CONTENT
01. Random variable and random process. Classes of random processes
(Overview).
02. Convergence with probability one and in probability. Other types of
convergence. Ergodic theorem.
03. Orthogonal projection. Conditional expectation in the wide sense.
04. Wiener filter.
05. Kalman filter.
06. Kalman filter implementation for linear algebraic equations. Karhunen
Loeve decomposition.
07. Independence. The conditional expectation.
08. Non linear filtering.
09. Gaussian random sequences.
10. Wiener process. Gaussian white noise.
11. Poisson process. Poisson white noise. Telegraphic signal.
12. Stochastic Itˆ
o integral.
1.
Random variable and random process. Classes of
random processes (Overview)
1.1.
Random variable.
Denote by Ω =
{
ω
}
a sample space. A func
tion
1
ξ
(
ω
),
ω
∈
Ω:
Ω
ξ
→
R
is called random variable. Typically, a description of the random vari
able
ξ
(
ω
) is given in term of its distribution function
F
(
x
) =
P
(
ω
:
ξ
(
ω
)
≤
x
)
, x
∈
R
,
where
P
, named probabilistic measure, is a function transforming sub
sets of Ω to the interval [0
,
1]:
Ω
P
→
[0
,
1]
and is such that
P
(Ω) = 1,
P
(
∅
) = 0.
1.2.
Random vector.
A vector
ξ
(
ω
) =
(
ξ
1
(
ω
)
, . . . , ξ
n
(
ω
)
)
.
with random variables as its entries is called the random vector. Also,
the random vector is characterized by distribution function:
F
(
x
1
,
· · ·
, x
n
) =
P
(
ω
:
ξ
1
(
ω
)
≤
x
1
,
· · ·
, ξ
n
≤
x
n
)
.
1
more exactly, measurable function w.r.t. some
σ
algebra
1
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1.3.
Random processes.
A random vector
ξ
(
ω
) =
(
ξ
1
(
ω
)
, . . . , ξ
n
(
ω
)
, . . .
)
might be interpreted as random sequence (process), if
k
= 1
, . . . , n, . . .
are understood as time moments.
Formally, in a case of random se
quence the distribution function of countable values of arguments is
considered
F
(
x
1
,
· · ·
, x
n
,
· · ·
,
) =
P
(
ω
:
ξ
1
(
ω
)
≤
x
1
,
· · ·
, ξ
n
≤
x
n
,
· · ·
,
)
.
For a continuous time case, a family of random variables
ξ
t
(
ω
) para
metrized by
t
≥
0 or
∞
< t <
∞
, is called continuous time random
process. For fixed
t
0
,
ξ
t
0
(
ω
) is the random variable while for fixed
ω
0
,
a function
ξ
t
(
ω
0
) of the argument
t
is called trajectory (or a path)
of the random process
ξ
t
(
ω
). This trajectory might be continuous or
discontinuous function. If all paths of a random process are continuous,
we say shortly ”continuous process”. Under consideration of continuous
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 Spring '10
 Rejaei
 Electromagnet, Probability theory, random process, Random Processes, linear theory

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