SCIENTIFIC ABSTRACT V.P. CHISTYAKOV - YU. N. CHISTYAKOV

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000308910013-6
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: June 12, 2000
• Sequence Number: 13
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

23580
S/052/61/oo6/ool/002/005
Transient phenomena in ... 0 111/ C 333
types is given by the vector ~M = (219-9 a ).and the
distribution at
n (k
the moment t by the vector ~) = ). Let*i ) be the vector
which describes at the moment t the distribution of the
descendants of
k-th particle of type T,~ Let -- be the-number of the.yeetore
different
(i (23i
from zero among the 9 ~A j 9 ,,.j (m6). For t -) aD and' 0
it holds
Y
0? 0 e
P Ij jAt
t. -
2m u e n
14 5j - Xj t)
46 m A +... -a- mrx knj s, (m u + + m u e Xt
j Oi j n n
where n I-e Y
4- o-o + Xn' Akj (t) - Mejj (t) and V j
Card 7/3

23580 -
B/052/61/006/001/002/005
Transient phenomena in 0 111/ 0 333
Let H (z 2 be the distribution fuxiction of the %/V-if V~> 0
1 n j J
and let 4' 1 ( t, .1, n) be the corresponding characteristic
function.
Theorem 3s If ak OD -4 0, t -i oo., where 0 k < OD (k
thenAt holdffuniformly relative to all ~f k~ ( Ks
sup I H.1 (z) - hI (Z9 0,
z
where tbe,distribution function h,(z,9 ... 9 z n9 X119 ... 9 4)
has the
characteristic function
1 2
9 Yn) = exp
ly ... 9L' n; a 1 e/ - I
IV 1+... +~T,'nq X - ~ 1 + ... + ~ a)
Let h(zgf) - h (z C-f 19-0t, z C. + 1, 19-9 v where
1 1 n CY
Card 8A

23580
S10521611006100110021005
Traaalmt phenomena in ... C 111/ C 333
62 0 2 (1 94) - 0-~ are the-second,contered moments.of h 1(ZqY
and, HJz) H (z kV* I a e* + I , where
1 1 n
ar*2 '= 2-(2+w)e-*. and w 20 t n a u
bCt t) i J
J.
Theorem 43 If Mk -4 OD (k- 1 n), 0, t -) oo then
sup I H(z) - h( ~z, 1-4 0
z
uniform- relative to all 0 49 -.4 CO and if k(x)16 K.
The author thanks B. A. SOTa13t'yaAQV-for subject and advices.
There are 6 Soviet-bloe and -1 non--Soviet-bloc reference. The
reference
to English-langaage publication reads as followas T.E. Harris,
Some
mathematical models for branching processes, Proceedings of the
second 3erkeley symposium 1951
SUBMITTEDs November 18, 1�59
Card 919

34774
S/052/62/007/001/004/005
10c) C111/C444
AUTHORS: Savin, A. A; Chistyakov, V. P.
TITLE: Some limit theorems for branching processes with a few
types of particles
PERIODICAL: Teoriya veroyatnostey i yeye primeneniye, V. 7,
no.
1962, 95-104
TEXTz Lot a particle of the type T k change in the time A t 0
W
with the probability + p WA t + 0 (At), where 1 for 43 1 ,
Wk k k
0 (1 / k) and .0 in other cases, into the set of particles
(U)l I .... 63n) of the types TV . . . 9 Tn. Let t1k i(t) be
the number
of the particles of the type T which in the time t originate
from a
particle of the type T Let I..., X ) - 7- a W, . X "'Tt I'/
and let exist k* fk(xl n ca 3P. x1 n
a = afI b~k)_ c~k) = ax a fk
i j ~X ij 8 Xiax j I X=11 :Ljl i 9Xi9X 1 1 X=1
Card 1/*

S/052/62/007/001/004/005
Some limit theorems for C111/C444
The class of the types and the degree of the class be
defined as in
(Ref. 1: B. A. Sevastlyanov, Teoriya vetvyashchikhsya
sluchaynykh
protsessov [Theory of the branching random processes I ,
Uspekhi matem.
nauk, vi, 6 (1951), 47-99). Let A be the characteristic
number of
11 ai-j 11 with the largest real part.
Considered is a class sequence by which with positive
probability one
can obtain from particles belonging to the class with
maximal degree r,
particles of the class with degree 0. This class sequence
is corres-
ponding to a sequence of irreducible matrices with elements
a... Let
k of these matrices have the characteristic number 0. Let p
=laax k
with respect to all class sequences which lead from the
class with
degree r into the classes with degree 0. Let
~ n
Qk(t) = F T- 64 kj (t) >
j=1
Theorem 1: Let in a degenerated branching process with 0
the aij
and b.(k) (I.,j,k = 1,---,n) are finite, then for the types
T
1j k'
Card 2/,f

S/052/62/007/001/004/005
Some limit theorem for C111/C444
belonging to the class with degree r, for t 4 oo there holds
Qk(t) ev q kt-21-p
where the constants q 0 are depending on a,j, b~k)
k 3.j
Theorem 2: If in a degenerated process with three types of
particles
1.) the types T1, T 2, T 3belong to the classes with degree
2, 1, 0
2.) b ~k) (i,jgk 1,2,3, a (3) exist
333
0, a 09 a 0, then for t --~- oo the distributions
22 33 3
2 IL
&2 Vj~- k t) > 0 $k = 1,2,3
eki < Y11 4- y Y &J(
P t t 2? 3 , 3
1 b33 t j=1
converge to the distribution
Card 38

S/052/62/007/001/004/005
Some limit theorem for . . . C111/C444
S(y ) = ~ 1 Y3 e-z F (1-2 1-P , 2,z)dz, for y0
3 2P-1 1 1 3
0
0 for y3< 0
where
F, (CL Z) 1 + WOW Z2 +
1 11 r(81 +1) 21
p '% 1 being the number of the zeros in the main diagonal
of the matrix
a;-.
The author mentions I. M, Samusenko.
There are 7 Soviet-bloc references and 1 non-Soviet-bloc
reference,
SUBMITTED: September 9. 1960
Card 4/4

42700
j
S/02 62/147/002/006/021
B112~B186
AUTHORS: Chistyakov, V. F., Markova, N. P.
TITLE: Certain theorems for unhomogeneous branching processes
PERIODICAL: Akademiya nauk SSSR. Doklady, v. 147i no. 2, 1962,
317-320
TEXT: A system of particles each of which decays into k
particles during
the time interval Lt with the probabi t
00 '"y Pk(t)6t + O(L\t) (Pk( ~>O, k i I,
P1(t )'< 0' ' Pk (t) = 0) is described by a function F(e,t,x)
-~- Pn sot xn
f=-O n=O
for the particle number at the instant t, the p~article number
at the instant
being equal to unity. This function satisfies the equation x
-aF(s,tox /as - f(s,F(s,t,x)) with the initial condition F(s,t
x
(0