# Brownian Bridge Fourier Expansions

Brownian bridges were described in a previous post, along with various different methods by which they can be constructed. Since a Brownian bridge on an interval ${[0,T]}$ is continuous and equal to zero at both endpoints, we can consider extending to the entire real line by partitioning the real numbers into intervals of length T and replicating the path of the process across each of these. This will result in continuous and periodic sample paths, suggesting another method of representing Brownian bridges. That is, by Fourier expansion. As we will see, the Fourier coefficients turn out to be independent normal random variables, giving a useful alternative method of constructing a Brownian bridge.

There are actually a couple of distinct Fourier expansions that can be used, which depends on precisely how we consider extending the sample paths to the real line. A particularly simple result is given by the sine series, which I describe first. This is shown for an example Brownian bridge sample path in figure 1 above, which plots the sequence of approximations formed by truncating the series after a small number of terms. This tends uniformly to the sample path, although it is quite slow to converge as should be expected when approximating such a rough path by smooth functions. Also plotted, is the series after the first 100 terms, by which time the approximation is quite close to the target. For simplicity, I only consider standard Brownian bridges, which are defined on the unit interval ${[0,1]}$. This does not reduce the generality, since bridges on an interval ${[0,T]}$ can be expressed as scaled versions of standard Brownian bridges.

Theorem 1 A standard Brownian bridge B can be decomposed as

 $\displaystyle B_t=\sum_{n=1}^\infty\frac{\sqrt2Z_n}{\pi n}\sin(\pi nt)$ (1)

over ${0\le t\le1}$, where ${Z_1,Z_2,\ldots}$ is an IID sequence of standard normals. This series converges uniformly in t, both with probability one and in the ${L^p}$ norm for all ${1\le p < \infty}$.

# Brownian Bridges

A Brownian bridge can be defined as standard Brownian motion conditioned on hitting zero at a fixed future time T, or as any continuous process with the same distribution as this. Rather than conditioning, a slightly easier approach is to subtract a linear term from the Brownian motion, chosen such that the resulting process hits zero at the time T. This is equivalent, but has the added benefit of being independent of the original Brownian motion at all later times.

Lemma 1 Let X be a standard Brownian motion and ${T > 0}$ be a fixed time. Then, the process

 $\displaystyle B_t = X_t - \frac tTX_T$ (1)

over ${0\le t\le T}$ is independent from ${\{X_t\}_{t\ge T}}$.

Proof: As the processes are joint normal, it is sufficient that there is zero covariance between them. So, for times ${s\le T\le t}$, we just need to show that ${{\mathbb E}[B_sX_t]}$ is zero. Using the covariance structure ${{\mathbb E}[X_sX_t]=s\wedge t}$ we obtain,

 $\displaystyle {\mathbb E}[B_sX_t]={\mathbb E}[X_sX_t]-\frac sT{\mathbb E}[X_TX_t]=s-\frac sTT=0$

as required. ⬜

This leads us to the definition of a Brownian bridge.

Definition 2 A continuous process ${\{B_t\}_{t\in[0,T]}}$ is a Brownian bridge on the interval ${[0,T]}$ if and only it has the same distribution as ${X_t-\frac tTX_T}$ for a standard Brownian motion X.

In case that ${T=1}$, then B is called a standard Brownian bridge.

There are actually many different ways in which Brownian bridges can be defined, which all lead to the same result.

• As a Brownian motion minus a linear term so that it hits zero at T. This is definition 2.
• As a Brownian motion X scaled as ${tT^{-1/2}X_{T/t-1}}$. See lemma 9 below.
• As a joint normal process with prescribed covariances. See lemma 7 below.
• As a Brownian motion conditioned on hitting zero at T. See lemma 14 below.
• As a Brownian motion restricted to the times before it last hits zero before a fixed positive time T, and rescaled to fit a fixed time interval. See lemma 15 below.
• As a Markov process. See lemma 13 below.
• As a solution to a stochastic differential equation with drift term forcing it to hit zero at T. See lemma 18 below.

There are other constructions beyond these, such as in terms of limits of random walks, although I will not cover those in this post. Continue reading “Brownian Bridges”

# Independence of Normals

A well known fact about joint normally distributed random variables, is that they are independent if and only if their covariance is zero. In one direction, this statement is trivial. Any independent pair of random variables has zero covariance (assuming that they are integrable, so that the covariance has a well-defined value). The strength of the statement is in the other direction. Knowing the value of the covariance does not tell us a lot about the joint distribution so, in the case that they are joint normal, the fact that we can determine independence from this is a rather strong statement.

Theorem 1 A joint normal pair of random variables are independent if and only if their covariance is zero.

Proof: Suppose that X,Y are joint normal, such that ${X\overset d= N(\mu_X,\sigma^2_X)}$ and ${Y\overset d=N(\mu_Y,\sigma_Y^2)}$, and that their covariance is c. Then, the characteristic function of ${(X,Y)}$ can be computed as

 \displaystyle \begin{aligned} {\mathbb E}\left[e^{iaX+ibY}\right] &=e^{ia\mu_X+ib\mu_Y-\frac12(a^2\sigma_X^2+2abc+b^2\sigma_Y^2)}\\ &=e^{-abc}{\mathbb E}\left[e^{iaX}\right]{\mathbb E}\left[e^{ibY}\right] \end{aligned}

for all ${(a,b)\in{\mathbb R}^2}$. It is standard that the joint characteristic function of a pair of random variables is equal to the product of their characteristic functions if and only if they are independent which, in this case, corresponds to the covariance c being zero. ⬜

To demonstrate necessity of the joint normality condition, consider the example from the previous post.

Example 1 A pair of standard normal random variables X,Y which have zero covariance, but ${X+Y}$ is not normal.

As their sum is not normal, X and Y cannot be independent. This example was constructed by setting ${Y={\rm sgn}(\lvert X\rvert -K)X}$ for some fixed ${K > 0}$, which is standard normal whenever X is. As explained in the previous post, the intermediate value theorem ensures that there is a unique value for K making the covariance ${{\mathbb E}[XY]}$ equal to zero. Continue reading “Independence of Normals”

# Multivariate Normal Distributions

I looked at normal random variables in an earlier post but, what does it mean for a sequence of real-valued random variables ${X_1,X_2,\ldots,X_n}$ to be jointly normal? We could simply require each of them to be normal, but this says very little about their joint distribution and is not much help in handling expressions involving more than one of the ${X_i}$ at once. In case that the random variables are independent, the following result is a very useful property of the normal distribution. All random variables in this post will be real-valued, except where stated otherwise, and we assume that they are defined with respect to some underlying probability space ${(\Omega,\mathcal F,{\mathbb P})}$.

Lemma 1 Linear combinations of independent normal random variables are again normal.

Proof: More precisely, if ${X_1,\ldots,X_n}$ is a sequence of independent normal random variables and ${a_1,\ldots,a_n}$ are real numbers, then ${Y=a_1X_1+\cdots+a_nX_n}$ is normal. Let us suppose that ${X_k}$ has mean ${\mu_k}$ and variance ${\sigma_k^2}$. Then, the characteristic function of Y can be computed using the independence property and the characteristic functions of the individual normals,

 \displaystyle \begin{aligned} {\mathbb E}\left[e^{i\lambda Y}\right] &={\mathbb E}\left[\prod_ke^{i\lambda a_k X_k}\right] =\prod_k{\mathbb E}\left[e^{i\lambda a_k X_k}\right]\\ &=\prod_ke^{-\frac12\lambda^2a_k^2\sigma_k^2+i\lambda a_k\mu_k} =e^{-\frac12\lambda^2\sigma^2+i\lambda\mu} \end{aligned}

where we have set ${\mu_k=\sum_ka_k\mu_k}$ and ${\sigma^2=\sum_ka_k^2\sigma_k^2}$. This is the characteristic function of a normal random variable with mean ${\mu}$ and variance ${\sigma^2}$. ⬜

The definition of joint normal random variables will include the case of independent normals, so that any linear combination is also normal. We use use this result as the defining property for the general multivariate normal case.

Definition 2 A collection ${\{X_i\}_{i\in I}}$ of real-valued random variables is multivariate normal (or joint normal) if and only if all of its finite linear combinations are normal.

# The Riemann Zeta Function and Probability Distributions

The famous Riemann zeta function was first introduced by Riemann in order to describe the distribution of the prime numbers. It is defined by the infinite sum

 \displaystyle \begin{aligned} \zeta(s) &=1+2^{-s}+3^{-s}+4^{-s}+\cdots\\ &=\sum_{n=1}^\infty n^{-s}, \end{aligned} (1)

which is absolutely convergent for all complex s with real part greater than one. One of the first properties of this is that, as shown by Riemann, it extends to an analytic function on the entire complex plane, other than a simple pole at ${s=1}$. By the theory of analytic continuation this extension is necessarily unique, so the importance of the result lies in showing that an extension exists. One way of doing this is to find an alternative expression for the zeta function which is well defined everywhere. For example, it can be expressed as an absolutely convergent integral, as performed by Riemann himself in his original 1859 paper on the subject. This leads to an explicit expression for the zeta function, scaled by an analytic prefactor, as the integral of ${x^s}$ multiplied by a function of x over the range ${ x > 0}$. In fact, this can be done in a way such that the function of x is a probability density function, and hence expresses the Riemann zeta function over the entire complex plane in terms of the generating function ${{\mathbb E}[X^s]}$ of a positive random variable X. The probability distributions involved here are not the standard ones taught to students of probability theory, so may be new to many people. Although these distributions are intimately related to the Riemann zeta function they also, intriguingly, turn up in seemingly unrelated contexts involving Brownian motion.

In this post, I derive two probability distributions related to the extension of the Riemann zeta function, and describe some of their properties. I also show how they can be constructed as the sum of a sequence of gamma distributed random variables. For motivation, some examples are given of where they show up in apparently unrelated areas of probability theory, although I do not give proofs of these statements here. For more information, see the 2001 paper Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions by Biane, Pitman, and Yor. Continue reading “The Riemann Zeta Function and Probability Distributions”

# Manipulating the Normal Distribution

The normal (or Gaussian) distribution is ubiquitous throughout probability theory for various reasons, including the central limit theorem, the fact that it is realistic for many practical applications, and because it satisfies nice properties making it amenable to mathematical manipulation. It is, therefore, one of the first continuous distributions that students encounter at school. As such, it is not something that I have spent much time discussing on this blog, which is usually concerned with more advanced topics. However, there are many nice properties and methods that can be performed with normal distributions, greatly simplifying the manipulation of expressions in which it is involved. While it is usually possible to ignore these, and instead just substitute in the density function and manipulate the resulting integrals, that approach can get very messy. So, I will describe some of the basic results and ideas that I use frequently.

Throughout, I assume the existence of an underlying probability space ${(\Omega,\mathcal F,{\mathbb P})}$. Recall that a real-valued random variable X has the standard normal distribution if it has a probability density function given by,

 $\displaystyle \varphi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}.$

For it to function as a probability density, it is necessary that it integrates to one. While it is not obvious that the normalization factor ${1/\sqrt{2\pi}}$ is the correct value for this to be true, it is the one fact that I state here without proof. Wikipedia does list a couple of proofs, which can be referred to. By symmetry, ${-X}$ and ${X}$ have the same distribution, so that they have the same mean and, therefore, ${{\mathbb E}[X]=0}$.

The derivative of the density function satisfies the useful identity

 $\displaystyle \varphi^\prime(x)=-x\varphi(x).$ (1)

This allows us to quickly verify that standard normal variables have unit variance, by an application of integration by parts.

 \displaystyle \begin{aligned} {\mathbb E}[X^2] &=\int x^2\varphi(x)dx\\ &= -\int x\varphi^\prime(x)dx\\ &=\int\varphi(x)dx-[x\varphi(x)]_{-\infty}^\infty=1 \end{aligned}

Another identity satisfied by the normal density function is,

 $\displaystyle \varphi(x+y)=e^{-xy - \frac{y^2}2}\varphi(x)$ (2)

This enables us to prove the following very useful result. In fact, it is difficult to overstate how helpful this result can be. I make use of it frequently when manipulating expressions involving normal variables, as it significantly simplifies the calculations. It is also easy to remember, and simple to derive if needed.

Theorem 1 Let X be standard normal and ${f\colon{\mathbb R}\rightarrow{\mathbb R}_+}$ be measurable. Then, for all ${\lambda\in{\mathbb R}}$,

 \displaystyle \begin{aligned} {\mathbb E}[e^{\lambda X}f(X)] &={\mathbb E}[e^{\lambda X}]{\mathbb E}[f(X+\lambda)]\\ &=e^{\frac{\lambda^2}{2}}{\mathbb E}[f(X+\lambda)]. \end{aligned} (3)

# Brownian Drawdowns

Here, I apply the theory outlined in the previous post to fully describe the drawdown point process of a standard Brownian motion. In fact, as I will show, the drawdowns can all be constructed from independent copies of a single ‘Brownian excursion’ stochastic process. Recall that we start with a continuous stochastic process X, assumed here to be Brownian motion, and define its running maximum as ${M_t=\sup_{s\le t}X_s}$ and drawdown process ${D_t=M_t-X_t}$. This is as in figure 1 above.

Next, ${D^a}$ was defined to be the drawdown ‘excursion’ over the interval at which the maximum process is equal to the value ${a \ge 0}$. Precisely, if we let ${\tau_a}$ be the first time at which X hits level ${a}$ and ${\tau_{a+}}$ be its right limit ${\tau_{a+}=\lim_{b\downarrow a}\tau_b}$ then,

 $\displaystyle D^a_t=D_{({\tau_a+t})\wedge\tau_{a+}}=a-X_{({\tau_a+t)}\wedge\tau_{a+}}.$

Next, a random set S is defined as the collection of all nonzero drawdown excursions indexed the running maximum,

 $\displaystyle S=\left\{(a,D^a)\colon D^a\not=0\right\}.$

The set of drawdown excursions corresponding to the sample path from figure 1 are shown in figure 2 below.

As described in the post on semimartingale local times, the joint distribution of the drawdown and running maximum ${(D,M)}$, of a Brownian motion, is identical to the distribution of its absolute value and local time at zero, ${(\lvert X\rvert,L^0)}$. Hence, the point process consisting of the drawdown excursions indexed by the running maximum, and the absolute value of the excursions from zero indexed by the local time, both have the same distribution. So, the theory described in this post applies equally to the excursions away from zero of a Brownian motion.

Before going further, let’s recap some of the technical details. The excursions lie in the space E of continuous paths ${z\colon{\mathbb R}_+\rightarrow{\mathbb R}}$, on which we define a canonical process Z by sampling the path at each time t, ${Z_t(z)=z_t}$. This space is given the topology of uniform convergence over finite time intervals (compact open topology), which makes it into a Polish space, and whose Borel sigma-algebra ${\mathcal E}$ is equal to the sigma-algebra generated by ${\{Z_t\}_{t\ge0}}$. As shown in the previous post, the counting measure ${\xi(A)=\#(S\cap A)}$ is a random point process on ${({\mathbb R}_+\times E,\mathcal B({\mathbb R}_+)\otimes\mathcal E)}$. In fact, it is a Poisson point process, so its distribution is fully determined by its intensity measure ${\mu={\mathbb E}\xi}$.

Theorem 1 If X is a standard Brownian motion, then the drawdown point process ${\xi}$ is Poisson with intensity measure ${\mu=\lambda\otimes\nu}$ where,

• ${\lambda}$ is the standard Lebesgue measure on ${{\mathbb R}_+}$.
• ${\nu}$ is a sigma-finite measure on E given by
 $\displaystyle \nu(f) = \lim_{\epsilon\rightarrow0}\epsilon^{-1}{\mathbb E}_\epsilon[f(Z^{\sigma})]$ (1)

for all bounded continuous continuous maps ${f\colon E\rightarrow{\mathbb R}}$ which vanish on paths of length less than L (some ${L > 0}$). The limit is taken over ${\epsilon > 0}$, ${{\mathbb E}_\epsilon}$ denotes expectation under the measure with respect to which Z is a Brownian motion started at ${\epsilon}$, and ${\sigma}$ is the first time at which Z hits 0. This measure satisfies the following properties,

• ${\nu}$-almost everywhere, there exists a time ${T > 0}$ such that ${Z > 0}$ on ${(0,T)}$ and ${Z=0}$ everywhere else.
• for each ${t > 0}$, the distribution of ${Z_t}$ has density
 $\displaystyle p_t(z)=z\sqrt{\frac 2{\pi t^3}}e^{-\frac{z^2}{2t}}$ (2)

over the range ${z > 0}$.

• over ${t > 0}$, ${Z_t}$ is Markov, with transition function of a Brownian motion stopped at zero.

# Drawdown Point Processes

For a continuous real-valued stochastic process ${\{X_t\}_{t\ge0}}$ with running maximum ${M_t=\sup_{s\le t}X_s}$, consider its drawdown. This is just the amount that it has dropped since its maximum so far,

 $\displaystyle D_t=M_t-X_t,$

which is a nonnegative process hitting zero whenever the original process visits its running maximum. By looking at each of the individual intervals over which the drawdown is positive, we can break it down into a collection of finite excursions above zero. Furthermore, the running maximum is constant across each of these intervals, so it is natural to index the excursions by this maximum process. By doing so, we obtain a point process. In many cases, it is even a Poisson point process. I look at the drawdown in this post as an example of a point process which is a bit more interesting than the previous example given of the jumps of a cadlag process. By piecing the drawdown excursions back together, it is possible to reconstruct ${D_t}$ from the point process. At least, this can be done so long as the original process does not monotonically increase over any nontrivial intervals, so that there are no intervals with zero drawdown. As the point process indexes the drawdown by the running maximum, we can also reconstruct X as ${X_t=M_t-D_t}$. The drawdown point process therefore gives an alternative description of our original process.

See figure 1 for the drawdown of the bitcoin price valued in US dollars between April and December 2020. As it makes more sense for this example, the drawdown is shown as a percent of the running maximum, rather than in dollars. This is equivalent to the approach taken in this post applied to the logarithm of the price return over the period, so that ${X_t=\log(B_t/B_0)}$. It can be noted that, as the price was mostly increasing, the drawdown consists of a relatively large number of small excursions. If, on the other hand, it had declined, then it would have been dominated by a single large drawdown excursion covering most of the time period.

For simplicity, I will suppose that ${X_0=0}$ and that ${M_t}$ tends to infinity as t goes to infinity. Then, for each ${a\ge0}$, define the random time at which the process first hits level ${a}$,

 $\displaystyle \tau_a=\inf\left\{t\ge 0\colon X_t\ge a\right\}.$

By construction, this is finite, increasing, and left-continuous in ${a}$. Consider, also, the right limits ${\tau_{a+}=\lim_{b\downarrow0}\tau_b}$. Each of the excursions on which the drawdown is positive is equal to one of the intervals ${(\tau_a,\tau_{a+})}$. The excursion is defined as a continuous stochastic process ${\{D^a_t\}_{t\ge0}}$ equal to the drawdown starting at time ${\tau_a}$ and stopped at time ${\tau_{a+}}$,

 $\displaystyle D^a_t=D_{(\tau_a+t)\wedge\tau_{a+}}=a-X_{(\tau_a+t)\wedge\tau_{a+}}.$

This is a continuous nonnegative real-valued process, which starts at zero and is equal to zero at all times after ${\tau_{a+}-\tau_a}$. Note that there uncountably many values for ${a}$ but, the associated excursion will be identically zero other than for the countably many times at which ${\tau_{a+} > \tau_a}$. We will only be interested in these nonzero excursions.

As usual, we work with respect to an underlying probability space ${(\Omega,\mathcal F,{\mathbb P})}$, so that we have one path of the stochastic process X defined for each ${\omega\in\Omega}$. Associated to this is the collection of drawdown excursions indexed by the running maximum.

 $\displaystyle S=\left\{(a,D^a)\colon a\ge0,\ D^a\not=0\right\}.$

As S is defined for each given sample path, it depends on the choice of ${\omega\in\Omega}$, so is a countable random set. The sample paths of the excursions ${D^a}$ lie in the space of continuous functions ${{\mathbb R}_+\rightarrow{\mathbb R}}$, which I denote by E. For each time ${t\ge0}$, I use ${Z_t}$ to denote the value of the path sampled at time t,

 \displaystyle \begin{aligned} &E=\left\{z\colon {\mathbb R}_+\rightarrow{\mathbb R}{\rm\ is\ continuous}\right\}.\\ &Z_t\colon E\rightarrow{\mathbb R},\\ & Z_t(z)=z_t. \end{aligned}

Use ${\mathcal E}$ to denote the sigma-algebra on E generated by the collection of maps ${\{Z_t\colon t\ge0\}}$, so that ${(E,\mathcal E)}$ is the measurable space in which the excursion paths lie. It can be seen that ${\mathcal E}$ is the Borel sigma-algebra generated by the open subsets of E, with respect to the topology of compact convergence. That is, the topology of uniform convergence on finite time intervals. As this is a complete separable metric space, it makes ${(E,\mathcal E)}$ into a standard Borel space.

Lemma 1 The set S defines a simple point process ${\xi}$ on ${{\mathbb R}_+\times E}$,

 $\displaystyle \xi(A)=\#(S\cap A)$

for all ${A\in\mathcal B({\mathbb R}_+)\otimes\mathcal E}$.

From the definition of point processes, this simply means that ${\xi(A)}$ is a measurable random variable for each ${A\in \mathcal B({\mathbb R}_+)\otimes\mathcal E}$ and that there exists a sequence ${A_n\in \mathcal B({\mathbb R}_+)\otimes\mathcal E}$ covering E such that ${\xi(A_n)}$ are almost surely finite. The set of drawdowns for the point process corresponding to the bitcoin prices in figure 1 are shown in figure 2 below.

# Criteria for Poisson Point Processes

If S is a finite random set in a standard Borel measurable space ${(E,\mathcal E)}$ satisfying the following two properties,

• if ${A,B\in\mathcal E}$ are disjoint, then the sizes of ${S\cap A}$ and ${S\cap B}$ are independent random variables,
• ${{\mathbb P}(x\in S)=0}$ for each ${x\in E}$,

then it is a Poisson point process. That is, the size of ${S\cap A}$ is a Poisson random variable for each ${A\in\mathcal E}$. This justifies the use of Poisson point processes in many different areas of probability and stochastic calculus, and provides a convenient method of showing that point processes are indeed Poisson. If the theorem applies, so that we have a Poisson point process, then we just need to compute the intensity measure to fully determine its distribution. The result above was mentioned in the previous post, but I give a precise statement and proof here. Continue reading “Criteria for Poisson Point Processes”

# Poisson Point Processes

The Poisson distribution models numbers of events that occur in a specific period of time given that, at each instant, whether an event occurs or not is independent of what happens at all other times. Examples which are sometimes cited as candidates for the Poisson distribution include the number of phone calls handled by a telephone exchange on a given day, the number of decays of a radio-active material, and the number of bombs landing in a given area during the London Blitz of 1940-41. The Poisson process counts events which occur according to such distributions.

More generally, the events under consideration need not just happen at specific times, but also at specific locations in a space E. Here, E can represent an actual geometric space in which the events occur, such as the spacial distribution of bombs dropped during the Blitz shown in figure 1, but can also represent other quantities associated with the events. In this example, E could represent the 2-dimensional map of London, or could include both space and time so that ${E=F\times{\mathbb R}}$ where, now, F represents the 2-dimensional map and E is used to record both time and location of the bombs. A Poisson point process is a random set of points in E, such that the number that lie within any measurable subset is Poisson distributed. The aim of this post is to introduce Poisson point processes together with the mathematical machinery to handle such random sets.

The choice of distribution is not arbitrary. Rather, it is a result of the independence of the number of events in each region of the space which leads to the Poisson measure, much like the central limit theorem leads to the ubiquity of the normal distribution for continuous random variables and of Brownian motion for continuous stochastic processes. A random finite subset S of a reasonably ‘nice’ (standard Borel) space E is a Poisson point process so long as it satisfies the properties,

• If ${A_1,\ldots,A_n}$ are pairwise-disjoint measurable subsets of E, then the sizes of ${S\cap A_1,\ldots,S\cap A_n}$ are independent.
• Individual points of the space each have zero probability of being in S. That is, ${{\mathbb P}(x\in S)=0}$ for each ${x\in E}$.

The proof of this important result will be given in a later post.

We have come across Poisson point processes previously in my stochastic calculus notes. Specifically, suppose that X is a cadlag ${{\mathbb R}^d}$-valued stochastic process with independent increments, and which is continuous in probability. Then, the set of points ${(t,\Delta X_t)}$ over times t for which the jump ${\Delta X}$ is nonzero gives a Poisson point process on ${{\mathbb R}_+\times{\mathbb R}^d}$. See lemma 4 of the post on processes with independent increments, which corresponds precisely to definition 5 given below. Continue reading “Poisson Point Processes”