Infinitesimal Calculus of Random Walk and Poisson Processes

Gauge Institute Journal, 

H. Vic Dannon 

Infinitesimal Calculus of 

Random Walk and Poisson 

Processes 

H. Vic Dannon 

vic0@comcast.net 

March, 2013 

Abstract We set up the Infinitesimal Calculus of Random 

Processes X(,) 

ζ t , and apply it to the Random Walk B(,) 

ζ t , 

and to the Poisson Process P(,) 

ζ t . 

Both Processes are Continuous, and have Derivative 

Processes with Delta Function Variance. 

The integral 

t= 

b 

∫ ftdB () (, ζ t) 

, of integrable f () t , with respect to 

t= 

a 

the Random Walk B(,) 

ζ t , and the integral ftdP () (, ζ t) 

of 

t= 

b 

∫ 

t= 

a 

integrable 

f () t , with respect to the Poisson Process 

P(,) 

ζ t 

are well-defined Random Variables. 

Keywords: Infinitesimal, Infinite-Hyper-real, Hyper-real, 

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H. Vic Dannon 

Calculus, Limit, Continuity, Derivative, Integral, Delta 

Function, Random Variable, Random Process, Random 

Signal, Stochastic Process, Stochastic Calculus, Probability 

Distribution, Bernoulli Random Variables, Binomial 

Distribution, Gaussian, Normal, Expectation, Variance, 

Random Walk, Poisson Process 

2000 Mathematics Subject Classification 26E35; 26E30; 

26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15; 

46S20; 97I40; 97I30. 

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H. Vic Dannon 

Contents 

Introduction 

1. Hyper-real Line 

2. Hyper-real Function 

3. Integral of a Hyper-real Function 

4. Delta Function 

5. Hyper-real Random Variable 

6. Normal Distribution, and Delta Function 

7. Hyper-real Random Signal X(,) 

ζ t 

8. Continuity of X(,) 

ζ t 

9. Derivative of X(,) 

ζ t 

10. Random Walk B(,) 

ζ t 

11. Random Walk is Continuous, has a Derivative with Delta 

Function Variance, and 

Variation. 

t= 

b 

∫ 

12. f () tdB(, ζ t) 

t= 

a 

13. Poisson Process P(,) 

ζ t 

EB [ ( ζ, t)] 

has unbounded 

14. Poisson Process is Continuous and has a Derivative with 

Delta Function Variance 

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H. Vic Dannon 

t= 

b 

∫ 

15. f () tdP(, ζ t) 

t= 

a 

References 

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H. Vic Dannon 

Introduction 

0.1 Infinitesimal Calculus 

Recently we have shown that when the Real Line is 

represented as the infinite dimensional space of all the 

Cauchy sequences of rational numbers, the hyper-reals are 

spanned by the constant hyper-reals, a family of 

infinitesimal hyper-reals, and the associated family of 

infinite hyper-reals. 

The infinitesimal hyper-reals are smaller than any real 

number, yet bigger than zero. 

The reciprocals of the infinitesimal hyper-reals are the 

infinite hyper-reals. They are greater than any real number, 

yet strictly smaller than infinity. 

A neighborhood of infinitesimals separates the zero hyperreal 

from the reals, and each real number is the center of an 

interval of hyper-reals, that includes no other real number. 

The Hyper-reals are totally ordered, and are lined up on a 

line, the hyper-real line. 

A hyper-real function is a mapping from the hyper-real line 

into the hyper-real line. 

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H. Vic Dannon 

Infinitesimal Calculus is the Calculus of hyper-real 

functions. 

Infinitesimal Calculus is far more effective than the 

εδ , 

Calculus, because being based on almost zero numbers, it 

allows us to deal with their reciprocals, the almost infinite 

numbers. We have no use for infinity by itself, but to 

comprehend the effects of singularities, we have use for the 

almost infinite. 

Infinitesimals are a precise tool compared to the vague limit 

concept, and the awkward 

εδ , 

statements. 

Random walks are made clearer with infinitesimals. 

Poisson Process can be derived only in Infinitesimal 

Calculus. 

0.2 Random Processes 

Probability Distributions are defined on Random Variables. 

Random Variables assign numerical vales to outcomes. 

Thus, maps outcomes into the real line. 

Random Variables that evolve in time are called Random 

Processes, in Mechanics, or Random Signals, in Electricity. 

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H. Vic Dannon 

Random Walk is the Random drift of a particle in fluid due 

to collisions with fluid molecules. 

Poisson Process models the Random arrival of radioactive 

particles at a counter. 

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H. Vic Dannon 

1. 

Hyper-real Line 

The minimal domain and range, needed for the definition 

and analysis of a hyper-real function, is the hyper-real line. 

Each real number α can be represented by a Cauchy 

sequence of rational numbers, ( r , r , r ,...) so that r → α . 

1 2 3 

The constant sequence ( ααα , , ,...) is a constant hyper-real. 

In [Dan2] we established that, 

1. Any totally ordered set of positive, monotonically 

n 

decreasing to zero sequences 

family of infinitesimal hyper-reals. 

( ι1, ι2, ι3,...) 

constitutes a 

2. The infinitesimals are smaller than any real number, 

yet strictly greater than zero. 

1 1 1 

3. Their reciprocals ( , , ,... 

ι 1 

ι 2 

ι 3 

) are the infinite hyperreals. 

4. The infinite hyper-reals are greater than any real 

number, yet strictly smaller than infinity. 

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H. Vic Dannon 

5. The infinite hyper-reals with negative signs are 

smaller than any real number, yet strictly greater than 

−∞. 

6. The sum of a real number with an infinitesimal is a 

non-constant hyper-real. 

7. The Hyper-reals are the totality of constant hyperreals, 

a family of infinitesimals, a family of 

infinitesimals with negative sign, a family of infinite 

hyper-reals, a family of infinite hyper-reals with 

negative sign, and non-constant hyper-reals. 

8. The hyper-reals are totally ordered, and aligned along 

a line: the Hyper-real Line. 

9. That line includes the real numbers separated by the 

non-constant hyper-reals. Each real number is the 

center of an interval of hyper-reals, that includes no 

other real number. 

10. In particular, zero is separated from any positive 

real by the infinitesimals, and from any negative real 

by the infinitesimals with negative signs, −dx . 

11. Zero is not an infinitesimal, because zero is not 

strictly greater than zero. 

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H. Vic Dannon 

12. We do not add infinity to the hyper-real line. 

13. The infinitesimals, the infinitesimals with 

negative signs, the infinite hyper-reals, and the infinite 

hyper-reals with negative signs are semi-groups with 

respect to addition. Neither set includes zero. 

∞ 

14. The hyper-real line is embedded in , and is 

not homeomorphic to the real line. There is no bicontinuous 

one-one mapping from the hyper-real onto 

the real line. 

15. In particular, there are no points on the real line 

that can be assigned uniquely to the infinitesimal 

hyper-reals, or to the infinite hyper-reals, or to the nonconstant 

hyper-reals. 

16. No neighbourhood of a hyper-real is 

n 

homeomorphic to an ball. Therefore, the hyperreal 

line is not a manifold. 

17. The hyper-real line is totally ordered like a line, 

but it is not spanned by one element, and it is not onedimensional. 

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H. Vic Dannon 

2. 

Hyper-real Function 

2.1 Definition of a hyper-real function 

f () x is a hyper-real function, iff it is from the hyper-reals 

into the hyper-reals. 

This means that any number in the domain, or in the range 

of a hyper-real f () x is either one of the following 

real 

real + infinitesimal 

real – infinitesimal 

infinitesimal 

infinitesimal with negative sign 

infinite hyper-real 

infinite hyper-real with negative sign 

Clearly, 

2.2 Every function from the reals into the reals is a hyperreal 

function. 

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H. Vic Dannon 

3. 

Integral of Hyper-real Function 

In [Dan3], we defined the integral of a Hyper-real Function. 

Let f () x be a hyper-real function on the interval [ ab] , . 

The interval may not be bounded. 

f () x may take infinite hyper-real values, and need not be 

bounded. 

At each 

a 

≤ 

x 

≤b, 

there is a rectangle with base 

dx 

[ x − , x + 2 

], height f () x , 

dx 

2 

and area 

f ( xdx. ) 

We form the Integration Sum of all the areas for the x ’s 

that start at x = a, and end at x = b, 

∑ f ( xdx ) . 

x∈[ a, b] 

If for any infinitesimal dx , the Integration Sum has the 

same hyper-real value, then f () x is integrable over the 

interval [ ab] , . 

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H. Vic Dannon 

Then, we call the Integration Sum the integral of f () x from 

x = a, to x = b, and denote it by 

x= 

b 

∫ f ( xdx ) . 

x= 

a 

If the hyper-real is infinite, then it is the integral over [, ab] , 

If the hyper-real is finite, 

x= 

b 

∫ fxdx ( ) = real part of the hyper-real . 

x= 

a 

3.1 The countability of the Integration Sum 

In [Dan1], we established the equality of all positive 

infinities: 

We proved that the number of the Natural Numbers, 

Card , equals the number of Real Numbers, 

2 Card 

Card = , and we have 

2 Card 

2 

Card 

Card = ( Card) = .... = 2 = 2 = ... ≡ ∞. 

In particular, we demonstrated that the real numbers may 

be well-ordered. 

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H. Vic Dannon 

Consequently, there are countably many real numbers in the 

interval 

[,] ab 

, and the Integration Sum has countably many 

terms. 

While we do not sequence the real numbers in the interval, 

the summation takes place over countably many f ( xdx. ) 

The Lower Integral is the Integration Sum where f ( x ) is 

replaced 

by its lowest value on each interval 

3.2 

∑ 

x∈[ a, b] 

⎛ 

⎜⎝ 

dx dx 

2 2 

[ x − , x + ] 

⎞ 

inf f ( t) 

dx 

⎠⎟ 

x− ≤t≤ x+ 

dx dx 

2 2 

The Upper Integral is the Integration Sum where f ( x ) is 

replaced by its largest value on each interval 

3.3 

∑ 

x∈[ a, b] 

⎛ 

⎜⎝ 

⎞ sup f ( t) 

dx 

⎠⎟ 

dx dx 

2 2 

x− ≤t≤ x+ 

[ x − , x + ] 

dx dx 

2 2 

If the integral is a finite hyper-real, we have 

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H. Vic Dannon 

3.4 A hyper-real function has a finite integral if and only if 

its upper integral and its lower integral are finite, and differ 

by an infinitesimal. 

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H. Vic Dannon 

4. 

Delta Function 

In [Dan4], we defined the Delta Function, and established its 

properties 

1. The Delta Function is a hyper-real function defined 

from the hyper-real line into the set of two hyper-reals 

⎧ 

⎪ 1 ⎫ 

⎨0, ⎪ 

⎬. The hyper-real 0 is the sequence 0, 0, 0,... . 

⎪⎩ 

dx 

⎭⎪ The infinite hyper-real 1 

dx 

depends on our choice of 

dx . 

2. We will usually choose the family of infinitesimals that 

is spanned by the sequences 

1 

n , 1 

2 

n 

, 

1 

n 

3 

,… It is a 

semigroup with respect to vector addition, and includes 

all the scalar multiples of the generating sequences 

that are non-zero. That is, the family includes 

infinitesimals with negative sign. Therefore, 

1 

dx 

will 

mean the sequence n . Alternatively, we may choose 

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H. Vic Dannon 

the family spanned by the sequences 

1 

2 n , 

1 

3 n , 

1 

4 n ,… Then, 1 

dx 

will mean the sequence 

2 n . Once we determined the basic infinitesimal dx , 

we will use it in the Infinite Riemann Sum that defines 

an Integral in Infinitesimal Calculus. 

3. The Delta Function is strictly smaller than ∞ 

4. We define, 

1 

χ δ ( x) ≡ dx ( ) 

, 

dx x 

dx 

⎡ ⎤ , 

⎢− 

⎣ 2 2 ⎥⎦ 

where 

χ ⎡ 

⎢− 

⎣ 

dx, 

dx 

2 2 

⎧ dx dx 

1, x ∈ ⎡− 

, ⎤ 

( x) 

= ⎪ ⎢ 2 2 ⎥ 

⎨ ⎣ ⎦ . 

⎪⎪ 0, otherwise 

⎩ 

⎤ 

⎥⎦ 

5. Hence, 

for x < 0 , δ ( x) = 0 

at 

for 

dx 

x =− , δ( x) 

jumps from 0 to 

2 

dx dx 

⎢ ⎣ 

, 

2 2 ⎥ ⎦ , 1 

( x) 

x ∈ ⎡− 

⎤ 

δ = . 

dx 

1 

dx , 

at x = 0 , 

δ (0) = 

1 

dx 

at 

dx 

x = , δ( x) 

drops from 

2 

for x > 0 , δ ( x) = 0. 

1 

dx 

to 0. 

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H. Vic Dannon 

xδ ( x) = 0 

6. If dx = 

1 

, ( x) = 1 1( x),2 1 1( x),3 1 1( x )... 

n 

[ − , ] [ − , ] [ − , ] 

δ χ χ χ 

2 2 4 4 6 6 

7. If dx = 

2 

, 

n 

8. If dx = 

1 

, 

n 

1 2 3 

δ ( x) = , , ,... 

2 2 2 

2 cosh x 2 cosh 2x 2 cosh 3x 

−x −2x −3x 

[0, ∞) [0, ∞) [0, ∞) 

δ( x) = e χ ,2 e χ , 3 e χ ,... 

x =∞ 

∫ 

9. δ( xdx ) = 1. 

x =−∞ 

k =∞ 

1 −ik( ξ−x 

) 

10. δξ ( − x) 

= e 

2π 

∫ dk 

k =−∞ 

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H. Vic Dannon 

5. 

Hyper-real Random Variable 

A Random Variable 

X() 

ζ 

is a real-valued function that maps any event (=outcome) ζ , 

in the Sample space S , into a real number x , in . 

S includes the non-event φ , and X( φ ) = 0. 

Example 

A ball is drawn from a container that has 

5 Red balls, and 4 

Black balls. 

The 

2 

possible outcomes, 

ζ = , ζ = , 

1 

B 

2 

R 

constitute the sample space, 

S 

= { ζ , ζ }. 

1 2 

The number of Red balls is a Random Variable, 

X() 

ζ 

with 

the values 

X( ζ ) = X( B) = 0, 

1 

X( ζ ) = X( R) = 1. 

2 

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H. Vic Dannon 

5.1 Hyper-real X() 

ζ 

X() 

ζ is Hyper-real Random Variable iff 

its values may 

include infinitesimals, and infinite hyper-reals. 

5.2 Hyper-real Probability Distribution of X() 

ζ 

Let 

X() 

ζ 

be Hyper-real, and define, 

dF( x) = Pr( x − dx ≤ X( ζ) ≤ x + dx) 

. 

1 1 

2 2 

Then, 

Fx ( ) = ∑ dFx ( ). 

x= X( ζ), ζ∈S 

is a Hyper-real Probability Distribution of X() 

ζ 

Example 

If a ball is drawn from a container that has 5 Red balls, and 

4 Black balls, and X() 

ζ is the number of Red balls, 

dF(0) = Pr( X( ζ) = 0) = 

4 

9 

dF(1) = Pr( X( ζ) = 1) = . 

5 

9 

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5.3 Hyper-real Probability Density of X() 

ζ 

Let 

X() 

ζ be Hyper-real. If there is Hyper-real f ( x ) so that 

Then 

dF( x) = f ( x) 

dx , 

fx ( ) = 

dF( x) 

dx 

is the Hyper-real Probability Density of X() 

ζ . 

5.4 Expectation of Hyper-real X() 

ζ 

is a Hyper-real number. 

If dF( x) = f ( x) 

dx , 

Example 

E[ X( ζ)] ≡ ∑ xdF( x) 

, 

x= X( ζ), ζ∈S 

EX [ ( ζ)] = ∑ xf( xdx ) . 

x= X( ζ), ζ∈S 




EX [ ( ζ)] = ∑ xdFx ( ) 

x= X( ζ), ζ∈S 

= 0 ⋅ dF(0) + 1 ⋅ dF(1) 

= 

5 . 

9 

4/9 5/9 

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5.5 2 nd Moment of Hyper-real X() 

ζ 


Example 

2 2 

EX [ ( ζ)] ≡ ∑ xdFx ( ) 

x= X( ζ), ζ∈S 




2 2 

EX [ ( ζ)] = ∑ xdFx ( ) 

x= X( ζ), ζ∈S 

2 2 5 

= 0 ⋅ dF(0) + 1 ⋅ dF(1) 

= 

 

. 

9 

4/9 5/9 

5.6 Variance of Hyper-real Random Variable X() 

ζ 


Example 

2 2 

Var[ X()] ζ ≡ E[ X ()] ζ −( E[ X()]) 

ζ 




2 2 

2 

Var[ X()] ζ = E[ X ()] ζ −( E[ X()]) 

ζ 

5 5 20 

= − () = . 

9 9 

81 

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6. 

Normal Distribution and Delta 

Function 

A Normal Random Variable N() 

ζ , with EN [ ( ζ )] = μ, and 

Var[ N( ζ )] = 

σ 2 

, has a probability density function 

The Variance of a Hyper-real 

or an infinite hyper real. 

1 − 

fx ( ) = e . 

2πσ 

N() 

ζ 

( x−μ) 2 

2σ2 

may be an infinitesimal, 

6.1 Infinite Hyper-real Variance 

Proof: 

1 

σ = ⇒ f ( x ) = infinitesimal 

dx 

fx ( ) = 

1 ( dxe ) 

2π 

x = finite hyper-real Then, 

2 2 

−1 

( x−μ 

) ( dx ) 

2 

(x − μ) is finite hyper-real, 

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( x − μ) 

dx is at most infinitesimal (it vanishes at x = μ ), 

− μ 2 

d ) 

2 

1 

( x ) ( x is at most infinitesimal, 

2 

e 

1 

2 2 

− ( x−μ 

) ( dx ) 2 1 2 2 

1 ( x μ) ( ) 

2 

infinitesimal 

≈ − − dx ≈ 1, 

f ( x) ≈ ( dx) = infinitesimal. 

1 

2π 

x = infinite hyper-real Then, 

x 

1 

= α , where α is finite hyper-real, 

dx 

1 2 2 1 1 

2 2 dx 

( ) 2 

2 

) 

( x − μ) ( dx) = α −μ 

( dx , 

e 

1 

2π 

1 2 1 2 

α αμ dx μ 

2 2 

= − ( ) + ( dx) 

2 , 

1 

α 2 

2 

≈ , 

1 2 2 

( x μ) ( dx) 

1 

2 2 

− − − 

− 

1 

2 

α 

2 

≈ e 

fx ( ) ≈ ( dxe ) = infinitesimal. 

α 

2 

, 

6.2 Infinitesimal Variance 

σ = dx ⇒ f ( x ) = Delta Function 


We’ll show that 

fx ( ) 

= 

1 e 

2πdx 

− 

1 

( x−μ 

) 2 

2 dx 

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H. Vic Dannon 

is a Delta Function. 

x 

= μ Then, 

e 

1 x−μ 

2 

dx 

− ( ) 2 0 

= e = 1, 

f ( μ) 

1 

= . 

2πdx 

That is, at x 

= μ , the density function peaks to 

1 

. 

2πdx 

x 

≠ μ Substituting 

e 

1 x−μ 

2 

2 

− ( ) 1 x−μ 

2 1 1 x−μ 

dx 4 

2 dx 2! 2 

2 dx 

= 1 + ( ) + ( ) + ..., 

fx ( ) 

⎧ 

⎫ 

1 1 1 

= ⎪ 

⎨ 

⎪ 

⎬ 

2 1 x 2 1 1 x 

π dx −μ −μ 

4 

1 + ( ) + ( ) + ... 

2 dx 2! 2 

⎪⎩ 

2 dx ⎪⎭ 

⎧ 

⎫ 

1 1 

= ⎪ 

⎨ 

⎪ 

⎬ 

1 2 1 1 1 4 

( ) ( ) 

1 

2π 

dx + x − μ + x − μ + .. 

2 dx 2! 2 3 

⎪ 

⎩ 

. 

2 ( dx) 

⎪⎭ 

⎧ 

⎫ 

1 1 

≈ ⎪ 

⎨ 

⎪ 

⎬ 

1 2 1 1 1 4 

( ) ( ) 

1 

2π 

x − μ + x − μ + .. 

2 dx 2! 2 3 

⎪ 

⎩ 

. 

2 ( dx) 

⎪⎭ 

= 1 1 

infinitesimal 

2π infinite hyper-real 

= 

Finally, for a normal density function, 

x=∞ x=∞ 

x=−∞ 

1 1 

( x−μ 

) 2 

− 

fxdx ( ) = e 

2 σ 

dx= 

1 

2πσ 

∫ ∫ . 

x=−∞ 

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H. Vic Dannon 

7. 

Hyper-real Random Signal 

A Random Signal (=Random Process) is a Random 

Variable that depends also on the time t : 

X(,) 

ζ t . 

Then, the outcome of a Black ball, 

ζ = 

B 

is identified with the outcome of drawing one Black ball, and 

one Red ball successively, 

BR , and RB , 

and with the drawing of one Black ball, and two Red balls 

successively, 

etc. 

BRR , RBR , RRB , 

For a given outcome ζ 0 

, 

X( ζ , t) = x ( t) 

, 

0 

is a function of t , a Sample Function, or Process Realization. 

Example 

ζ 

0 

At time 

t = 1, a ball is drawn from a container that has 5 

Red balls, and 4 Black balls, and X(,1) 

ζ is the number of 

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H. Vic Dannon 

Red balls at t = 1. 

At time 

t = 2, another ball is drawn from the container that 

now has 8 Red, and Black balls, and X(,2) 



At time 


now has 7 Red, and Black balls, and X((3)) 



The outcome of no Red balls, appears once at 

t = 1, once at 

t = 2, and once at t = 3: 

X(no R,1) = X(no R,2) = X(no R, 3) = 1. 

The outcome of one Red ball, appears once at 

t = 1, twice at 

t = 2, and 3 times at t = 3, 

X(1 R ,1) = 1 , 

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H. Vic Dannon 

X(1 R , 2) = 2 , 

X(1 R , 3) = 3 . 

The outcome of two Red balls, appears once at 

four times at t = 3, 

X(2 R ,1) = 0 , 

X(2 R , 2) = 1, 

X(2 R , 3) = 4 . 

t = 2, and 

The outcome of three Red balls, appears once at t = 3, 

X(3 R ,1) = 0 , 

X(3 R ,2) = 0 , 

X(3 R , 3) = 1 . 

The sample space of the process is 

{0 R,1 R, 2 R, 3 R } . 

7.1 Hyper-real X(,) 

ζ t 

A Random Signal is Hyper-real iff the time variable t , and 

the values of 

hyper-reals. 

X(,) 

ζ t 

may include infinitesimals, and infinite 

7.2 Hyper-real Probability Distribution of X(,) 

ζ t 

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H. Vic Dannon 

ζ 

0 

Let X(,) 

t be Hyper-real, fix t = t , and define, 

dF (, x t ) = Pr( x − dx ≤ X (, ζ t ) < x + dx ) . 

1 1 

0 2 0 

2 

Then, 

Fxt (, ) = ∑ dFxt (, ). 

0 0 

x= X(, ζ t ), ζ∈S 

0 

is a Hyper-real Probability Distribution of X(, ζ t ). 

Example 

0 

At time 

t = 1, a ball is drawn from a container that has 5 

Red balls, and 4 Black balls, and X(,1) 

ζ is the number of Red 

balls at t = 1. 

At time 


now has 8 Red, and Black balls, and X(,2) 



dF(0,2) = Pr( X( ζ,2) = 0) = ⋅ = 

4 3 1 

9 8 6 

dF(1, 2) = Pr( X( ζ, 2) = 1) = ⋅ + ⋅ = 

4 5 5 4 5 

9 8 9 8 9 

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H. Vic Dannon 

dF(2, 2) = Pr( X( ζ, 2) = 2) = ⋅ = . 

5 4 5 

9 8 18 

7.3 Hyper-real Probability Density of X(,) 

ζ t 

ζ 

0 

Let X(,) 

t be Hyper-real, and fix t = t . If there is Hyperreal 

f (, xt 

0) 

so that 

dF(, x t ) = f (, x t ) dx , 

0 0 

Then 

fxt (, ) 

0 

= 

dF(, x t ) 

dx 

0 

is the Hyper-real Probability Density of X(, ζ t ). 

0 

7.4 Expectation of Hyper-real X(,) 

ζ t 

ζ 

0 

Let X(,) 

t be Hyper-real, fix t = t , and define 

If dF(, x t ) = f (, x t ) dx , 

Example 

E[ X( ζ, t )] ≡ ∑ xdF( x, t ), 

0 0 

0 0 

x= X(, ζ t ), ζ∈S 

EX [ ( ζ, t )] = ∑ xf( xt , ) dx. 

0 0 

x= X(, ζ t ), ζ∈S 

0 

0 

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H. Vic Dannon 

E[ X( ζ,2)] = ∑ xdF( x,2) 

x= X(,2), ζ ζ∈S 

= 0 ⋅ dF(0, 2) + 1 ⋅ dF(1, 2) + 2 ⋅ dF(2, 2) = 

 

1/6 5/9 5/18 

10 

9 

. 

7.5 2 nd Moment of Hyper-real X(,) 

ζ t 

Example 

2 2 

EX [ ( ζ, t)] ≡ ∑ xdFxt ( , ). 

2 2 

x= X(,), ζ t ζ∈S 

x= X(,), ζ t ζ∈S 

EX [ ( ζ, t)] = ∑ xdFxt ( , ) 

2 2 2 5 

= 0 ⋅ dF(0) + 1 ⋅ dF(1) + 2 ⋅ dF(2) 

= 

 

. 

3 

1/6 5/9 5/18 

7.6 Variance of Hyper-real Random Variable 

2 2 

Var[ X( ζ, t)] ≡ E[ X ( ζ, t)] −( E[ X( ζ , t)]) 

. 

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H. Vic Dannon 

Example 

2 2 

Var[ X( ζ, t)] = E[ X ( ζ, t)] −( E[ X( ζ, t)]) 

9 6 2 

() 

9 

5 5 25 

= − = . 

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H. Vic Dannon 

8. 

Continuity of X(,) 

ζ t 

ζ t t0 


t is continuous at 

= iff for any dt , 

⇔ 

E{[ X( ζ, t + dt) − X( ζ, t )] } = infinitesimal , 

∑ 

X( ζ, t ), ζ∈S 

0 

0 0 

2 

2 

0 

ζ 

0 0 

[ X( ζ, t + dt) − X( , t )] dF( x, t ) = infinitesimal 

If dF(, x t ) = f (, x t ) dx , 

⇔ 

∑ 

X(, ζ t ), ζ∈S 

0 

0 0 

2 

0 

ζ 

0 0 

[ X( ζ, t + dt) − X( , t )] f( x, t ) dx = infinitesimal 

ζ = 

0 

0 

8.2 X(,) 

t is continuous at t t ⇒ EX [ ( ζ, t )] is continuous 


0 ≤ E[{[ X(, ζ t + dt) −X(, ζ t )] − E[ X(, ζ t + dt) −X(, ζ t0)]}] 

0 0 0 

= E{[ X( ζ, t + dt) −X( ζ, t )] } 

0 0 

2 

− 2 E{[ X( ζ, t + dt) − X( ζ, t )] E[ X( ζ, t + dt) −X( ζ, t )]} 

0 0 0 0 

+ { EX [ ( ζ, t + dt) −X( ζ, t )]} 

0 0 

2 

2 2 

0 0 0 

0 

= E{[ X(, ζ t + dt) −X(, ζ t )]} − { E[ X(, ζ t + dt) −X(, ζ t )]} 

Therefore, 

2 

33


H. Vic Dannon 

2 2 

0 

+ − 

0 

≤ 

0 

+ − 

0 

{ EX [ ( ζ, t dt) X( ζ, t )]} E{[ X( ζ, t dt) X( ζ, t )] } 

 

Hence, 

≥0 ifinitesimal 

{ EX [ ( ζ, t + dt) − X( ζ, t )]} = infinitesimal, 

0 0 

EX [ ( ζ, t + dt) − X( ζ, t )] = infinitesimal. 

0 0 

2 

34


H. Vic Dannon 

9. 

Derivative of X(,) 

ζ t 


ζ t has derivative with respect to t at 

t = t 0 

iff there is a Random Signal X '( ζ, t) =∂tX( ζ, t) 

, so 

that for any dt , 

⎡ 

2 ⎤ 

⎡X(, ζ t0 + dt) −X(, ζ t0) 

⎤ 

E − X '( ζ, t0) = infinitesimal , 

⎢⎢ 

dt 

⎥ 

⎣ 

⎦ ⎥ 

⎣ 

⎦ 

⇔ 

∑ 

x= X( ζ, t ), ζ∈S 

0 

⎡x(, ζ t0 + dt) −x(, ζ t0) 

⎤ 

− x'( ζ, t0) dF( x, t0) = infinitesimal 

⎢ dt 

⎥ 

⎣ 

⎦ 

2 

If dF(, x t ) = f (, x t ) dx , 

0 0 

⇔ 

∑ 

x= X( ζ, t ), ζ∈S 

0 

⎡x(, ζ t0 + dt) −x(, ζ t0) 

⎤ 

− x'( ζ, t0) f( x, t0) dx = infinitesimal 

⎢ dt 

⎥ 

⎣ 

⎦ 

2 

35


H. Vic Dannon 

10. 

Random Walk 

The Random Walk of small particles in fluid is named after 

Brown, who first observed it, Brownian Motion. It models 

other processes, such as the fluctuations of a stock price. 

In a volume of fluid, the path of a particle is in any direction 

in the volume, and of variable size 

10.1 The Bernoulli Random Variables of the Walk 

We restrict the Walk here to the line, in uniform 

infinitesimal size steps dx : 

To the left, with probability 

1 

p = , 

2 

36


H. Vic Dannon 

or to the right, with probability 

At fixed time t , after 

1 

q = . 

2 

N infinitesimal time intervals dt , 

N = 

t 

dt 

the particle have made 

, is a fixed infinite hyper-real, 

and 

K infinitesimal steps of size dx to the right, 

L infinitesimal steps of size dx to the left, 

and is at the point 

x = ( K 

− L) 

dx = Mdx. 

M 

KLM, , , are infinite hyper-reals. 

At the i th step we define the Bernoulli Random Variable, 

Bi (right step) = dx , ζ 

1 

= right step . 

Bi (left step) 

where i = 1,2,..., N . 

=−dx, ζ 

2 

= left step . 

Pr( B = dx) 

= p = , 

i 

1 

2 

Pr( B =− dx) 

= q = , 

i 

EB [ ] = dx⋅ + ( −dx) ⋅ = 0, 

i 

1 1 

2 2 

2 2 1 2 1 

i 

2 2 

2 

E[ B ] = ( dx) ⋅ + ( −dx) ⋅ = ( dx) 

1 

2 

37


H. Vic Dannon 

Var[ Bi] = E[ Bi ] − ( E[ B ]) ( ) 

i 

= dx 

 

2 

( dx ) 

2 2 

0 

2 . 

10.2 The Binomial Distribution of the Walk 

B( ζ , t) = B + B + ... + BN 

1 2 

is a Random Process with 

EB [ ( ζ , t)] = 0, 

distributed Binomially 

Var[ B( ζ , t)] = N( dx) 

, 

⎛ 

Pr ( , ) N ⎞ − ≤ ≤ + = ⎜ 

⎟ N 

( x 

1dx B t x 

1dx) 

ζ 

1 

2 2 ⎜ M+ 

N 

⎟ 2 ⎜⎝ 2 ⎠ 

2 

Proof: Since the 

B i 

are independent, 

EB [ ( ζ , t)] = EB [ ] + ... + EB [ N 

] = 0 

 

1 

0 0 

Var[ B( ζ , t)] = Var[ B ] + ... + Var[ BN 

] = N( dx) 

 

1 

( dx ) ( dx ) 

2 2 

B(,) 

ζ t has a Binomial distribution, 

⎛N 

⎞ 

1 1 

Pr ( x − dx ≤ X( ζ, t) 

≤ x + dx) 

= p q 

2 2 ⎜K 

⎜⎝ ⎠⎟ 

2 

K N−K 

, 

38


H. Vic Dannon 

From 

N 

= K +L 

, 

⎛N 

⎞ 

= ⎜K 

⎜⎝ ⎠⎟ 

⎛N 

⎞ 

= ⎜K 

⎜⎝ ⎠⎟ 

K 

1 1 

( ) ( ) 

1 

2 N 

2 2 

. 

N−K , 

we have 

M = K − L, 

K 

N+ 

M 

2 

= , 

L 

= . 

N−M 

2 

Thus, 

⎛ 

Pr ( , ) N ⎞ − ≤ ≤ + = ⎜ 

⎟ 

1 . 

N 

1 1 

( x dx B t x dx) 

ζ 

2 2 ⎜ M+ 

N 2 ⎜⎝ ⎟ 

2 ⎠ 

10.3 The Gaussian Distribution of the Walk 

If 

2 

( dx) = 2 D( dt) 

, where the Drift Coefficient D is a constant 

Then, the Binomial distribution of 

B(,) 

ζ t 

is infinitesimally 

close to a Gaussian distribution of a Random Signal with 

μ = 0, 

σ = t2D 

= Ndx. 

fxt (,) ≈ 

1 

e 

2π 

t2D 

2 

1 x 

− 

2t2D 

39


H. Vic Dannon 


( x dx X ζ t x dx) 

Pr − ≤ ( , ) ≤ + = 

 

N ! 

1 1 1 

2 2 N+ M 2 

( )! N−M 

( )! N 

dF( x, t) 

2 2 

. 

− 

N N 

Substituting N! ≈ 2πNN 

e from Sterling’s Formula for 

infinite hyper-real N , 

≈ 2πNN e 

1 

, 

N M N M N M N M N 

2 

2 π ( ) + − 

+ 2 2 

2 ( ) 

− − 

2 

e 

− 

2 

N+ M N+ M 

e π 

N−M N−M 

2 2 2 2 

N 

−N 

= 

2 

π 

N 

N + 

N+ M+ 1 N+ M+ 1 N− M+ 1 

N− M+ 

1 

2 M 2 2 M 2 

(1 + ) N (1 − ) 

N 

N 

1 

2 

N 

, 

= 

2 1 

πN 

(1 + 

M) (1 − ) 

N 

N + M + 1 N − M + 1 

2 M 2 

N 

. 

Then, up to an infinitesimal, 

⎡ ⎤ ≈ − + − − 

2 N+ M+ 1 M N− M+ 

1 

M 

log 

⎣ 

dF( x, t) ⎦ 

log log(1 ) log(1 ) 

πN 

2 N 2 

N 

Since 0 < 

M 

< 1, 

N 

2 

+ 

M 

≈ 

M 

− 

1 M 

, 

2 

log(1 ) 

N N 2 N 

2 

− 

M 

≈ −M 

− 

1 M , 

2 

log(1 ) 

N N 2 N 

log ⎡dF( x, t) ⎤ 

⎣ ⎦ 

≈ log 

2 

πN 

N+ M+ 1 M N+ M+ 

1 M 

2 N 4 N 

− + 

2 

2 

40


H. Vic Dannon 

N− M+ 1 M N− M+ 

1 M 

2 N 4 N 

+ + 

2 

2 

= 

log 

2 

πN 

2 2 3 

−M − 

M 

− 

M 

+ 

M 

+ 

M 

+ 

M 

2 

2 2 

2 2N 2N 4N 4N 4N 

2 2 3 

+ 

M 

− 

M 

+ 

M 

+ 

M 

− 

M 

+ 

M 

2 

2 2 

2 2N 2N 4N 4N 4N 

2 2 

2 M M 

log 

πN 2N 2N 

= − + 

= log 

2 

− 

M 

(1 − 

1) 

πN 2N N 

2 

≈1 

2 

This would give 

1 1 

N 2π 

− 

1 M2 

2 N 

= log 2 e . 

1M 

1 − 

(,) 2 2 N 

2 

dF x t 

≈ 

for negative M , and x , we have 

dF(,) 

x t ≈ 

1 

2π 

π 

e 

N 

e 

N 

2 

1M 

− 

2 N 

2 

, but accounting 

= 

1 

2π 

t 

dt 

e 

− 

x2 

1 ( dx ) 

2 

2 t 

dt 

= 

dt 

2π 

e 

t 

1 dt 

− 

2( dx) 

2 

2 

x 

t 

Thus, we need to assume that 

2 

( dx) 

, and dt are proportional, 

41


H. Vic Dannon 

2 

( dx) = 2 D( 

dt), 

where the Drift Coefficient D is a constant of the Walk. 

Then, 

2 

1 x 

− 

2t2D 

1 

dF(,) 

x t ≈ e dx . 

2π 

t2D 

Hence, the probability density of the Walk is 

with 

2 

1 x 

− 

2t2D 

dF(,) x t 1 

fxt (,) = ≈ e , 

dx 2π 

t2D 

μ = 0, 

σ = 2tD = Ndx . 

2 

10.4 f (,) xt solves the parabolic wave equation ∂ f = D∂ f . 

Proof: By substitution. 

t 

x 

10.5 Increments of Random Walk 

2 

If ( dx) = 2 D( 

dt) 

, 

Then 

1) For any τ > 0, the distribution of B(, ζ t + τ) −B(,) 

ζ t is 

infinitesimally close to a Gaussian distribution that has 

μ = 0, 

42


H. Vic Dannon 

2 

σ = τ2D 

, 

and depends only on 

τ 

(Stationary Process). 

2) For fixed t , and any dt , the Random Variables 


B(,) ζ t −B(, ζ t − dt) 

, 

B(, ζ t −dt) −B(, ζ t − 2 dt) 

, 

……………………………., 

B(, ζ dt) − B(,0) 

ζ , 

are independent, random variables. 

1) Let T 

= . Then, as in 10.2, the Binomial distribution of 

τ 

dt 

B( ζ, t τ) B( ζ , t) B 

+ 

B 

+ 

... B 

+ − = 

N 1 

+ 

N 2 

+ + 

N+ T 

, 

is infinitesimally close to a Gaussian distribution with 

2 

μ = 0, and σ = τ2D 

, that depends only on τ . 

2) B(,) ζ t −B(, ζ t −dt) 

is precisely one Bernoulli Random Variable that is 

statistically independent of the precisely one Bernoulli 

Random Variable that equals 

B(, ζ t −dt) −B(, ζ t − 2 dt) 

43


H. Vic Dannon 

11. 

Random Walk is Continuous, 

has a Derivative with Delta 

Function Variance, and EB [ ( ζ , t)] 

has unbounded Variation 

2 

11.1 ( dx) = (2 D) 

dt ⇒ Random Walk is Continuous 


E[{ B( ζ, t + dt) − B( ζ, t)} ] = 

2 

= Var[ B(, ζ t + dt) − B(,)] ζ t + ( E[ B(, ζ t + dt) −B(,)]) ζ t 

2 , 

 

B 

i 

B 

i 

where 

B i 

is a Bernoulli Random Variable, 

2 

= Var[ Bi] + ( E[ B ]) = (2 D) 

. 

i 

dt 

2 0 

( dx ) = (2 D) 

dt 

11.2 If 

2 

( dx) = (2 D) 

dt 

44


H. Vic Dannon 

Then The Derivativ e of Random Walk is 

1 

B = B i 

, 

dt 

where (1) 

B = B(, ζ t + dt) − B(, ζ t ), is a Bernoulli 

i 

0 0 

Random Variable. 

(2) EB [ ] = 0, 

(3) Var[ B ] = 2Dδ( t0) 

, 


(1) For each t = t 0 

, we need to find a Random Signal 

B (, ζ t ) , so that for any dt , 

0 

⎡ 

2 ⎤ 

⎡B(, ζ t0 + dt) −B( ζ, 

t0) 

⎤ 

E − B ( ζ, t0) = infinitesimal , 

⎢⎢ 

dt 

⎥ 

⎣ 

⎦ ⎥ 

⎣ 

⎦ 

Since 

B(, ζ t + dt) −B(, ζ t ), is a Bernoulli Random Variable 

0 0 

B 

i 

, 

⎡ 2 ⎤ ⎡ 

2 ⎤ 

XBt ( , dt) B( , t) 

Bi 

E ⎪ 

⎧ + − ζ 

⎫ ⎧ ⎫ 

B(,) ζ t ⎨ 

− ⎪ 

⎬ = E ⎪ −B⎪ 

⎨ 

 

⎬ 

⎢⎪⎩ dt 

⎪⎭ ⎥ ⎢ 

⎪⎩dt 

⎣ ⎦ ⎪⎭ 

⎥ 

⎣ ⎦ 

Therefore, at time 

t = t 0 

, the Random Variable 

1 

B 

i 

, 

dt 

is the derivative of the Random Walk B(, ζ t0) 

. 

45


H. Vic Dannon 

(2) 

1 

EB [ ] = EB [ ] 0 

dt i 

= . 

(3) Var[ B] = E[ B 

] −( E 

 

[ B]) 

0 

2 2 

0 

= 

= 

1 

2 

( dt) 

2 

( dx) 1 

dt 

dt 

2D 

2 

i 

EB [ ] 

, 

2 

( dx ) 

= (2 D) δ( t ). 

0 

11.3 EB( [ ζ , t)] has unbounded Variation in [ ab ,] 


2 Db− ( a) = (2 Ddt ) + (2 Ddt ) + ... + (2 Ddt ) 

 

2 2 

( dx) ( dx) ( dx) 2 

⎡ 2⎤ ⎡ 

2 

= E ⎢{ B(,) ζ b −B(, ζ b − dt) } ⎥ + .. + E ⎢{ 

B(, ζ a + dt) −B(, ζ a) 

} ⎤ ⎥ 

⎣ ⎦ ⎣ 

⎦ 

≤ max B(, ζ t + dt) −B(,) ζ t E ⎡ B(,) ζ b −B(, ζ b −dt) ⎤ .. 

a≤≤ 

t b 

⎣ 

⎦ 

+ 

 

infinitesimal 

... + max B(, ζ t + dt) − B(,) ζ t E ⎡ B(, ζ a + dt) − B(, ζ a) 

⎤ 

⎣ 

⎦ 

= 

a≤≤ 

t b 

= infinitesimal{ EB( ζ, b) −B( ζ, b− dt) + ... + EB( ζ, a+ dt) − B( ζ, a) }, 

46


H. Vic Dannon 

{ E ⎡ B ζ b −B ζ b − dt + + B ζ a + dt −B ζ a ⎤} 

=infinitesimal 

⎣ 

( , ) ( , ) ... ( , ) ( , ) 

⎦ , 

since the Bernoulli Random Variables are independent. 

Therefore, 

(2 )( ) 

E ⎡ 

D b − a 

⎣ 

B(, ζ b) 

−B(, ζ b − dt) + ... + B(, ζ a + dt) − B(, ζ a) 

⎤ 

⎦ 

= , 

infinitesimal 

is infinite hyper-real, and EB [ ( ζ, t)] 

has unbounded variati on 

in [ ab. ,] 

47


H. Vic Dannon 

12. 

t= 

b 

∫ 

t= 

a 

ftdB () (, ζ t) 

While EB [ ( ζ , t)] 

has unbounded Variation in [,] ab, integration 

with respect to 

B(,) 

ζ t 

is possible. 

Let 

f () t be a hyper-real function on the bounded time 

interval [,] ab . f () t need not be bounded. 

At each a ≤ t ≤ b, 

there is a Bernoulli Random Variable 

dB(,) ζ t = B(, ζ t + dt) − B(,) ζ t = B (,) ζ t = B (,) ζ t dt. 

We form the Integration Sum 

For any dt , 

t= b t= b t= 

b 

∑ ∑ ∑ 

f () tdB(, ζ t) = ftB () (, ζ t) = ftB () (, ζ tdt ) 

i 

t= a t= a t= 

a 

(1) the First Moment of the Integration Sum is 

⎡t= b ⎤ t= 

b 

E f() t B(, ζ t) dt = f() t E[ B 

∑ 

∑ (, ζ t)] dt = 0. 

 

⎢⎣t= a ⎥⎦ 

t= 

a 

(2) the Second Moment of the Integration sum is 

i 

0 

48


H. Vic Dannon 

E 

⎡ ⎛ 

⎞ 

2 

t= b ⎤ t= b τ= 

b 

⎢⎜ 

f () t B (, ) i 

t ⎡⎛ ⎞⎛ E f () t B i(, t ⎞⎤ 

ζ ζ ) f ( τ) B (, ζ τ) 

∑ = 

∑ ∑ j 

⎜ ⎝t= a ⎠⎟ ⎢⎝⎜t= a ⎠⎝ ⎟⎜ 

⎣ 

τ= 

a 

⎠⎟ 

⎢ 

⎥ 

⎥ 

⎣ 

⎦ 

⎦ 

t= b τ= 

b 

= ∑∑ 

t= a τ= 

a 

f ()( tfτ) EB [ (, ζ τ) B(, ζ t)] 

Since the Bernoulli Random Variables are independent, 

EB [ ( ζτ , ) B( ζ, t)] = EB [ ( ζ, t)] = ( dx) 

j 

2 2 

j i i 

i 

only for t 

= τ . Then, 

⎡ 

⎤ 

⎛ 

E ⎜ f t B t f t 

⎢ 

⎜⎝ 

⎣ 

⎥⎦ 

2 

2 

t= b ⎞ t= 

b 

2 2 

∑ () 

i(, ζ ) ⎟ = ∑ ()( dx) 

 

, 

t= a ⎠⎟ 

⎥ 

t= 

a 

(2 Ddt ) 

t= 

b 

2 

= 2 D∑ 

f ( t) dt, 

t= 

a 

t= 

b 

2 

= 2 D∫ 

f ( t) dt. 

t= 

a 

assuming (d x) = (2 D) 

dt , and f () t integrable 

Thus, for any dt , the Integration Sum is a 

unique well- 

defined hyper-real Random Variable I() 

ζ 

. 

We call 

I() 

ζ the integral of f () t , with respect to 

B(, 

ζ t) 

from 

x 

t= 

b 

= a, to x = b, and denote it by f () tdB(, ζ t) 

. 

∫ 

t= 

a 

49


H. Vic Dannon 

13. 

Poisson Process 

The arrival at rate 

λ , of radioactive particles at a counter is 

modeled by the Poisson Process. It models other processes, 

such as the arrival of phone calls at rate 

λ , to an operator. 

13.1 The Bernoulli Random Variables of the Process 

We assume that 

an arrival probability in time 

dt 

is 

and no arrival probability in time 

p 

= λdt 

, 

dt 

is 

q 

= 1 −λdt. 

At fixed time t , after 

N infinitesimal t ime intervals dt , 

N 

= 

t 

dt 

, is an infinite hyper-real, 

there are 

and 

k 

k arrivals, 

is a finite hyper-real 

N 

− k no arrivals, 

50


H. Vic Dannon 

N 

− k is an infinite Hyper-real 

At the i th step we define the Bernoulli Random Variable, 

P 

i (arrival) = 1 , ζ 

1 

= arrival 

P (no-arrival) = 0 , 

i 2 

ζ = 

no-arrival 

where i = 1,2,..., N . 

Pr( P = 1) = p = λdt, 

i 

Pr( P = 0) = q = 1 −λdt, 

i 

E[ P ] = 1⋅ λdt + 0 ⋅(1 − λdt) 

= λdt 

, 

i 

2 2 2 

E[ P ] = 1 ⋅ λdt + 0 ⋅(1 − λdt) 

= λdt 

, 

i 

2 

Var[ P ]) 2 

i] = E[ Pi ] −( E[ 

Pi 

, 

 

λdt 

λdt 

= λdt (1 −λdt) ≈ λdt 

 

. 

≈1 

13.2 The Binomial Distribution of the Process 

P( 

ζ, t ) = P 1 

+ P 2 

+ ... + P N 

is a Random Process with 

EP [ ( ζ, t)] 

= λt, 

Var[ P( ζ, t)] 

= λt, 

distributed Binomially 

51


H. Vic Dannon 

⎛N 

⎞ 

k 

Pr ( P( ζ, t) k) ( λdt) ( 1 λdt) N − 

= = 

⎜ 

− 

k 

⎜⎝k ⎠ 

⎟ 

P i 

Proof: Since the are independent, 

EP [ ( ζ, t)] = EP [ 

1 ] + ... + EP [ ] 

N 

= λNdt 

 

λdt 

λdt 

Var[ P( ζ, t)] = Var[ P1 

] + ... + Var[ PN 

] ≈ λNdt 

 

≈λdt 

≈λdt 

t 

t 

P(,) 

ζ t 

has a Binomial distribution, 

⎛N 

⎞ 

k N−k 

Pr ( P( ζ, t) 

= k) 

= ⎜ 

p q , 

k 

⎜⎝ ⎠⎟ 

⎛N 

⎞ 

= k 

⎜ 

λdt 

−λdt 

k 

1 

⎜⎝ ⎠⎟ 

( ) ( ) 

N−k 

. 

13.3 The Poisson Distribution of the Process 

The Binomial distribution of 

P(,) 

ζ t 

is infinitesimally 

close to a Poisson distribution of a Random Signal with 

μ = λt 

, 

2 

σ 

= λt 

. 


1 k −λ 

Pr[ P( ζ, t) = k] ≈ ( λt) 

e 

k ! 

t 

52


H. Vic Dannon 

N ! 

Pr ( , ) 1 

k!( N − k)! 

( P ζ t = k) = ( λdt) ( −λdt) 

k N−k 

. 

Substituting 

1 

2 

N + − N 

N! ≈ 2πN 

e from Sterling’s Formula for 

infinite hyper-rea l N , 

≈ 

2πN 

1 

2 

1 

2 

k! 2 π( N − k) 

e 

N + N 

e − 

N−k , 

N− k+ − N + k 

k 

( λdt 

) ( 1 −λdt 

) 

N + 

1 

2 

1 N 

k N−k 

= 

1 1 

( λ 

t 

) ( 1 − λ 

t 

) , 

k ! N+−k N k N 

N 

2 

(1 

k −+ 

N 

) 

2 e 

k 

− 

N 

1 k 1 1 

−k 

( ) 

( ) N 

k k− 

1 

(1 ) 

2 λt 

= λt 

e − 1 − 

, 

k ! (1 

k N 

) N 

N 

k 

− 

λt 

N 

1 

−λt 

( 1 − 

≈ 

) 

≈e 

N 

≈e −k 

≈1 ≈1 

since N is an infinite Hyper-real. 

≈1 

13.4 Increment of Poisson Process 

1) F or any τ > 0, the distribution of P(, ζ t + τ) − P(,) 

ζ t is 

infinitesimally close to a Poisson distribution of a Random 

Signal with 

μ 

= 

λτ, 

2 

σ 

= 

λτ. 

1 k −λτ 

Pr[ P( ζ, t + τ) − P( ζ, t) = k] ≈ ( λτ) 

e 

k ! 

53


H. Vic Dannon 

that depends only on 

τ 

(Stationary Process). 

2) For fixed t , and any dt , the Random Variables 

P(,) ζ t −P(, ζ t −dt) 

, 

P(, ζ t −dt) −P(, ζ t − 2 dt) 

, 

……………………………., 

P(, ζ dt) − P(,0) 

ζ , 

are independent, random variables. 


1) Let T 

= . Then, as in 12.2, the Binomial distribution of 

τ 

dt 

P( ζ, t + τ) − P( ζ , t) = P 

+ 

+ P 

+ 

+ ... + P 

N 1 N 2 

N+ T 

, 

is infinitesimally close to a Poisson distribution with μ = λτ, 

2 

and σ 

= λτ, that depends only on τ . 

2 ) 

P(,) ζ t −P(, ζ t − dt) 

is precisely one Bernoulli Random 

Variable that is 

statistically independent of the precisely one Bernoulli 

Random Variable that equals P(, ζ t −dt) −P(, ζ t − 2 dt) 

. 

54


H. Vic Dannon 

14. 

Poisson Process is Continuous 

and has a Derivative with Delta 

Function Variance 

14.1 Poisson Process is Continuous 


E[{ P( ζ, t + dt) −P( ζ, t)} ] = 

P 

i 

2 

= Var[ P( 

ζ, t + dt) − P( ζ, t)] + ( E[ P( 

ζ, t + dt) −P( ζ, t)]) 

, 

 

 

where 

X 

i 

is a Bernoulli Random Variable, 

= Var[ Pi 

] + ( [ ]) = 

 

infinitesimal . 

≈λdt 

E P 

2 

i 

λdt 

P 

i 

2 

14.2 The Derivative of the Poisson process is 

1 

P = P 

dt i 

, 

where (1) 

P = P(, ζ t + dt) − P(, ζ t ), 

is a Bernoulli 

i 

0 0 

55


H. Vic Dannon 

Random Variable. 

(2) EP [ ] = λ , 

(3) Var[ P 

] = λδ( t0) 


(1) For each t = t0 

, we need to find a Random Signal 

P (, ζ t ) , so that for any dt , 

0 

⎡ 

2 ⎤ 

⎡P(, ζ t0 + dt) −P(, ζ t0) 

⎤ 

E − P ( ζ, t0) = infinitesimal, 

⎢⎢ 

dt 

⎥ 

⎣ 

⎦ ⎥ 

⎣ 

⎦ 

Since 

P(, ζ t + dt) −P(, ζ t ), is a Bernoulli Random Variable 

0 0 

P i 

, 

⎡ 2 ⎤ ⎡ 

2 ⎤ 

P(, t dt) P(,) 

t 

Pi 

E ⎪ 

⎧ ζ + − ζ 

⎫ ⎧ ⎫ 

P(,) ζ t ⎨ 

− ⎪ 

⎬ = E ⎪ −P⎪ 

⎨ 

 

⎬ 

⎢⎪⎩ dt 

⎪⎭ ⎥ ⎢⎪⎩ 

dt ⎪⎭ 

⎣ ⎦ 

⎥ 

⎣ ⎦ 

Therefore, at time t = t0 

, the Random Variable 

1 

dt 

is the derivative of the Random Walk P(, ζ t0) 

. 

(2) 

P i 

EP 1 

[ ] = EP [ ] 

dt i 

= λ . 

λdt 

(3) Var[ P] = E[ P 

] −( E 

 

[ P]) 

, 

2 2 

λ 

56


H. Vic Dannon 

1 2 

= EP [ ] −λ 

 

2 

2 i 

( dt) 

1 

= λ 

dt 

2 2 

λdt+ 

λ ( dt) 

= 

λδ ( t ), 

0 

By [Dan4]. 

57


H. Vic Dannon 

15. 

t= 

b 

∫ 

t= 

a 

ftdP () (, ζ t) 

Let f () t be a hyper-real function on the bounded time 

interval [,] ab . f () t need not be bounded. 

At each 

a ≤ t ≤ b, 

there is a Bernoulli Random Variable 

dP(,) ζ t = P(, ζ t + dt) − P(,) ζ t = P(,) ζ t = P (,) ζ t dt. 

We form the Integration Sum 

F or any dt , 

t= b t= b t= 

b 

∑ ∑ ∑ 

f () tdP(, ζ t) = ftP () (, ζ t) = ftP () (, ζ tdt ) 

i 

t= a t= a t= 

a 

(1) the First Moment of the Integration Sum is 

⎡t= b ⎤ t= 

b 

t= 

b 

E f () t P(,) ζ t dt = f () t E[ P 

∑ ∑ (,)] ζ t dt = λ f () t dt , 

∫ 

⎢⎣t= a ⎥⎦ 

t= a t= 

a 

λ 

i 

assuming 

f () t integrable. 

(2) the Second Moment of the Integration sum is 

58


H. Vic Dannon 

E 

⎡ ⎛ 

⎞ 

2 

t= b ⎤ t= b τ= 

b 

⎢⎜ 

f () t P (, ) i 

t ⎡⎛ ⎞⎛ E f () t P i(, t ⎞⎤ 

ζ ζ ) f ( τ) P = 

(, ζ τ) 

j 

∑ ∑ ∑ 

⎜ ⎝t= a ⎠⎟ ⎢⎝⎜t= a ⎠⎝ ⎟⎜ 

⎣ 

τ= 

a 

⎠⎟ 

⎢ 

⎥ 

⎥ 

⎣ 

⎦ 

⎦ 

t= b τ= 

b 

= ∑∑ 

t= a τ= 

a 

f ()( tfτ) EP [ (, ζ τ) P(, ζ t)] 

Since the Bernoulli Random Variables are independent, 

only for t 

2 

j i i 

EP [ ( ζτ , ) P( ζ, t)] = EP [ ( ζ, t)] = λdt(1 + λdt) 

= 

τ . Then, 

⎡ 

⎛ 

E f t P t = f t 

⎜ 

⎢⎝ 

⎠ 

⎣ 

⎥⎦ 

j 

2 ⎤ 

∑ t= b ⎞ t= 

b 

2 

() 

i(, ζ ) λ () dt 

⎥ 

∑ t= a ⎟ 

, 

t= 

a 

i 

t= 

b 

= λ ∫ f 

2 () tdt, 

t= 

a 

≈1 

assuming f () t integrable. 

Thus, assuming f () t integrable, for any dt , the Integration 

Sum is a unique well-defined hyper-real Random Variable 

I() 

ζ . We call I() ζ the integral of f () t , with respect to P(,) 

ζ t 

from x 

= a, to x = b, and denote it by 

t= 

b 

∫ f () tdP(, ζ t) 

. 

t= 

a 

59


H. Vic Dannon 

References 

[Ben oit] Eric Benoit “Random Walks and Stochastic Differential 

Equations” in “Nonstandard Analysis in Practice” edited by Francine 

Diener, and Marc Diener, Springer, 1995. 

[Chandrasekhar] S. Chandrasekhar, “Stochastic Problems in Physics 

and Astronomy” Reviews of Modern Physics, Volume 15, Number1, 

January 1943. 

Reprinted in “Selected Papers on Noise and Stochastic Processes” edited 

by Nelson Wax, Dover, 1954 

[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all 

Infinities, and the Continuum Hypothesis” in Gauge 

Institute Journal 

Vol.6 No 2, May 2010; 

[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal 

Vol.6 No 4, November 2010; 

[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute 

Journal Vol.7 No 4, November 2011; 

[Dan4] Dannon, H. Vic, “The Delta Function” 

in Gauge Institute 

Journal Vol.8 No 1, February 2012; 

[Hoe l/Port/Stone] Paul Hoel, Sidney Port, Charles Stone, “Introduction 

to Stochastic Processes” Houghton Mifflin, 1972. 

[Hsu] 

Hwei Hsu, “Probability, Random Variables, & Random 

Processes”, Schaum’s Outlines, McGraw-Hill, 1997. 

[Karlin/Taylor] Howard Taylor, Samuel Karlin, “An Introduction to 

Stochastic Modeling”, Academic Press, 1984. 

60


H. Vic Dannon 

[Larson/Shubert] Harold Larson, Bruno Shubert, “Probabilistic Models 

in Engineering Sciences, Volume II, Random Noise, Signals, and 

Dynamic Systems”, Wiley, 1979. 

61

Infinitesimal Calculus of Random Walk and Poisson Processes

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