Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
Infinitesimal Calculus of Random Walk and Poisson Processes
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
Pro<strong>of</strong>:<br />
( x dx X ζ t x dx)<br />
Pr − ≤ ( , ) ≤ + =<br />
<br />
N !<br />
1 1 1<br />
2 2 N+ M 2<br />
( )! N−M<br />
( )! N<br />
dF( x, t)<br />
2 2<br />
.<br />
−<br />
N N<br />
Substituting N! ≈ 2πNN<br />
e from Sterling’s Formula for<br />
infinite hyper-real N ,<br />
≈ 2πNN e<br />
1<br />
,<br />
N M N M N M N M N<br />
2<br />
2 π ( ) + −<br />
+ 2 2<br />
2 ( )<br />
− −<br />
2<br />
e<br />
−<br />
2<br />
N+ M N+ M<br />
e π<br />
N−M N−M<br />
2 2 2 2<br />
N<br />
−N<br />
=<br />
2<br />
π<br />
N<br />
N +<br />
N+ M+ 1 N+ M+ 1 N− M+ 1<br />
N− M+<br />
1<br />
2 M 2 2 M 2<br />
(1 + ) N (1 − )<br />
N<br />
N<br />
1<br />
2<br />
N<br />
,<br />
=<br />
2 1<br />
πN<br />
(1 +<br />
M) (1 − )<br />
N<br />
N + M + 1 N − M + 1<br />
2 M 2<br />
N<br />
.<br />
Then, up to an infinitesimal,<br />
⎡ ⎤ ≈ − + − −<br />
2 N+ M+ 1 M N− M+<br />
1<br />
M<br />
log<br />
⎣<br />
dF( x, t) ⎦<br />
log log(1 ) log(1 )<br />
πN<br />
2 N 2<br />
N<br />
Since 0 <<br />
M<br />
< 1,<br />
N<br />
2<br />
+<br />
M<br />
≈<br />
M<br />
−<br />
1 M<br />
,<br />
2<br />
log(1 )<br />
N N 2 N<br />
2<br />
−<br />
M<br />
≈ −M<br />
−<br />
1 M ,<br />
2<br />
log(1 )<br />
N N 2 N<br />
log ⎡dF( x, t) ⎤<br />
⎣ ⎦<br />
≈ log<br />
2<br />
πN<br />
N+ M+ 1 M N+ M+<br />
1 M<br />
2 N 4 N<br />
− +<br />
2<br />
2<br />
40