Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality

We prove two theorems on upper and lower bounds for probabilities in the multidimensional case. We generalize and improve the Prokhorov multidimensional analog of the Chebyshev inequality and establish a multidimensional analog of the generalized Kolmogorov probability estimate.

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Бібліографічні деталі
Дата:	2006
Автори:	Sokolov, N. V., Соколов, Н. В.
Формат:	Стаття
Мова:	Російська Англійська
Опубліковано:	Institute of Mathematics, NAS of Ukraine 2006
Онлайн доступ:	https://umj.imath.kiev.ua/index.php/umj/article/view/3476
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Назва журналу:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Sokolov, N. V. Соколов, Н. В. Соколов, Н. В.
author_facet	Sokolov, N. V. Соколов, Н. В. Соколов, Н. В.
author_sort	Sokolov, N. V.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2020-03-18T19:55:25Z
description	We prove two theorems on upper and lower bounds for probabilities in the multidimensional case. We generalize and improve the Prokhorov multidimensional analog of the Chebyshev inequality and establish a multidimensional analog of the generalized Kolmogorov probability estimate.
first_indexed	2026-03-24T02:43:14Z
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fulltext	UDK 519.21 N.�V.�Sokolov (Yn-t vod. probl. RAN, Moskva, Rossyq) OBOBWENYE PROXOROVSKOHO MNOHOMERNOHO ANALOHA NERAVENSTVA ÇEBÁÍEVA We prove two theorems on upper and lower bounds of probabilities in the multidimensional case. We generalize and correct the Prokhorov multidimensional analog of the Chebyshev inequality. We obtain a multidimensional analog for the generalization of the Kolmogorov probability estimation. Dovedeno dvi teoremy pro verxng ta nyΩng ocinky jmovirnostej u bahatovymirnomu vypadku. Uzahal\neno j utoçneno proxorovs\kyj bahatovymirnyj analoh nerivnosti Çebyßova. Znajdeno bahatovymirnyj analoh uzahal\nennq ocinky jmovirnosti Kolmohorova. G.2V.2Proxorov ustanovyl mnohomern¥j analoh neravenstva P.2L.2Çeb¥ßeva, yspol\zuq kvadratyçn¥e form¥ [1]. Obobwym eho v¥raΩenye na osnove ranee poluçennoho avtorom rezul\tata po utoçnenyg nyΩnyx ocenok veroqtnostej A.2N.2Kolmohorova y dal\nejßemu razvytyg ydej Çeb¥ßeva [2]. V rezul\tate πtoj rabot¥ b¥ly dokazan¥ dve teorem¥ ocenky veroqtnostej prynqtyq znaçenyj mnohomern¥x sluçajn¥x velyçyn. Teorema o verxnej ocenke veroqtnostej v mnohomernom sluçae. Pust\ ξ = ξ ( ω ) — dejstvytel\naq mnohomernaq sluçajnaq velyçyna, ϕ( ξ ( ω ) ) — dej- stvytel\naq odnoznaçnaq skalqrnaq funkcyq ot mnohomernoj sluçajnoj vely- çyn¥, α — ne�ravn¥j nulg pokazatel\ stepeny (cel¥j yly drobn¥j, poloΩy- tel\n¥j yly otrycatel\n¥j). Tohda v¥polnqetsq neravenstvo P { ξ ∈ A } ≤ E bord bord bord ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω α ω ξ ω α ω α ξ ( ) ) ) ) ( ) ( ) ( ) : ( ) [ ] − (    (         − (    ∈ ∈ ∈ i i A i Ω Θ Ω∩ , v kotorom v sluçae neopredelennostej yspol\zugtsq predel\n¥e perexod¥. Teorema o nyΩnej ocenke veroqtnostej v mnohomernom sluçae. Pust\ ξ = ξ ( ω ) — dejstvytel\naq mnohomernaq sluçajnaq velyçyna, ϕ( ξ ( ω ) ) — dej- stvytel\naq odnoznaçnaq skalqrnaq funkcyq ot mnohomernoj sluçajnoj vely- çyn¥, α — ne�ravn¥j nulg pokazatel\ stepeny (cel¥j yly drobn¥j, poloΩy- tel\n¥j yly otrycatel\n¥j). Tohda v¥polnqetsq neravenstvo P { ξ ∈ A } ≥ E bord bord bord ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω ξ ω α ω α ω ξ ω α ξ ξ ( ) ) ) ) : ( ) : ( ) ( ) ( ) ( ) [ ] − (         (    − (         ∈ ∈ ∈ i A i i A Θ Ω Θ ∩ ∩ , v kotorom v sluçaqx neopredelennostej yspol\zugtsq predel\n¥e perexod¥. V pryvedenn¥x v¥ße sootnoßenyqx yspol\zovan¥ sledugwye oboznaçenyq: bordi ω ϕ ξ ω ∈ ( Ω ( )) — operator hrany, i — yndeks operatora hranyc¥, i = 1 — nyΩ- nqq hran\, i = 2 — verxnqq hran\, 0 ≤ bord1 ω ϕ ξ ω ∈ ( Ω ( )) ≤ inf ω ϕ ξ ω ∈ ( Ω ( )) y 0 ≤ bord1 ω ξ ω ξ ϕ ξ ω : ( ) ( )) ∈ ( Θ ∩ A ≤ inf ω ξ ω ξ ϕ ξ ω : ( ) ( )) ∈ ( Θ ∩ A , bord sup2 ω ξ ω ω ξ ωξ ξ ϕ ξ ω ϕ ξ ω : ( ) : ( ) ( ) ( )) ) ∈ ∈ ( ≥ ( Θ Θ∩ ∩A A , © N.2V.2SOKOLOV, 2006 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4 573 574 N.2V.2SOKOLOV 0 ≤ bord inf1 ω ξ ω ω ξ ω ξ ξ ϕ ξ ω ϕ ξ ω : ( ) : ( ) ( ) ( )) ) ∈ ∈ ( ≤ ( Θ Θ∩ ∩A A , bord sup2 ω ξ ω ω ξ ωξ ξ ϕ ξ ω ϕ ξ ω : ( ) : ( ) ( ) ( )) ) ∈ ∈ ( ≥ ( Θ Θ∩ ∩A A , bord sup2 ω ω ϕ ξ ω ϕ ξ ω ∈ ∈ ( ≥ ( Ω Ω ( ) ( )) ) , krome toho, bord1 ω ξ ω ξ ϕ ξ ω : ( ) ( )) ∈ ( Θ ∩ A ≥ bord1 ω ϕ ξ ω ∈ ( Ω ( )) , bord bord2 2 ω ξ ω ωξ ϕ ξ ω ϕ ξ ω : ( ) ( ) ( )) ) ∈ ∈ ( ≤ ( Θ Ω∩ A , bord bord1 1 ω ξ ω ωξ ϕ ξ ω ϕ ξ ω : ( ) ( ) ( )) ) ∈ ∈ ( ≥ ( Θ Ω∩ A , bord bord2 2 ω ξ ω ωξ ϕ ξ ω ϕ ξ ω : ( ) ( ) ( )) ) ∈ ∈ ( ≤ ( Θ Ω∩ A , A — mnohomernoe mnoΩestvo dejstvytel\n¥x çysel, A — dopolnenye mno- Ωestva dejstvytel\n¥x çysel A , P { ξ ∈ A } — veroqtnost\ prynadleΩnosty sluçajnoj velyçyn¥ mnoΩestvu dejstvytel\n¥x çysel A , ω — πlementarnoe sob¥tye yz prostranstva πlementarn¥x sob¥tyj Ω , Θξ — mnohomernoe mno- Ωestvo vsex znaçenyj velyçyn¥ ξ ( ω ) , prynymaem¥x eg s nenulevoj veroqt- nost\g, E — operator matematyçeskoho oΩydanyq. Dokazatel\stva. Zametym, çto pry beskoneçn¥x matematyçeskyx oΩyda- nyqx y vsex neopredelennostqx vyda 0 / 0, ∞ / ∞ , nulevom znamenatele v rezul\- tate predel\n¥x perexodov poluçaem tryvyal\n¥e, no spravedlyv¥e ocenky veroqtnostej. Pry nulev¥x znaçenyqx znamenatelej moΩet potrebovat\sq predel\n¥j pe- rexod, pryvodqwyj k odnovremennoj smene znakov çyslytelq y znamenatelq. Pry πtom formal\no poluçagtsq ocenky P { ξ ∈ A } ≥ – ∞ y (yly) P { ξ ∈ A } ≤ ∞ , çto dokaz¥vaet spravedlyvost\ dvux teorem v takyx sytuacyqx (pry beskoneç- n¥x matematyçeskyx oΩydanyqx y vsex neopredelennostqx vyda 0 / 0, ∞ / ∞ , nulevom znamenatele), tak kak P { ξ ∈ A } ≥ 0 y P { ξ ∈ A } ≤ 1. Sledovatel\no, dlq dokazatel\stva teorem sleduet podtverdyt\ yx spraved- lyvost\ pry nenulev¥x znamenatelqx y otsutstvyy neopredelennostej (t.2e. y v ostal\n¥x sluçaqx). S πtoj cel\g zapyßem vozmoΩn¥e sootnoßenyq pry α ≠ 0, koneçn¥x mate- matyçeskyx oΩydanyqx y pry uslovyqx, çto inf ω ξ ω ξ ϕ ξ ω : ( ) ( )) ∈ ( Θ ∩ A > inf ω ϕ ξ ω ∈ ( Ω ( )) y sup ω ϕ ξ ω ∈ ( Ω ( )) > sup ω ξ ω ξ ϕ ξ ω : ( ) ( )) ∈ ( Θ ∩ A : inf P inf P E ω ξ ω α ω α α ξ ϕ ξ ω ξ ϕ ξ ω ξ ϕ ξ : ( ) ( ) ( )) { } ) { } ( ) ∈ \| \| ∈ \| \| \| \|(    ∈ + (    − ∈[ ] ≤ [ ]Θ Ω∩ A A A1 , (1) inf P inf P E ω ξ ω α ω α α ξ ϕ ξ ω ξ ϕ ξ ω ξ ϕ ξ : ( ) ( ) ( )) { } ) { } ( ) ∈ −\| \| ∈ −\| \| −\| \|(    ∈ + (    − ∈[ ] ≥ [ ]Θ Ω∩ A A A1 , (2) sup P P sup E ω α ω ξ ω α αϕ ξ ω ξ ξ ϕ ξ ω ϕ ξ ξ∈ \| \| ∈ \| \| \| \|(    ∈ + − ∈[ ] (         ≥ [ ] Ω Θ ( ) ( )) { } { } ) ( ) : ( ) A A A 1 ∩ (3) y ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4 OBOBWENYE PROXOROVSKOHO MNOHOMERNOHO ANALOHA NERAVENSTVA … 575 sup P P sup E ω α ω ξ ω α αϕ ξ ω ξ ξ ϕ ξ ω ϕ ξ ξ∈ −\| \| ∈ −\| \| −\| \|(    ∈ + − ∈[ ] (         ≤ [ ] Ω Θ ( ) ( )) { } { } ) ( ) : ( ) A A A 1 ∩ . (4) Yz sootnoßenyq (1) neposredstvenno poluçaem v¥raΩenye P { ξ ∈ A } ≤ E inf inf inf ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω α ω ξ ω α ω α ξ ( ) ) ) ) ( ) ( ) ( ) : ( ) \| \| ∈ \| \| ∈ \| \| ∈ \| \| [ ] − (    (    − (    Ω Θ Ω∩ A . (5) Neravenstvo (2) daet vozmoΩnost\ poluçyt\ takΩe druhoe sootnoßenye P { ξ ∈ A } ≤ inf E inf inf ω α α ω α ω ξ ω α ϕ ξ ω ϕ ξ ϕ ξ ω ϕ ξ ω ξ ∈ −\| \| −\| \| ∈ −\| \| ∈ −\| \| (    − [ ] (    (    Ω Ω Θ ( ) ( ) ( ) ) ( ) ) – ) : ( ) ∩ A . (6) Yz sootnoßenyq (3) neposredstvenno naxodym P { ξ ∈ A } ≥ E sup sup sup ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω ξ ω α ω α ω ξ ω α ξ ξ ( ) ) ) ) : ( ) : ( ) ( ) ( ) ( ) \| \| ∈ \| \| ∈ \| \| ∈ \| \| [ ] − (         (    − (         Θ Ω Θ ∩ ∩ A A . (7) Neravenstvo (4) pozvolqet poluçyt\ takΩe sootnoßenye P { ξ ∈ A } ≥ sup E sup sup ω ξ ω α α ω ξ ω α ω α ξ ξ ϕ ξ ω ϕ ξ ϕ ξ ω ϕ ξ ω : ( ) : ( ) ( ) ( ) ( ) ) ( ) ) ) ∈ −\| \| −\| \| ∈ −\| \| ∈ −\| \| (         − [ ] (         − (    Θ Θ Ω ∩ ∩ A A . (8) Neravenstva (5) – (8) moΩno obæedynyt\ v dva s pokazatelqmy stepeny, prynyma- gwymy kak poloΩytel\n¥e, tak y otrycatel\n¥e znaçenyq: P { ξ ∈ A } ≤ E inf inf inf ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω α ω ξ ω α ω α ξ ( ) ) ) ) ( ) ( ) ( ) : ( ) [ ] − (    (    − (    ∈ ∈ ∈ Ω Θ Ω∩ A (9) y P { ξ ∈ A } ≥ E sup sup sup ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω ξ ω α ω α ω ξ ω α ξ ξ ( ) ) ) ) : ( ) : ( ) ( ) ( ) ( ) [ ] − (         (    − (         ∈ ∈ ∈ Θ Ω Θ ∩ ∩ A A . (10) Esly dopolnytel\no predpoloΩyt\, çto sup sup ω ξ ω ωξ ϕ ξ ω ϕ ξ ω : ( ) ( ) ( )) ) ∈ ∈ ( < ( Θ Ω∩ A y inf inf ω ω ξ ω ϕ ξ ω ϕ ξ ω ξ∈ ∈ ( < ( Ω Θ ( ) ( )) ) : ( ) ∩ A , to, prymenyv (9) y (10) k funkcyy, ravnoj 1 / ϕ ( ξ ( ω ) ), y yspol\zovav pokazatel\ stepeny, ravn¥j – α, najdem ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4 576 N.2V.2SOKOLOV P { ξ ∈ A } ≤ E sup sup sup ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω α ω ξ ω α ω α ξ ( ) ) ) ) ( ) ( ) ( ) : ( ) [ ] − (    (         − (    ∈ ∈ ∈ Ω Θ Ω∩ A (11) y P { ξ ∈ A } ≥ E inf inf inf ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω ξ ω α ω α ω ξ ω α ξ ξ ( ) ) ) ) : ( ) : ( ) ( ) ( ) ( ) [ ] − (       (    − (       ∈ ∈ ∈ Θ Ω Θ ∩ ∩ A A . (12) Takym obrazom, neravenstva (9) y (10) ymegt paradoksal\noe svojstvo: ony soxranqgt znak pry formal\noj zamene vsex inf na sup y naoborot! Zametym, çto dlq sluçaev otsutstvyq beskoneçnosty v pravoj çasty nera- venstv (11) y (12) ony mohut b¥t\ dokazan¥ yz sledugwyx sootnoßenyj: sup P sup P E ω ξ ω α ω α α ξ ϕ ξ ω ξ ϕ ξ ω ξ ϕ ξ : ( ) ( ) ( )) { } ) { } ( ) ∈ \| \| ∈ \| \| \| \|(         ∈ + (    − ∈[ ] ≥ [ ] Θ Ω∩ A A A1 , (13) sup P sup P E ω ξ ω α ω α α ξ ϕ ξ ω ξ ϕ ξ ω ξ ϕ ξ : ( ) ( ) ( )) { } ) { } ( ) ∈ −\| \| ∈ −\| \| −\| \|(         ∈ + (    − ∈[ ] ≤ [ ] Θ Ω∩ A A A1 , (14) inf P P inf E ω α ω ξ ω α αϕ ξ ω ξ ξ ϕ ξ ω ϕ ξ ξ∈ \| \| ∈ \| \| \| \|(    ∈ + − ∈[ ] (       ≤ [ ]Ω Θ ( ) ( )) { } { } ) ( ) : ( ) A A A 1 ∩ (15) y inf P P inf E ω α ω ξ ω α αϕ ξ ω ξ ξ ϕ ξ ω ϕ ξ ξ∈ −\| \| ∈ −\| \| −\| \|(    ∈ + − ∈[ ] (       ≥ [ ]Ω Θ ( ) ( )) { } { } ) ( ) : ( ) A A A 1 ∩ . (16) V¥raΩenyqmy (1) – (4) y (13) – (16) ysçerp¥vagtsq vse vozmoΩn¥e sluçay. Obæedynym sootnoßenyq (9) – (12) v dva, a zatem yspol\zuem svojstvo pra- vyl\n¥x drobej (men\ßyx edynyc¥) uvelyçyvat\ svoe znaçenye pry odnovre- mennom vozrastanyy na odno y to Ωe çyslo çyslytelq y znamenatelq. Tohda yz (9) – (12) budet sledovat\ spravedlyvost\ sformulyrovann¥x v naçale stat\y teorem dlq sluçaev nenulev¥x znamenatelej y otsutstvyq nepredelennostej. Poskol\ku πty teorem¥ spravedlyv¥, kak b¥lo pokazano, neposredstvenno y pry beskoneçn¥x matematyçeskyx oΩydanyqx y vsex neopredelennostqx vyda 0 / 0, ∞ / ∞ y nulevom znamenatele, sledovatel\no, ony spravedlyv¥ vo vsex sluçaqx. Takym obrazom, teorem¥ dokazan¥. Otmetym, çto v¥raΩenye (10) qvlqetsq mnohomern¥m analohom sdelannoho v rabote [2] obobwenyq y utoçnenyq nyΩnej kolmohorovskoj ocenky veroqt- nosty. Zameçanye 1. Esly toçnaq verxnqq hran\ neyzvestna yly ravna beskoneç- nosty, to yz neravenstva (8) sleduet P { ξ ∈ A } ≥ 1 – sup E ω ξ ω α α ξ ϕ ξ ω ϕ ξ : ( ) ( )) ( ) ∈ \| \| −\| \|(         [ ] Θ ∩ A . (17) ∏tot rezul\tat takΩe sleduet yz teorem¥ o nyΩnej ocenke veroqtnostej v mnohomernom sluçae pry otrycatel\n¥x znaçenyqx pokazatelq stepeny. Pry πtom ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 4 P { ξ ∈ A } ≥ E bord bord bord ϕ ξ ϕ ξ ω ϕ ξ ω ϕ ξ ω α ω ξ ω α ω α ω ξ ω α ξ ξ ( ) ) ) ) : ( ) : ( ) ( ) ( ) ( ) −\| \| ∈ −\| \| ∈ −\| \| ∈ −\| \| [ ] − (         (    − (         2 2 2 Θ Ω Θ ∩ ∩ A A ≥ ≥ 1 – bord E2 ω ξ ω α α ξ ϕ ξ ω ϕ ξ : ( ) ( )) ( ) ∈ \| \| −\| \|(         [ ] Θ ∩ A . (18) V¥raΩenye (17) qvlqetsq çastn¥m sluçaem sootnoßenyq (18). Zameçanye 2. Esly v kaçestve funkcyy yspol\zovat\ kvadratyçn¥e form¥, to yz teorem¥ 1 kak ee sledstvye najdem proxorovskyj mnohomern¥j analoh neravenstva Çeb¥ßeva. V samom dele, dlq sluçaq pravyl\n¥x drobej pry α = 1 y i = 1 yz teorem¥ 1 sleduet P { X ∉ D } ≤ E inf inf inf ϕ ϕ ω ϕ ω ϕ ω ω ω ω ω ( ) ) ) ) ( ) ( ) ( ) : ( ) X X X X X DX [ ] − (    (    − (    ∈ ∈ ∈ Ω Θ Ω∩ ≤ ≤ E inf ϕ ϕ ω ω ω ( ) ) : ( ) ( ) X X DX X [ ] (   ∈Θ ∩ ≤ E inf ϕ ϕ ω ω ω ( ) ) : ( ) ( ) X X DX X ∈ (         Θ ∩ ≤ E ˜( )Q X , (19) t.2e. yz teorem¥ 1 v kaçestve çastnoho sluçaq v¥tekaet v¥raΩenye P { X ∉ D } ≤ E ˜( )Q X , (20) pryvedennoe v obzore G.2V.2Proxorova [1], posvqwennom mnohomern¥m analoham neravenstva Çeb¥ßeva. Zdes\ X — sluçajn¥j vektor, D — prqmouhol\naq ob- last\, ˜ ( )Q X — neotrycatel\no opredelennaq kvadratyçnaq forma, pry vsex X ∉ D prynymagwaq znaçenyq ne2men\ße 1 ( pry πtom ϕ ω( ))X( = = inf ω ω ϕ ω : ( ) ( )) X DX X ∈ ( Θ ∩ p r y v s e x X ∉ D y inf ω ϕ ω ∈ ( Ω ( ))X < < inf ω ω ϕ ω : ( ) ( )) X DX X ∈ ( Θ ∩ , a ϕ ω ϕ ω ω ω ( ) ( )) ) : ( ) X X X DX ( ( ∈ inf Θ ∩ ≤ ˜ ( )Q X pry vsex X ∈ D ). Mnohomern¥j analoh neravenstva Çeb¥ßeva (20) moΩno takΩe neposred- stvenno obosnovat\ s pomow\g v¥raΩenyq (5). V dokazann¥x teoremax mnoΩestvo A moΩet sostoqt\ yz otdel\n¥x toçek yly otdel\n¥x vektorov, çto osobenno vaΩno dlq sluçaev dyskretn¥x sluçaj- n¥x mnohomern¥x velyçyn. Pokazatel\ stepeny α obæedynqet dve vetvy nera- venstv s poloΩytel\n¥my y otrycatel\n¥my çyslytelqmy y znamenatelqmy y predostavlqet vozmoΩnost\ optymyzacyy ocenok v sluçae fyksacyy vyda funkcyy ϕ ( ξ ( ω ) ) . Pry πtom vozmoΩn¥ otrycatel\n¥e y drobn¥e znaçenyq α . 1. Proxorov�G.�V. Mnohomern¥e raspredelenyq: neravenstva y predel\n¥e teorem¥ // Ytohy nauky y texnyky. Teoryq veroqtnostej. Mat. statystyka. Teoret. kybernetyka / VYNYTY. – 1972. – 10. – S.25 – 24. 2. Sokolov�N.�V. Rasßyrenye vozmoΩnostej neravenstv çeb¥ßevskoho typa y ocenky Kolmo- horova // Dokl. RAN. – 2002. – 384, #23. – S.2308 – 311. Poluçeno 17.06.2005
id	umjimathkievua-article-3476
institution	Ukrains’kyi Matematychnyi Zhurnal
keywords_txt_mv	keywords
language	rus English
last_indexed	2026-03-24T02:43:14Z
publishDate	2006
publisher	Institute of Mathematics, NAS of Ukraine
record_format	ojs
resource_txt_mv	umjimathkievua/2e/1999bdaea4871f6803893aeaf9c7f92e.pdf
spelling	umjimathkievua-article-34762020-03-18T19:55:25Z Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality Обобщение прохоровского многомерного аналога неравенства Чебышева Sokolov, N. V. Соколов, Н. В. Соколов, Н. В. We prove two theorems on upper and lower bounds for probabilities in the multidimensional case. We generalize and improve the Prokhorov multidimensional analog of the Chebyshev inequality and establish a multidimensional analog of the generalized Kolmogorov probability estimate. Доведено дві теореми про верхню та нижню оцінки ймовірностей у багатовимірному випадку. Узагальнено й уточнено прохоровський багатовимірний аналог нерівності Чебишова. Знайдено багатовимірний аналог узагальнення оцінки ймовірності Колмогорова. Institute of Mathematics, NAS of Ukraine 2006-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3476 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 4 (2006); 573–576 Український математичний журнал; Том 58 № 4 (2006); 573–576 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3476/3689 https://umj.imath.kiev.ua/index.php/umj/article/view/3476/3690 Copyright (c) 2006 Sokolov N. V.
spellingShingle	Sokolov, N. V. Соколов, Н. В. Соколов, Н. В. Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality
title	Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality
title_alt	Обобщение прохоровского многомерного аналога неравенства Чебышева
title_full	Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality
title_fullStr	Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality
title_full_unstemmed	Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality
title_short	Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality
title_sort	generalization of the prokhorov multidimensional analog of the chebyshev inequality
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3476
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Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality

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