $0 \leqslant s < t < \infty$, i.e., the variables $W(t)-W(s)$ have the Gaussian (normal) distribution with parameters 0 and $t-s$.

Doob's theorem (Theorem 14 of Chapter V). Let a process $X = \left\{ X_{t}, t \geqslant 0\right\}$ take values in $\mathbb {R}^{m}$ for each $t$ and have independent increments. If the sample paths of the process are continuous a.s., then the process $Y = \left\{ Y_{t} = X_{t}-X_{0}, t \geqslant 0\right\}$ is Gaussian.

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

Prove that if $S = \left\{ S_{n}, n \geqslant 0\right\}$ is a recurrent random walk in $\mathbb {R}^{m}$, then for every $k \in \mathbb {N}$ the walk $S^{(k)} = \left\{ S_{n k}, n \geqslant 0\right\}$ is recurrent as well.

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

Let the random variable $\xi_{1}$ have a symmetric stable distribution on $\mathbb {R}$ with exponent $\alpha \in (0,2]$, i.e., let

Prove that such an $\alpha$-stable random walk is recurrent for $\alpha \geqslant 1$ and transient for $\alpha < 1$.

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

Let $S = \left\{ S_{n}, n \geqslant 0\right\}$ be a random walk in $\mathbb {R}^{m}(m \geqslant 2)$ such that the components of the vector $\xi_{1}$ are independent and have an $\alpha$-stable distribution (see Exercise 8). Investigate for which values of $\alpha$ the random walk is recurrent and for which it is transient.

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

The symmetrization of a random walk $S = \left\{ S_{n}, n \geqslant 0\right\}$ in $\mathbb {R}^{m}$ is a sequence $\widetilde{S} = \left\{ \widetilde{S}_{n}, n \geqslant 0\right\}$ with the same distribution as $\left\{ S_{n}-S_{n}^{\prime }, n \geqslant 0\right\}$, where $S_{n}^{\prime } = \left\{ S_{n}^{\prime }, n \geqslant 0\right\}$ is an independent copy of the sequence $\left\{ S_{n}\right\}$ (one may assume that for this purpose an independent copy $\left\{ \xi_{n}^{\prime }, n \geqslant 0\right\}$ of the sequence $\left\{ \xi_{n}, n \geqslant 0\right\}$ is constructed and $S_{0}^{\prime } = 0, S_{n}^{\prime } = \sum_{j = 1}^{n} \xi_{j}^{\prime }, n \in \mathbb {N}$).

Prove that if the random walk $S = \left\{ S_{n}, n \geqslant 0\right\}$ is recurrent, then the symmetrized random walk $\widetilde{S} = \left\{ \widetilde{S}_{n}, n \geqslant 0\right\}$ is recurrent as well.

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

Prove that a random walk $S = \left\{ S_{n}, n \geqslant 0\right\}$ in $\mathbb {R}^{m}$ is recurrent if

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

Definition 16. The support of an arbitrary measure $\lambda$ on the Borel $\sigma$-algebra of a topological space $\mathrm{S}$ is the smallest closed set $F \subset \mathrm{S}$ such that $\lambda (\mathrm{S} \backslash F) = 0$.

Let $S = \left\{ S_{n}, n \geqslant 0\right\}$ be a random walk in $\mathbb {R}$, where $\xi_{1}$ has a symmetric nondegenerate distribution with bounded support. Prove that such a random walk is recurrent.

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

Definition 16. The support of an arbitrary measure $\lambda$ on the Borel $\sigma$-algebra of a topological space $\mathrm{S}$ is the smallest closed set $F \subset \mathrm{S}$ such that $\lambda (\mathrm{S} \backslash F) = 0$.

Definition 17. The effective dimension of the random walk (I.9) is the dimension of the subspace of $\mathbb {R}^{m}$ generated by the linear span of the support of the measure $\mathbb {P}_{\xi_{1}}$.

Prove that if the effective dimension of a random walk is at least three, then this random walk is transient.

Consider the random walk in $\mathbb {R}^{m}$ from Example 1 of Chapter I, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors. We assume $S_{0} = 0$.

Example 1 of Chapter I. Let $\xi_{0}, \xi_{1}, \ldots$ be independent random vectors defined on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in the space $\mathbb {R}^{m}$ ($m \geqslant 1$). The (discrete-time) stochastic process

is customarily called a random walk.

Definition 13. The (random) occupation measure of the process $S = \left\{ S_{n}, n \geqslant 0\right\}$ in a Borel set $B$ of $\mathbb {R}^{m}$ is the quantity

The intensity measure $\nu = \nu (\cdot )$ corresponding to this measure $\mu = \mu (\cdot )$ is defined by the formula

Definition 14. Denoting by $V_{\varepsilon }(x)$ the open ball in $\mathbb {R}^{m}$ with center at the point $x$ and radius $\varepsilon > 0$, we introduce the following sets:

Theorem 10 (dichotomy theorem; see, e.g., [146; p. 137]). For a random walk $S$ in $\mathbb {R}^{m}$ defined according to (I.9), only one of the following two possibilities is realized:

1. $R = M = A$ and this set is a closed subgroup of $\mathbb {R}^{m}$,

2. $R = M = \varnothing$ and $\left|S_{n}\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Definition 15. A random walk is called recurrent if property 1) holds in Theorem 10 above, and non-recurrent (or transient) if property 2) holds.

We call a walk in $\mathbb {R}^{m}$ the simple symmetric random walk if

where the $i$-th coordinate of the vector $e_{i}$ equals 1, and the other coordinates equal 0. This walk can be regarded as a model of the motion of a "particle" that, with equal probability, moves by one unit in either direction along any coordinate axis.

The result of Exercise 13 implies the well-known Pólya criterion (see [78; vol. 1, p. 374]) stating that the simple symmetric random walk in $\mathbb {R}^{m}$ is non-recurrent for $m \geqslant 3$. It is also proved there ([78]) that for $m = 1$ and $m = 2$ such a walk is recurrent.

(compare with the result on random walks given above). Prove that the sample paths of a particle performing the simple symmetric random walk in $\mathbb {R}^{m}$ have, with probability 1, infinitely many self-intersections, i.e., with probability 1 the particle will return to positions that it occupied earlier.

Consider the random walk in $\mathbb {R}^{m}$, i.e., the process of partial sums $S = \left\{ S_{n}, n \geqslant 0\right\}$, where $S_{n} = S_{0}+\xi_{1}+\cdots +\xi_{n}, n \geqslant 1$, and $\xi_{1}, \xi_{2}, \ldots$ are independent identically distributed vectors.

Prove that if $\mathbb {P}\left(\xi_{1} \neq 0\right) > 0$, then the one-dimensional random walk exits every strip $(a, b)$ with boundaries $a < 0 < b$ with probability one (recall that $S_{0} = 0$ ).

Let $X = \left\{ X_{t}, t \in [a, b]\right\}$ be a real process that is stochastically continuous at every point $t \in [a, b]$. Then the process $X$ is uniformly stochastically continuous on $[a, b]$, i.e. for any $\varepsilon , \gamma > 0$ there exists $\Delta = \Delta (\varepsilon , \gamma ) > 0$ such that

Moreover, the process $X$ is bounded in probability, i.e.

(Ottaviani's inequality). Let $Y_{1}, \ldots , Y_{n}$ be independent real random variables such that for some $\alpha \in [0,1), r \geqslant 0$

where $S_{k} = \sum_{j = 1}^{k} Y_{j}$. Prove that for all $c \geqslant 0$

Let a process $X = \left\{ X_{t}, t \in [0, \infty )\right\}$ have independent increments and be stochastically continuous on $[0, \infty )$. Prove that

where $M$ is the set of dyadic rational points of the line.

Prove that the Poisson process is stochastically continuous on $[0, \infty )$.

Show how Lévy's formula is derived from Khinchin's formula and vice versa. What is the relation between the parameters $a, \nu$ and $b, \sigma^{2}, \widetilde{\nu }$ ?

Theorem 14 (Khinchin's formula; see, e.g., [39; Part 1, §16]). Let a stochastically continuous real process $X = \left\{ X_{t}, t \geqslant 0\right\}$ have independent increments. Then for any $t \geqslant 0$ and all $\lambda \in \mathbb {R}$ the characteristic function

where $a \in \mathbb {R}$, $\nu$ is a finite measure on $\mathscr {B}(\mathbb {R})$ and

Moreover, the representation (41) is unique, i.e. in this formula the number $a$ and the measure $\nu$ are uniquely determined.

Theorem 15 (Lévy's formula; see, e.g., [39; Part 1, §16]). If the conditions of the previous theorem hold, then for any $t \geqslant 0$ and all $\lambda \in \mathbb {R}$

where $b \in \mathbb {R}, \sigma \geqslant 0, \widetilde{\nu }$ is a measure on $\mathscr {B}(\mathbb {R})$ (taking, generally speaking, infinite values) such that

Moreover, $b, \sigma$ and $\widetilde{\nu }$, appearing in the representation (44), are uniquely determined.

What does Lévy's formula look like for the process $W_{x}(t) = x+W(t), t \geqslant 0$, where $W = \left\{ W(t), t \geqslant 0\right\}$ is a Wiener process and $x \in \mathbb {R}$?

Theorem 14 (Khinchin's formula; see, e.g., [39; Part 1, §16]). Let a stochastically continuous real process $X = \left\{ X_{t}, t \geqslant 0\right\}$ have independent increments. Then for any $t \geqslant 0$ and all $\lambda \in \mathbb {R}$ the characteristic function

where $a \in \mathbb {R}$, $\nu$ is a finite measure on $\mathscr {B}(\mathbb {R})$ and

Moreover, the representation (41) is unique, i.e. in this formula the number $a$ and the measure $\nu$ are uniquely determined.

Theorem 15 (Lévy's formula; see, e.g., [39; Part 1, §16]). If the conditions of the previous theorem hold, then for any $t \geqslant 0$ and all $\lambda \in \mathbb {R}$

where $b \in \mathbb {R}, \sigma \geqslant 0, \widetilde{\nu }$ is a measure on $\mathscr {B}(\mathbb {R})$ (taking, generally speaking, infinite values) such that