id sid tid token lemma pos ajrhss-1432 1 1 american american PROPN ajrhss-1432 1 2 journal journal PROPN ajrhss-1432 1 3 of of ADP ajrhss-1432 1 4 research research NOUN ajrhss-1432 1 5 in in ADP ajrhss-1432 1 6 humanities humanity NOUN ajrhss-1432 1 7 and and CCONJ ajrhss-1432 1 8 social social ADJ ajrhss-1432 1 9 sciences science NOUN ajrhss-1432 1 10 issn issn PROPN ajrhss-1432 1 11 ( ( PUNCT ajrhss-1432 1 12 e e NOUN ajrhss-1432 1 13 ): ): PUNCT ajrhss-1432 1 14 2832 2832 NUM ajrhss-1432 1 15 - - SYM ajrhss-1432 1 16 8019 8019 NUM ajrhss-1432 1 17 volume volume NOUN ajrhss-1432 1 18 18 18 NUM ajrhss-1432 1 19 , , PUNCT ajrhss-1432 1 20 | | ADV ajrhss-1432 1 21 november november PROPN ajrhss-1432 1 22 , , PUNCT ajrhss-1432 1 23 2023 2023 NUM ajrhss-1432 1 24 p p NOUN ajrhss-1432 1 25 a a DET ajrhss-1432 1 26 g g NOUN ajrhss-1432 1 27 e e NOUN ajrhss-1432 1 28 | | ADV ajrhss-1432 1 29 8 8 NUM ajrhss-1432 1 30 www.americanjournal.org www.americanjournal.org NOUN ajrhss-1432 1 31 on on ADP ajrhss-1432 1 32 the the DET ajrhss-1432 1 33 behaviour behaviour NOUN ajrhss-1432 1 34 of of ADP ajrhss-1432 1 35 some some DET ajrhss-1432 1 36 probabilistic probabilistic ADJ ajrhss-1432 1 37 characteristics characteristic NOUN ajrhss-1432 1 38 of of ADP ajrhss-1432 1 39 the the DET ajrhss-1432 1 40 output output NOUN ajrhss-1432 1 41 of of ADP ajrhss-1432 1 42 multidimensional multidimensional ADJ ajrhss-1432 1 43 random random ADJ ajrhss-1432 1 44 walk walk NOUN ajrhss-1432 1 45 from from ADP ajrhss-1432 1 46 expanding expand VERB ajrhss-1432 1 47 sets set NOUN ajrhss-1432 1 48 gafurov gafurov ADJ ajrhss-1432 1 49 m. m. NOUN ajrhss-1432 1 50 u. u. PROPN ajrhss-1432 1 51 tashkent tashkent PROPN ajrhss-1432 1 52 state state PROPN ajrhss-1432 1 53 transport transport PROPN ajrhss-1432 1 54 university university PROPN ajrhss-1432 1 55 mgafurov@rambler.ru mgafurov@rambler.ru VERB ajrhss-1432 1 56 a a DET ajrhss-1432 1 57 b b PROPN ajrhss-1432 1 58 s s ADP ajrhss-1432 1 59 t t PROPN ajrhss-1432 1 60 r r NOUN ajrhss-1432 1 61 a a DET ajrhss-1432 1 62 c c NOUN ajrhss-1432 1 63 t t NOUN ajrhss-1432 1 64 k k X ajrhss-1432 1 65 e e PROPN ajrhss-1432 1 66 y y PROPN ajrhss-1432 1 67 w w NOUN ajrhss-1432 1 68 o o NOUN ajrhss-1432 1 69 r r NOUN ajrhss-1432 1 70 d d PROPN ajrhss-1432 1 71 s s VERB ajrhss-1432 1 72 this this DET ajrhss-1432 1 73 paper paper NOUN ajrhss-1432 1 74 establishes establish VERB ajrhss-1432 1 75 an an DET ajrhss-1432 1 76 analog analog NOUN ajrhss-1432 1 77 of of ADP ajrhss-1432 1 78 the the DET ajrhss-1432 1 79 well well ADV ajrhss-1432 1 80 - - PUNCT ajrhss-1432 1 81 known know VERB ajrhss-1432 1 82 theorem theorem NOUN ajrhss-1432 1 83 of of ADP ajrhss-1432 1 84 p. p. PROPN ajrhss-1432 1 85 j. j. PROPN ajrhss-1432 1 86 bickel bickel PROPN ajrhss-1432 1 87 and and CCONJ ajrhss-1432 1 88 j. j. PROPN ajrhss-1432 1 89 a. a. PROPN ajrhss-1432 1 90 yahav yahav PROPN ajrhss-1432 1 91 on on ADP ajrhss-1432 1 92 the the DET ajrhss-1432 1 93 number number NOUN ajrhss-1432 1 94 of of ADP ajrhss-1432 1 95 exits exit NOUN ajrhss-1432 1 96 of of ADP ajrhss-1432 1 97 a a DET ajrhss-1432 1 98 multidimensional multidimensional ADJ ajrhss-1432 1 99 random random ADJ ajrhss-1432 1 100 walk walk NOUN ajrhss-1432 1 101 from from ADP ajrhss-1432 1 102 expanding expand VERB ajrhss-1432 1 103 sets set NOUN ajrhss-1432 1 104 . . PUNCT ajrhss-1432 2 1 this this DET ajrhss-1432 2 2 theorem theorem ADJ ajrhss-1432 2 3 and and CCONJ ajrhss-1432 2 4 related related ADJ ajrhss-1432 2 5 problems problem NOUN ajrhss-1432 2 6 are be AUX ajrhss-1432 2 7 carried carry VERB ajrhss-1432 2 8 forward forward ADV ajrhss-1432 2 9 for for ADP ajrhss-1432 2 10 the the DET ajrhss-1432 2 11 moment moment NOUN ajrhss-1432 2 12 of of ADP ajrhss-1432 2 13 the the DET ajrhss-1432 2 14 first first ADJ ajrhss-1432 2 15 exit exit NOUN ajrhss-1432 2 16 . . PUNCT ajrhss-1432 3 1 multidimensional multidimensional ADJ ajrhss-1432 3 2 random random ADJ ajrhss-1432 3 3 walk walk NOUN ajrhss-1432 3 4 ; ; PUNCT ajrhss-1432 3 5 expanding expand VERB ajrhss-1432 3 6 sets set NOUN ajrhss-1432 3 7 ; ; PUNCT ajrhss-1432 3 8 moment moment NOUN ajrhss-1432 3 9 of of ADP ajrhss-1432 3 10 the the DET ajrhss-1432 3 11 first first ADJ ajrhss-1432 3 12 exit exit NOUN ajrhss-1432 3 13 ; ; PUNCT ajrhss-1432 3 14 number number NOUN ajrhss-1432 3 15 of of ADP ajrhss-1432 3 16 exits exit NOUN ajrhss-1432 3 17 . . PUNCT ajrhss-1432 4 1 introduction introduction NOUN ajrhss-1432 4 2 let let VERB ajrhss-1432 4 3 𝑋1 𝑋1 NOUN ajrhss-1432 4 4 … … PUNCT ajrhss-1432 4 5 𝑋𝑛 𝑋𝑛 ADJ ajrhss-1432 4 6 , , PUNCT ajrhss-1432 4 7 be be AUX ajrhss-1432 4 8 independent independent ADJ ajrhss-1432 4 9 identically identically ADV ajrhss-1432 4 10 distributed distribute VERB ajrhss-1432 4 11 random random ADJ ajrhss-1432 4 12 variables variable NOUN ajrhss-1432 4 13 ( ( PUNCT ajrhss-1432 4 14 rv rv NOUN ajrhss-1432 4 15 's 's PART ajrhss-1432 4 16 ) ) PUNCT ajrhss-1432 4 17 with with ADP ajrhss-1432 4 18 values value NOUN ajrhss-1432 4 19 in in ADP ajrhss-1432 4 20 𝑅𝑑 𝑅𝑑 PROPN ajrhss-1432 4 21 , , PUNCT ajrhss-1432 4 22 𝑑 𝑑 PROPN ajrhss-1432 4 23 β‰₯ β‰₯ NOUN ajrhss-1432 4 24 1 1 NUM ajrhss-1432 4 25 . . PUNCT ajrhss-1432 4 26 define define VERB ajrhss-1432 4 27 𝑆0 𝑆0 PROPN ajrhss-1432 4 28 = = SYM ajrhss-1432 4 29 0 0 PUNCT ajrhss-1432 4 30 and and CCONJ ajrhss-1432 4 31 𝑆𝑛 𝑆𝑛 PROPN ajrhss-1432 4 32 = = PUNCT ajrhss-1432 4 33 βˆ‘ βˆ‘ PUNCT ajrhss-1432 4 34 𝑋𝑖 𝑋𝑖 PROPN ajrhss-1432 4 35 𝑛 𝑛 ADP ajrhss-1432 4 36 1 1 NUM ajrhss-1432 4 37 for for ADP ajrhss-1432 4 38 nβ‰₯1 nβ‰₯1 NOUN ajrhss-1432 4 39 . . PUNCT ajrhss-1432 5 1 for for ADP ajrhss-1432 5 2 any any DET ajrhss-1432 5 3 borel borel NOUN ajrhss-1432 5 4 set set VERB ajrhss-1432 5 5 𝐴 𝐴 PROPN ajrhss-1432 6 1 βŠ‚ βŠ‚ PROPN ajrhss-1432 7 1 𝑅𝑑 𝑅𝑑 ADV ajrhss-1432 7 2 we we PRON ajrhss-1432 7 3 set set VERB ajrhss-1432 7 4 ( ( PUNCT ajrhss-1432 7 5 formally formally ADV ajrhss-1432 7 6 ) ) PUNCT ajrhss-1432 7 7 𝑁(𝐴 𝑁(𝐴 NOUN ajrhss-1432 7 8 ) ) PUNCT ajrhss-1432 8 1 = = PUNCT ajrhss-1432 8 2 βˆ‘ βˆ‘ PROPN ajrhss-1432 8 3 𝐼 𝐼 PROPN ajrhss-1432 8 4 ( ( PUNCT ajrhss-1432 8 5 𝑆𝑛 𝑆𝑛 PROPN ajrhss-1432 8 6 ∞ ∞ PROPN ajrhss-1432 8 7 𝑛=1 𝑛=1 PROPN ajrhss-1432 8 8 β‹² β‹² PROPN ajrhss-1432 8 9 𝐴 𝐴 PROPN ajrhss-1432 8 10 ) ) PUNCT ajrhss-1432 8 11 , , PUNCT ajrhss-1432 8 12 𝑇(𝐴 𝑇(𝐴 NOUN ajrhss-1432 8 13 ) ) PUNCT ajrhss-1432 8 14 = = SYM ajrhss-1432 8 15 inf inf PROPN ajrhss-1432 8 16 { { PUNCT ajrhss-1432 8 17 𝑛 𝑛 PROPN ajrhss-1432 8 18 , , PUNCT ajrhss-1432 8 19 𝑆𝑛 𝑆𝑛 PROPN ajrhss-1432 8 20 βˆ‰ βˆ‰ PROPN ajrhss-1432 8 21 a a PRON ajrhss-1432 8 22 } } PUNCT ajrhss-1432 8 23 there there PRON ajrhss-1432 8 24 are be VERB ajrhss-1432 8 25 many many ADJ ajrhss-1432 8 26 references reference NOUN ajrhss-1432 8 27 dealing deal VERB ajrhss-1432 8 28 with with ADP ajrhss-1432 8 29 the the DET ajrhss-1432 8 30 study study NOUN ajrhss-1432 8 31 of of ADP ajrhss-1432 8 32 the the DET ajrhss-1432 8 33 rv rv PROPN ajrhss-1432 8 34 's 's PART ajrhss-1432 8 35 n(a n(a NOUN ajrhss-1432 8 36 ) ) PUNCT ajrhss-1432 8 37 and and CCONJ ajrhss-1432 8 38 t(a t(a NOUN ajrhss-1432 8 39 ) ) PUNCT ajrhss-1432 8 40 in in ADP ajrhss-1432 8 41 the the DET ajrhss-1432 8 42 case case NOUN ajrhss-1432 8 43 d=1 d=1 PROPN ajrhss-1432 8 44 ( ( PUNCT ajrhss-1432 8 45 see see VERB ajrhss-1432 8 46 , , PUNCT ajrhss-1432 8 47 for for ADP ajrhss-1432 8 48 example example NOUN ajrhss-1432 8 49 , , PUNCT ajrhss-1432 8 50 the the DET ajrhss-1432 8 51 monograph monograph NOUN ajrhss-1432 9 1 [ [ X ajrhss-1432 9 2 1 1 NUM ajrhss-1432 9 3 ] ] PUNCT ajrhss-1432 9 4 ) ) PUNCT ajrhss-1432 9 5 . . PUNCT ajrhss-1432 10 1 in in ADP ajrhss-1432 10 2 the the DET ajrhss-1432 10 3 general general ADJ ajrhss-1432 10 4 case case NOUN ajrhss-1432 10 5 , , PUNCT ajrhss-1432 10 6 when when SCONJ ajrhss-1432 10 7 d d PROPN ajrhss-1432 10 8 > > X ajrhss-1432 10 9 1 1 NUM ajrhss-1432 10 10 , , PUNCT ajrhss-1432 10 11 it it PRON ajrhss-1432 10 12 was be AUX ajrhss-1432 10 13 proved prove VERB ajrhss-1432 10 14 in in ADP ajrhss-1432 10 15 ( ( PUNCT ajrhss-1432 10 16 2 2 X ajrhss-1432 10 17 ] ] PUNCT ajrhss-1432 10 18 that that SCONJ ajrhss-1432 10 19 if if SCONJ ajrhss-1432 10 20 𝐸𝑁(𝐴 𝐸𝑁(𝐴 NOUN ajrhss-1432 10 21 ) ) PUNCT ajrhss-1432 10 22 = = PUNCT ajrhss-1432 11 1 βˆ‘ βˆ‘ PUNCT ajrhss-1432 11 2 𝑃(𝑆𝑛 𝑃(𝑆𝑛 NUM ajrhss-1432 11 3 β‹² β‹² PROPN ajrhss-1432 11 4 𝐴 𝐴 PROPN ajrhss-1432 11 5 ) ) PUNCT ajrhss-1432 11 6 ∞ ∞ PROPN ajrhss-1432 11 7 𝑛=1 𝑛=1 NOUN ajrhss-1432 11 8 < < X ajrhss-1432 11 9 ∞ ∞ PROPN ajrhss-1432 11 10 for for ADP ajrhss-1432 11 11 any any DET ajrhss-1432 11 12 bounded bounded ADJ ajrhss-1432 11 13 set set NOUN ajrhss-1432 11 14 a a PRON ajrhss-1432 11 15 , , PUNCT ajrhss-1432 11 16 then then ADV ajrhss-1432 11 17 𝐸𝑒π‘₯𝑝 𝐸𝑒π‘₯𝑝 PROPN ajrhss-1432 11 18 { { PUNCT ajrhss-1432 11 19 𝑑 𝑑 PROPN ajrhss-1432 11 20 𝑁(𝐴 𝑁(𝐴 NOUN ajrhss-1432 11 21 ) ) PUNCT ajrhss-1432 11 22 } } PUNCT ajrhss-1432 11 23 < < X ajrhss-1432 11 24 ∞ ∞ PROPN ajrhss-1432 11 25 for for ADP ajrhss-1432 11 26 all all PRON ajrhss-1432 11 27 |𝑑| |𝑑| ADP ajrhss-1432 11 28 ≀ ≀ ADJ ajrhss-1432 11 29 𝑑0 𝑑0 PROPN ajrhss-1432 11 30 where where SCONJ ajrhss-1432 11 31 𝑑0 𝑑0 NOUN ajrhss-1432 11 32 , , PUNCT ajrhss-1432 11 33 is be AUX ajrhss-1432 11 34 some some DET ajrhss-1432 11 35 positive positive ADJ ajrhss-1432 11 36 number number NOUN ajrhss-1432 11 37 . . PUNCT ajrhss-1432 12 1 the the DET ajrhss-1432 12 2 asymptotic asymptotic ADJ ajrhss-1432 12 3 behavior behavior NOUN ajrhss-1432 12 4 of of ADP ajrhss-1432 12 5 the the DET ajrhss-1432 12 6 moments moment NOUN ajrhss-1432 12 7 of of ADP ajrhss-1432 12 8 n(a n(a NOUN ajrhss-1432 12 9 ) ) PUNCT ajrhss-1432 12 10 for for ADP ajrhss-1432 12 11 an an DET ajrhss-1432 12 12 expanding expand VERB ajrhss-1432 12 13 set set NOUN ajrhss-1432 12 14 a a PRON ajrhss-1432 12 15 was be AUX ajrhss-1432 12 16 investigated investigate VERB ajrhss-1432 12 17 in in ADP ajrhss-1432 12 18 the the DET ajrhss-1432 12 19 same same ADJ ajrhss-1432 12 20 paper paper NOUN ajrhss-1432 12 21 . . PUNCT ajrhss-1432 13 1 as as ADV ajrhss-1432 13 2 far far ADV ajrhss-1432 13 3 as as SCONJ ajrhss-1432 13 4 the the DET ajrhss-1432 13 5 author author NOUN ajrhss-1432 13 6 knows know VERB ajrhss-1432 13 7 , , PUNCT ajrhss-1432 13 8 the the DET ajrhss-1432 13 9 distribution distribution NOUN ajrhss-1432 13 10 of of ADP ajrhss-1432 13 11 t(a t(a NOUN ajrhss-1432 13 12 ) ) PUNCT ajrhss-1432 13 13 when when SCONJ ajrhss-1432 13 14 d d PROPN ajrhss-1432 13 15 > > X ajrhss-1432 13 16 l l NOUN ajrhss-1432 13 17 has have AUX ajrhss-1432 13 18 not not PART ajrhss-1432 13 19 yet yet ADV ajrhss-1432 13 20 been be AUX ajrhss-1432 13 21 studied study VERB ajrhss-1432 13 22 in in ADP ajrhss-1432 13 23 depth depth NOUN ajrhss-1432 13 24 . . PUNCT ajrhss-1432 14 1 our our PRON ajrhss-1432 14 2 purpose purpose NOUN ajrhss-1432 14 3 is be AUX ajrhss-1432 14 4 to to PART ajrhss-1432 14 5 determine determine VERB ajrhss-1432 14 6 the the DET ajrhss-1432 14 7 asymptotic asymptotic ADJ ajrhss-1432 14 8 behavior behavior NOUN ajrhss-1432 14 9 of of ADP ajrhss-1432 14 10 the the DET ajrhss-1432 14 11 moments moment NOUN ajrhss-1432 14 12 of of ADP ajrhss-1432 14 13 t(a t(a NOUN ajrhss-1432 14 14 ) ) PUNCT ajrhss-1432 14 15 on on ADP ajrhss-1432 14 16 sets set NOUN ajrhss-1432 14 17 of of ADP ajrhss-1432 14 18 the the DET ajrhss-1432 14 19 form form NOUN ajrhss-1432 14 20 𝐴 𝐴 NOUN ajrhss-1432 14 21 = = PUNCT ajrhss-1432 15 1 𝐴π‘₯ 𝐴π‘₯ NOUN ajrhss-1432 15 2 = = PUNCT ajrhss-1432 15 3 { { PUNCT ajrhss-1432 15 4 𝑦 𝑦 NOUN ajrhss-1432 15 5 β‹² β‹² X ajrhss-1432 15 6 𝑅𝑑 𝑅𝑑 PROPN ajrhss-1432 15 7 , , PUNCT ajrhss-1432 15 8 ||𝑦|| ||𝑦|| PROPN ajrhss-1432 15 9 < < X ajrhss-1432 15 10 π‘₯ π‘₯ X ajrhss-1432 15 11 } } PUNCT ajrhss-1432 15 12 , , PUNCT ajrhss-1432 15 13 where where SCONJ ajrhss-1432 15 14 ||Ξ‡|| ||Ξ‡|| NOUN ajrhss-1432 15 15 is be AUX ajrhss-1432 15 16 any any DET ajrhss-1432 15 17 norm norm NOUN ajrhss-1432 15 18 in in ADP ajrhss-1432 15 19 𝑅𝑑 𝑅𝑑 PROPN ajrhss-1432 15 20 , , PUNCT ajrhss-1432 15 21 as as ADV ajrhss-1432 15 22 well well ADV ajrhss-1432 15 23 as as ADP ajrhss-1432 15 24 the the DET ajrhss-1432 15 25 behavior behavior NOUN ajrhss-1432 15 26 of of ADP ajrhss-1432 15 27 the the DET ajrhss-1432 15 28 β€œ " PUNCT ajrhss-1432 15 29 first first ADJ ajrhss-1432 15 30 flight flight NOUN ajrhss-1432 15 31 of of ADP ajrhss-1432 15 32 stairs stair NOUN ajrhss-1432 15 33 ” " PUNCT ajrhss-1432 15 34 𝑆𝑇(𝐴π‘₯ 𝑆𝑇(𝐴π‘₯ NOUN ajrhss-1432 15 35 ) ) PUNCT ajrhss-1432 15 36 π‘Žπ‘  π‘Žπ‘  ADP ajrhss-1432 15 37 π‘₯ π‘₯ PROPN ajrhss-1432 15 38 β†’ β†’ SYM ajrhss-1432 15 39 ∞ ∞ PROPN ajrhss-1432 15 40 , , PUNCT ajrhss-1432 15 41 in in ADP ajrhss-1432 15 42 this this DET ajrhss-1432 15 43 connection connection NOUN ajrhss-1432 15 44 we we PRON ajrhss-1432 15 45 have have AUX ajrhss-1432 15 46 proved prove VERB ajrhss-1432 15 47 the the DET ajrhss-1432 15 48 following following ADJ ajrhss-1432 15 49 statements statement NOUN ajrhss-1432 15 50 . . PUNCT ajrhss-1432 16 1 theorem theorem NOUN ajrhss-1432 16 2 1 1 NUM ajrhss-1432 16 3 . . PUNCT ajrhss-1432 17 1 if if SCONJ ajrhss-1432 17 2 𝐸 𝐸 PROPN ajrhss-1432 17 3 ||𝑋1|| ||𝑋1|| X ajrhss-1432 17 4 < < X ajrhss-1432 17 5 ∞ ∞ PROPN ajrhss-1432 17 6 , , PUNCT ajrhss-1432 17 7 then then ADV ajrhss-1432 17 8 for for ADP ajrhss-1432 17 9 all all PRON ajrhss-1432 17 10 k k PROPN ajrhss-1432 17 11 β‰₯0 β‰₯0 PROPN ajrhss-1432 17 12 lim lim PROPN ajrhss-1432 17 13 π‘₯β†’βˆž π‘₯β†’βˆž NUM ajrhss-1432 17 14 πΈπ‘‡π‘˜(𝐴π‘₯ πΈπ‘‡π‘˜(𝐴π‘₯ PROPN ajrhss-1432 17 15 ) ) PUNCT ajrhss-1432 17 16 π‘₯π‘˜ π‘₯π‘˜ NOUN ajrhss-1432 17 17 = = NOUN ajrhss-1432 17 18 1 1 NUM ajrhss-1432 17 19 ||𝐸𝑋1||π‘˜ ||𝐸𝑋1||π‘˜ NOUN ajrhss-1432 17 20 . . PUNCT ajrhss-1432 18 1 this this PRON ajrhss-1432 18 2 theorem theorem ADJ ajrhss-1432 18 3 complements complement NOUN ajrhss-1432 18 4 a a DET ajrhss-1432 18 5 result result NOUN ajrhss-1432 18 6 in in ADP ajrhss-1432 18 7 [ [ X ajrhss-1432 18 8 2 2 NUM ajrhss-1432 18 9 ] ] PUNCT ajrhss-1432 18 10 on𝑁(𝐴π‘₯ on𝑁(𝐴π‘₯ PROPN ajrhss-1432 18 11 ) ) PUNCT ajrhss-1432 18 12 . . PUNCT ajrhss-1432 19 1 american american PROPN ajrhss-1432 19 2 journal journal PROPN ajrhss-1432 19 3 of of ADP ajrhss-1432 19 4 research research NOUN ajrhss-1432 19 5 in in ADP ajrhss-1432 19 6 humanities humanity NOUN ajrhss-1432 19 7 and and CCONJ ajrhss-1432 19 8 social social ADJ ajrhss-1432 19 9 sciences science NOUN ajrhss-1432 19 10 volume volume NOUN ajrhss-1432 19 11 17 17 NUM ajrhss-1432 19 12 oct oct PROPN ajrhss-1432 19 13 . . PROPN ajrhss-1432 19 14 , , PUNCT ajrhss-1432 19 15 2023 2023 NUM ajrhss-1432 19 16 p p NOUN ajrhss-1432 19 17 a a PRON ajrhss-1432 19 18 g g NOUN ajrhss-1432 19 19 e e NOUN ajrhss-1432 19 20 | | ADV ajrhss-1432 19 21 9 9 NUM ajrhss-1432 19 22 www.americanjournal.org www.americanjournal.org NOUN ajrhss-1432 19 23 let let VERB ajrhss-1432 19 24 us we PRON ajrhss-1432 19 25 consider consider VERB ajrhss-1432 19 26 a a DET ajrhss-1432 19 27 nondecreasing nondecrease VERB ajrhss-1432 19 28 positive positive ADJ ajrhss-1432 19 29 function function NOUN ajrhss-1432 19 30 Ο†(x Ο†(x NOUN ajrhss-1432 19 31 ) ) PUNCT ajrhss-1432 19 32 on on ADP ajrhss-1432 19 33 [ [ X ajrhss-1432 19 34 0 0 NUM ajrhss-1432 19 35 , , PUNCT ajrhss-1432 19 36 ∞ ∞ PROPN ajrhss-1432 19 37 ) ) PUNCT ajrhss-1432 19 38 that that PRON ajrhss-1432 19 39 is be AUX ajrhss-1432 19 40 representable representable ADJ ajrhss-1432 19 41 in in ADP ajrhss-1432 19 42 the the DET ajrhss-1432 19 43 form form NOUN ajrhss-1432 19 44 πœ‘(π‘₯ πœ‘(π‘₯ NOUN ajrhss-1432 19 45 ) ) PUNCT ajrhss-1432 19 46 = = SYM ajrhss-1432 19 47 π‘₯𝑙𝐻(π‘₯ π‘₯𝑙𝐻(π‘₯ CCONJ ajrhss-1432 19 48 ) ) PUNCT ajrhss-1432 19 49 where where SCONJ ajrhss-1432 19 50 𝑙 𝑙 PRON ajrhss-1432 19 51 β‰₯ β‰₯ NOUN ajrhss-1432 19 52 0 0 NUM ajrhss-1432 19 53 and and CCONJ ajrhss-1432 19 54 h(x h(x PROPN ajrhss-1432 19 55 ) ) PUNCT ajrhss-1432 19 56 is be AUX ajrhss-1432 19 57 a a DET ajrhss-1432 19 58 slowly slowly ADV ajrhss-1432 19 59 varying vary VERB ajrhss-1432 19 60 function function NOUN ajrhss-1432 19 61 in in ADP ajrhss-1432 19 62 the the DET ajrhss-1432 19 63 sense sense NOUN ajrhss-1432 19 64 of of ADP ajrhss-1432 19 65 karamata karamata NOUN ajrhss-1432 19 66 . . PUNCT ajrhss-1432 20 1 theorem theorem NOUN ajrhss-1432 20 2 2 2 NUM ajrhss-1432 20 3 . . PUNCT ajrhss-1432 20 4 suppose suppose VERB ajrhss-1432 20 5 that that SCONJ ajrhss-1432 20 6 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 20 7 β‰  β‰  PROPN ajrhss-1432 20 8 0 0 NUM ajrhss-1432 20 9 and and CCONJ ajrhss-1432 20 10 𝐸||𝑋1|| 𝐸||𝑋1|| PROPN ajrhss-1432 20 11 2 2 NUM ajrhss-1432 20 12 πœ‘(||𝑋1|| πœ‘(||𝑋1|| PROPN ajrhss-1432 20 13 ) ) PUNCT ajrhss-1432 20 14 < < X ajrhss-1432 20 15 ∞ ∞ PROPN ajrhss-1432 20 16 , , PUNCT ajrhss-1432 20 17 then then ADV ajrhss-1432 20 18 for for ADP ajrhss-1432 20 19 any any DET ajrhss-1432 20 20 Ξ΅ Ξ΅ PROPN ajrhss-1432 20 21 > > X ajrhss-1432 20 22 0 0 NUM ajrhss-1432 20 23 ∫ ∫ PROPN ajrhss-1432 20 24 Ο†(π‘₯ Ο†(π‘₯ PROPN ajrhss-1432 20 25 ) ) PUNCT ajrhss-1432 21 1 ∞ ∞ PROPN ajrhss-1432 21 2 0 0 NUM ajrhss-1432 22 1 𝑃{𝑆 𝑃{𝑆 NOUN ajrhss-1432 22 2 οΏ½ οΏ½ PROPN ajrhss-1432 22 3 Μ… Μ… NOUN ajrhss-1432 22 4 οΏ½ οΏ½ NOUN ajrhss-1432 22 5 (𝐴π‘₯ (𝐴π‘₯ NUM ajrhss-1432 22 6 ) ) PUNCT ajrhss-1432 22 7 β‹² β‹² PUNCT ajrhss-1432 23 1 π΄πœ€ π΄πœ€ PROPN ajrhss-1432 23 2 𝑇(𝐴π‘₯ 𝑇(𝐴π‘₯ NOUN ajrhss-1432 23 3 ) ) PUNCT ajrhss-1432 23 4 𝑐 𝑐 NOUN ajrhss-1432 23 5 } } PUNCT ajrhss-1432 23 6 𝑑π‘₯ 𝑑π‘₯ VERB ajrhss-1432 23 7 < < X ajrhss-1432 23 8 ∞ ∞ PROPN ajrhss-1432 23 9 here here ADV ajrhss-1432 23 10 and and CCONJ ajrhss-1432 23 11 in in ADP ajrhss-1432 23 12 what what PRON ajrhss-1432 23 13 follows follow VERB ajrhss-1432 23 14 𝑆 𝑆 PROPN ajrhss-1432 23 15 οΏ½ οΏ½ NOUN ajrhss-1432 23 16 Μ… Μ… NOUN ajrhss-1432 23 17 οΏ½ οΏ½ NOUN ajrhss-1432 23 18 = = SYM ajrhss-1432 23 19 βˆ‘ βˆ‘ PUNCT ajrhss-1432 23 20 ( ( PUNCT ajrhss-1432 23 21 𝑋𝑖 𝑋𝑖 PROPN ajrhss-1432 23 22 βˆ’ βˆ’ PROPN ajrhss-1432 23 23 𝐸𝑋𝑖 𝐸𝑋𝑖 NOUN ajrhss-1432 23 24 ) ) PUNCT ajrhss-1432 23 25 𝑛 𝑛 DET ajrhss-1432 23 26 1 1 NUM ajrhss-1432 23 27 , , PUNCT ajrhss-1432 23 28 and and CCONJ ajrhss-1432 23 29 𝐴𝑒 𝐴𝑒 PROPN ajrhss-1432 23 30 𝑐 𝑐 PROPN ajrhss-1432 23 31 is be AUX ajrhss-1432 23 32 the the DET ajrhss-1432 23 33 complement complement NOUN ajrhss-1432 23 34 of of ADP ajrhss-1432 23 35 𝐴𝑒. 𝐴𝑒. PROPN ajrhss-1432 23 36 remark remark NOUN ajrhss-1432 23 37 . . PUNCT ajrhss-1432 24 1 by by ADP ajrhss-1432 24 2 retracing retrace VERB ajrhss-1432 24 3 the the DET ajrhss-1432 24 4 course course NOUN ajrhss-1432 24 5 of of ADP ajrhss-1432 24 6 the the DET ajrhss-1432 24 7 proof proof NOUN ajrhss-1432 24 8 of of ADP ajrhss-1432 24 9 theorem theorem NOUN ajrhss-1432 24 10 2 2 NUM ajrhss-1432 24 11 it it PRON ajrhss-1432 24 12 can can AUX ajrhss-1432 24 13 be be AUX ajrhss-1432 24 14 shown show VERB ajrhss-1432 24 15 that that SCONJ ajrhss-1432 24 16 ( ( PUNCT ajrhss-1432 24 17 1 1 X ajrhss-1432 24 18 ) ) PUNCT ajrhss-1432 24 19 ∫ ∫ NOUN ajrhss-1432 24 20 πœ‘(π‘₯ πœ‘(π‘₯ PROPN ajrhss-1432 24 21 ) ) PUNCT ajrhss-1432 24 22 𝑃{𝑆 𝑃{𝑆 NOUN ajrhss-1432 24 23 οΏ½ οΏ½ PROPN ajrhss-1432 24 24 Μ… Μ… NOUN ajrhss-1432 24 25 οΏ½ οΏ½ NOUN ajrhss-1432 24 26 (𝐴π‘₯ (𝐴π‘₯ NUM ajrhss-1432 24 27 ) ) PUNCT ajrhss-1432 25 1 ∞ ∞ NUM ajrhss-1432 25 2 0 0 PUNCT ajrhss-1432 26 1 β‹² β‹² X ajrhss-1432 27 1 π΄πœ€ π΄πœ€ PROPN ajrhss-1432 27 2 π‘₯ π‘₯ PROPN ajrhss-1432 27 3 𝑐 𝑐 PROPN ajrhss-1432 27 4 } } PUNCT ajrhss-1432 27 5 𝑑π‘₯ 𝑑π‘₯ VERB ajrhss-1432 27 6 < < X ajrhss-1432 27 7 ∞ ∞ NUM ajrhss-1432 27 8 it it PRON ajrhss-1432 27 9 is be AUX ajrhss-1432 27 10 easy easy ADJ ajrhss-1432 27 11 to to PART ajrhss-1432 27 12 see see VERB ajrhss-1432 27 13 that that SCONJ ajrhss-1432 27 14 if if SCONJ ajrhss-1432 27 15 Ξ΅β†’0 Ξ΅β†’0 NOUN ajrhss-1432 27 16 , , PUNCT ajrhss-1432 27 17 then then ADV ajrhss-1432 27 18 the the DET ajrhss-1432 27 19 left left ADJ ajrhss-1432 27 20 - - PUNCT ajrhss-1432 27 21 hand hand NOUN ajrhss-1432 27 22 side side NOUN ajrhss-1432 27 23 of of ADP ajrhss-1432 27 24 ( ( PUNCT ajrhss-1432 27 25 1 1 X ajrhss-1432 27 26 ) ) PUNCT ajrhss-1432 27 27 converges converge NOUN ajrhss-1432 27 28 to to ADP ajrhss-1432 27 29 ∞ ∞ PROPN ajrhss-1432 27 30 , , PUNCT ajrhss-1432 27 31 and and CCONJ ajrhss-1432 27 32 the the DET ajrhss-1432 27 33 asymptotic asymptotic ADJ ajrhss-1432 27 34 behavior behavior NOUN ajrhss-1432 27 35 of of ADP ajrhss-1432 27 36 the the DET ajrhss-1432 27 37 integral integral ADJ ajrhss-1432 27 38 with with ADP ajrhss-1432 27 39 respect respect NOUN ajrhss-1432 27 40 to to ADP ajrhss-1432 27 41 Ξ΅ Ξ΅ PROPN ajrhss-1432 27 42 is be AUX ajrhss-1432 27 43 of of ADP ajrhss-1432 27 44 interest interest NOUN ajrhss-1432 27 45 . . PUNCT ajrhss-1432 28 1 let let VERB ajrhss-1432 28 2 b b X ajrhss-1432 28 3 be be AUX ajrhss-1432 28 4 the the DET ajrhss-1432 28 5 covariance covariance NOUN ajrhss-1432 28 6 matrix matrix NOUN ajrhss-1432 28 7 of of ADP ajrhss-1432 28 8 the the DET ajrhss-1432 28 9 𝑅𝑉 𝑅𝑉 PROPN ajrhss-1432 28 10 𝑋1 𝑋1 NOUN ajrhss-1432 28 11 . . PUNCT ajrhss-1432 29 1 in in ADP ajrhss-1432 29 2 this this DET ajrhss-1432 29 3 case case NOUN ajrhss-1432 29 4 we we PRON ajrhss-1432 29 5 have have VERB ajrhss-1432 29 6 the the DET ajrhss-1432 29 7 following follow VERB ajrhss-1432 29 8 assertion assertion NOUN ajrhss-1432 29 9 , , PUNCT ajrhss-1432 29 10 which which PRON ajrhss-1432 29 11 extends extend VERB ajrhss-1432 29 12 results result NOUN ajrhss-1432 29 13 in in ADP ajrhss-1432 29 14 [ [ X ajrhss-1432 29 15 3 3 NUM ajrhss-1432 29 16 ] ] PUNCT ajrhss-1432 29 17 . . PUNCT ajrhss-1432 30 1 theorem theorem NOUN ajrhss-1432 30 2 3 3 X ajrhss-1432 30 3 . . PUNCT ajrhss-1432 30 4 suppose suppose VERB ajrhss-1432 30 5 that that SCONJ ajrhss-1432 30 6 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 30 7 β‰  β‰  PROPN ajrhss-1432 30 8 0 0 NUM ajrhss-1432 30 9 and and CCONJ ajrhss-1432 30 10 𝐸||𝑋1||𝑑+2 𝐸||𝑋1||𝑑+2 PROPN ajrhss-1432 30 11 < < X ajrhss-1432 30 12 ∞. ∞. PROPN ajrhss-1432 30 13 then then ADV ajrhss-1432 30 14 lim lim PROPN ajrhss-1432 30 15 πœ€β†’0 πœ€β†’0 PROPN ajrhss-1432 30 16 πœ€2(1+𝑙 πœ€2(1+𝑙 PROPN ajrhss-1432 30 17 ) ) PUNCT ajrhss-1432 30 18 ∫ ∫ PROPN ajrhss-1432 31 1 π‘₯1𝑃{𝑆 π‘₯1𝑃{𝑆 PROPN ajrhss-1432 31 2 οΏ½ οΏ½ PROPN ajrhss-1432 31 3 Μ… Μ… NOUN ajrhss-1432 31 4 οΏ½ οΏ½ NOUN ajrhss-1432 31 5 (𝐴π‘₯ (𝐴π‘₯ NUM ajrhss-1432 31 6 ) ) PUNCT ajrhss-1432 31 7 β‹² β‹² PUNCT ajrhss-1432 32 1 𝐴𝑒 𝐴𝑒 AUX ajrhss-1432 32 2 𝑐 𝑐 PROPN ajrhss-1432 32 3 ∞ ∞ PROPN ajrhss-1432 32 4 0 0 NUM ajrhss-1432 32 5 } } PUNCT ajrhss-1432 32 6 𝑑π‘₯ 𝑑π‘₯ NOUN ajrhss-1432 32 7 = = SYM ajrhss-1432 32 8 ∫ ∫ NOUN ajrhss-1432 32 9 π‘₯π‘™π‘ƒπœ‚ π‘₯π‘™π‘ƒπœ‚ NOUN ajrhss-1432 32 10 ∞ ∞ PROPN ajrhss-1432 32 11 0 0 PUNCT ajrhss-1432 32 12 β‹² β‹² PROPN ajrhss-1432 32 13 𝐴 𝐴 PROPN ajrhss-1432 32 14 √π‘₯ √π‘₯ ADP ajrhss-1432 32 15 𝑐 𝑐 PROPN ajrhss-1432 32 16 } } PUNCT ajrhss-1432 32 17 𝑑π‘₯ 𝑑π‘₯ VERB ajrhss-1432 32 18 where where SCONJ ajrhss-1432 32 19 Ξ· Ξ· PROPN ajrhss-1432 32 20 is be AUX ajrhss-1432 32 21 a a DET ajrhss-1432 32 22 normal normal ADJ ajrhss-1432 32 23 rv rv NOUN ajrhss-1432 32 24 with with ADP ajrhss-1432 32 25 expectation expectation NOUN ajrhss-1432 32 26 the the DET ajrhss-1432 32 27 zero zero NUM ajrhss-1432 32 28 vector vector NOUN ajrhss-1432 32 29 and and CCONJ ajrhss-1432 32 30 covariance covariance NOUN ajrhss-1432 32 31 matrix matrix NOUN ajrhss-1432 32 32 ||𝐸𝑋1||βˆ’1𝐡. ||𝐸𝑋1||βˆ’1𝐡. NOUN ajrhss-1432 32 33 we we PRON ajrhss-1432 32 34 mention mention VERB ajrhss-1432 32 35 some some DET ajrhss-1432 32 36 consequences consequence NOUN ajrhss-1432 32 37 of of ADP ajrhss-1432 32 38 theorem theorem NOUN ajrhss-1432 32 39 3 3 NUM ajrhss-1432 32 40 when when SCONJ ajrhss-1432 32 41 d d NOUN ajrhss-1432 32 42 = = SYM ajrhss-1432 32 43 1 1 X ajrhss-1432 32 44 . . PUNCT ajrhss-1432 32 45 corollary corollary ADJ ajrhss-1432 32 46 1 1 PROPN ajrhss-1432 32 47 . . PUNCT ajrhss-1432 32 48 suppose suppose VERB ajrhss-1432 32 49 that that SCONJ ajrhss-1432 32 50 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 32 51 β‰  β‰  PROPN ajrhss-1432 32 52 0 0 NUM ajrhss-1432 32 53 and and CCONJ ajrhss-1432 32 54 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 32 55 𝑙+2 𝑙+2 X ajrhss-1432 33 1 < < X ajrhss-1432 33 2 ∞ ∞ PROPN ajrhss-1432 33 3 then then ADV ajrhss-1432 33 4 lim lim PROPN ajrhss-1432 33 5 πœ€β†’0 πœ€β†’0 PROPN ajrhss-1432 33 6 πœ€2(𝑙+2 πœ€2(𝑙+2 PROPN ajrhss-1432 33 7 ) ) PUNCT ajrhss-1432 33 8 ∫ ∫ PROPN ajrhss-1432 33 9 π‘₯𝑙𝑃{|𝑆 π‘₯𝑙𝑃{|𝑆 PUNCT ajrhss-1432 33 10 οΏ½ οΏ½ NOUN ajrhss-1432 33 11 Μ… Μ… NOUN ajrhss-1432 33 12 οΏ½ οΏ½ NOUN ajrhss-1432 33 13 (𝐴π‘₯ (𝐴π‘₯ NUM ajrhss-1432 33 14 ) ) PUNCT ajrhss-1432 33 15 > > X ajrhss-1432 34 1 πœ€π‘₯ πœ€π‘₯ ADP ajrhss-1432 34 2 ∞ ∞ NOUN ajrhss-1432 34 3 0 0 NUM ajrhss-1432 34 4 } } PUNCT ajrhss-1432 34 5 𝑑π‘₯ 𝑑π‘₯ NOUN ajrhss-1432 34 6 = = NOUN ajrhss-1432 34 7 2 2 NUM ajrhss-1432 34 8 Π³ Π³ PROPN ajrhss-1432 34 9 ( ( PUNCT ajrhss-1432 34 10 𝑙 𝑙 PROPN ajrhss-1432 34 11 + + NUM ajrhss-1432 34 12 3 3 NUM ajrhss-1432 34 13 2 2 NUM ajrhss-1432 34 14 ) ) PUNCT ajrhss-1432 34 15 βˆšΟ€(𝑙 βˆšΟ€(𝑙 NOUN ajrhss-1432 35 1 + + NOUN ajrhss-1432 35 2 1 1 X ajrhss-1432 35 3 ) ) PUNCT ajrhss-1432 35 4 ( ( PUNCT ajrhss-1432 35 5 𝐷𝑋1 𝐷𝑋1 PROPN ajrhss-1432 35 6 |𝐸𝑋1| |𝐸𝑋1| PROPN ajrhss-1432 35 7 ) ) PUNCT ajrhss-1432 35 8 𝑙+1 𝑙+1 X ajrhss-1432 35 9 corollary corollary ADJ ajrhss-1432 35 10 2 2 NUM ajrhss-1432 35 11 . . PUNCT ajrhss-1432 36 1 if if SCONJ ajrhss-1432 36 2 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 36 3 β‰  β‰  PROPN ajrhss-1432 36 4 0 0 NUM ajrhss-1432 36 5 and and CCONJ ajrhss-1432 36 6 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 36 7 2 2 NUM ajrhss-1432 36 8 < < X ajrhss-1432 36 9 ∞ ∞ NUM ajrhss-1432 36 10 then then ADV ajrhss-1432 36 11 lim lim PROPN ajrhss-1432 36 12 πœ€β†’0 πœ€β†’0 PROPN ajrhss-1432 36 13 πœ€2 πœ€2 PROPN ajrhss-1432 36 14 { { PUNCT ajrhss-1432 36 15 ∫ ∫ PROPN ajrhss-1432 36 16 π‘₯𝑙𝑃{|𝑆 π‘₯𝑙𝑃{|𝑆 PUNCT ajrhss-1432 36 17 οΏ½ οΏ½ NOUN ajrhss-1432 36 18 Μ… Μ… NOUN ajrhss-1432 36 19 οΏ½ οΏ½ NOUN ajrhss-1432 36 20 (𝐴π‘₯ (𝐴π‘₯ NUM ajrhss-1432 36 21 ) ) PUNCT ajrhss-1432 36 22 > > X ajrhss-1432 37 1 πœ€π‘₯ πœ€π‘₯ ADP ajrhss-1432 37 2 ∞ ∞ NOUN ajrhss-1432 37 3 0 0 NUM ajrhss-1432 37 4 } } PUNCT ajrhss-1432 37 5 βˆ’ βˆ’ ADP ajrhss-1432 37 6 𝑃{𝑆 𝑃{𝑆 NOUN ajrhss-1432 37 7 οΏ½ οΏ½ PROPN ajrhss-1432 37 8 Μ… Μ… NOUN ajrhss-1432 37 9 οΏ½ οΏ½ NOUN ajrhss-1432 37 10 (𝐴π‘₯)}]𝑑π‘₯ (𝐴π‘₯)}]𝑑π‘₯ NOUN ajrhss-1432 37 11 } } PUNCT ajrhss-1432 37 12 = = SYM ajrhss-1432 37 13 0 0 NUM ajrhss-1432 37 14 the the DET ajrhss-1432 37 15 following follow VERB ajrhss-1432 37 16 lemma lemma PROPN ajrhss-1432 37 17 , , PUNCT ajrhss-1432 37 18 which which PRON ajrhss-1432 37 19 is be AUX ajrhss-1432 37 20 also also ADV ajrhss-1432 37 21 of of ADP ajrhss-1432 37 22 independent independent ADJ ajrhss-1432 37 23 interest interest NOUN ajrhss-1432 37 24 , , PUNCT ajrhss-1432 37 25 can can AUX ajrhss-1432 37 26 be be AUX ajrhss-1432 37 27 used use VERB ajrhss-1432 37 28 to to PART ajrhss-1432 37 29 prove prove VERB ajrhss-1432 37 30 the the DET ajrhss-1432 37 31 theorems theorem NOUN ajrhss-1432 37 32 given give VERB ajrhss-1432 37 33 above above ADV ajrhss-1432 37 34 . . PUNCT ajrhss-1432 38 1 lemma lemma PROPN ajrhss-1432 38 2 . . PUNCT ajrhss-1432 39 1 a a PRON ajrhss-1432 39 2 ) ) PUNCT ajrhss-1432 39 3 suppose suppose VERB ajrhss-1432 39 4 that that SCONJ ajrhss-1432 39 5 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 39 6 β‰  β‰  PROPN ajrhss-1432 39 7 0 0 NUM ajrhss-1432 39 8 . . PUNCT ajrhss-1432 40 1 then then ADV ajrhss-1432 40 2 for for ADP ajrhss-1432 40 3 any any DET ajrhss-1432 40 4 πœ€ πœ€ NOUN ajrhss-1432 40 5 > > X ajrhss-1432 40 6 0 0 PUNCT ajrhss-1432 40 7 lim lim NOUN ajrhss-1432 40 8 π‘₯β†’βˆž π‘₯β†’βˆž NUM ajrhss-1432 40 9 𝑃 𝑃 PROPN ajrhss-1432 40 10 { { PUNCT ajrhss-1432 40 11 | | NOUN ajrhss-1432 40 12 𝑇(𝐴π‘₯ 𝑇(𝐴π‘₯ NOUN ajrhss-1432 40 13 ) ) PUNCT ajrhss-1432 41 1 π‘₯ π‘₯ PRON ajrhss-1432 41 2 βˆ’ βˆ’ PROPN ajrhss-1432 41 3 ||𝐸𝑋1|| ||𝐸𝑋1|| NOUN ajrhss-1432 41 4 βˆ’1 βˆ’1 NOUN ajrhss-1432 41 5 | | ADV ajrhss-1432 41 6 > > X ajrhss-1432 41 7 πœ€ πœ€ PROPN ajrhss-1432 41 8 } } PUNCT ajrhss-1432 41 9 = = SYM ajrhss-1432 41 10 0 0 NUM ajrhss-1432 41 11 b b X ajrhss-1432 41 12 ) ) PUNCT ajrhss-1432 41 13 if if SCONJ ajrhss-1432 41 14 𝐸𝑋1 𝐸𝑋1 PROPN ajrhss-1432 41 15 = = SYM ajrhss-1432 41 16 0 0 NUM ajrhss-1432 41 17 , , PUNCT ajrhss-1432 41 18 then then ADV ajrhss-1432 41 19 for for ADP ajrhss-1432 41 20 sufficiently sufficiently ADV ajrhss-1432 41 21 large large ADJ ajrhss-1432 41 22 c>0 c>0 NOUN ajrhss-1432 41 23 lim lim PROPN ajrhss-1432 41 24 π‘₯β†’βˆž π‘₯β†’βˆž NUM ajrhss-1432 41 25 𝑃|𝑇(𝐴π‘₯ 𝑃|𝑇(𝐴π‘₯ NOUN ajrhss-1432 41 26 ) ) PUNCT ajrhss-1432 41 27 > > PUNCT ajrhss-1432 42 1 𝐢π‘₯ 𝐢π‘₯ PROPN ajrhss-1432 42 2 = = SYM ajrhss-1432 42 3 1 1 NUM ajrhss-1432 42 4 remark remark NOUN ajrhss-1432 42 5 . . PUNCT ajrhss-1432 43 1 by by ADP ajrhss-1432 43 2 using use VERB ajrhss-1432 43 3 the the DET ajrhss-1432 43 4 results result NOUN ajrhss-1432 43 5 in in ADP ajrhss-1432 43 6 ( ( PUNCT ajrhss-1432 43 7 4 4 X ajrhss-1432 43 8 ] ] PUNCT ajrhss-1432 43 9 it it PRON ajrhss-1432 43 10 can can AUX ajrhss-1432 43 11 be be AUX ajrhss-1432 43 12 shown show VERB ajrhss-1432 43 13 that that SCONJ ajrhss-1432 43 14 the the DET ajrhss-1432 43 15 assertions assertion NOUN ajrhss-1432 43 16 of of ADP ajrhss-1432 43 17 the the DET ajrhss-1432 43 18 lemma lemma PROPN ajrhss-1432 43 19 remain remain VERB ajrhss-1432 43 20 in in ADP ajrhss-1432 43 21 force force NOUN ajrhss-1432 43 22 when when SCONJ ajrhss-1432 43 23 the the DET ajrhss-1432 43 24 rv rv X ajrhss-1432 43 25 π‘‹π‘˜ π‘‹π‘˜ PROPN ajrhss-1432 43 26 takes take VERB ajrhss-1432 43 27 values value NOUN ajrhss-1432 43 28 in in ADP ajrhss-1432 43 29 a a DET ajrhss-1432 43 30 separable separable ADJ ajrhss-1432 43 31 banach banach NOUN ajrhss-1432 43 32 space space NOUN ajrhss-1432 43 33 . . PUNCT ajrhss-1432 44 1 references reference NOUN ajrhss-1432 44 2 1 1 NUM ajrhss-1432 44 3 . . PUNCT ajrhss-1432 44 4 frank frank PROPN ajrhss-1432 44 5 spritzer spritzer PROPN ajrhss-1432 44 6 , , PUNCT ajrhss-1432 44 7 principles principle NOUN ajrhss-1432 44 8 of of ADP ajrhss-1432 44 9 random random ADJ ajrhss-1432 44 10 walk walk NOUN ajrhss-1432 44 11 , , PUNCT ajrhss-1432 44 12 van van PROPN ajrhss-1432 44 13 nostrand nostrand PROPN ajrhss-1432 44 14 , , PUNCT ajrhss-1432 44 15 princeton princeton PROPN ajrhss-1432 44 16 , , PUNCT ajrhss-1432 44 17 n. n. PROPN ajrhss-1432 44 18 j. j. PROPN ajrhss-1432 44 19 , , PUNCT ajrhss-1432 44 20 1964 1964 NUM ajrhss-1432 44 21 . . PUNCT ajrhss-1432 45 1 2 2 X ajrhss-1432 45 2 . . PUNCT ajrhss-1432 45 3 p. p. NOUN ajrhss-1432 45 4 j. j. PROPN ajrhss-1432 45 5 bickel bickel PROPN ajrhss-1432 45 6 and and CCONJ ajrhss-1432 45 7 j. j. PROPN ajrhss-1432 45 8 a. a. PROPN ajrhss-1432 45 9 yahav yahav PROPN ajrhss-1432 45 10 , , PUNCT ajrhss-1432 45 11 israel israel PROPN ajrhss-1432 45 12 j. j. PROPN ajrhss-1432 45 13 math math PROPN ajrhss-1432 45 14 . . PUNCT ajrhss-1432 46 1 3 3 NUM ajrhss-1432 46 2 ( ( PUNCT ajrhss-1432 46 3 1965 1965 NUM ajrhss-1432 46 4 ) ) PUNCT ajrhss-1432 46 5 , , PUNCT ajrhss-1432 46 6 181 181 NUM ajrhss-1432 46 7 . . NOUN ajrhss-1432 47 1 3 3 X ajrhss-1432 47 2 . . X ajrhss-1432 47 3 m. m. NOUN ajrhss-1432 47 4 u. u. PROPN ajrhss-1432 47 5 gafurov gafurov PROPN ajrhss-1432 47 6 and and CCONJ ajrhss-1432 47 7 s. s. PROPN ajrhss-1432 47 8 h. h. PROPN ajrhss-1432 47 9 sirazdinov sirazdinov PROPN ajrhss-1432 47 10 , , PUNCT ajrhss-1432 47 11 kybernetika kybernetika X ajrhss-1432 47 12 ( ( PUNCT ajrhss-1432 47 13 prague prague PROPN ajrhss-1432 47 14 ) ) PUNCT ajrhss-1432 47 15 15 15 NUM ajrhss-1432 47 16 ( ( PUNCT ajrhss-1432 47 17 1979 1979 NUM ajrhss-1432 47 18 ) ) PUNCT ajrhss-1432 47 19 , , PUNCT ajrhss-1432 47 20 272 272 NUM ajrhss-1432 47 21 ( ( PUNCT ajrhss-1432 47 22 russian russian PROPN ajrhss-1432 47 23 ) ) PUNCT ajrhss-1432 47 24 . . PUNCT ajrhss-1432 48 1 4 4 X ajrhss-1432 48 2 . . PUNCT ajrhss-1432 48 3 t. t. PROPN ajrhss-1432 48 4 a. a. PROPN ajrhss-1432 48 5 azlarov azlarov PROPN ajrhss-1432 48 6 and and CCONJ ajrhss-1432 48 7 n. n. PROPN ajrhss-1432 48 8 a. a. PROPN ajrhss-1432 48 9 volodin volodin PROPN ajrhss-1432 48 10 , , PUNCT ajrhss-1432 48 11 limit limit NOUN ajrhss-1432 48 12 theorems theorem NOUN ajrhss-1432 48 13 , , PUNCT ajrhss-1432 48 14 random random ADJ ajrhss-1432 48 15 processes process NOUN ajrhss-1432 48 16 and and CCONJ ajrhss-1432 48 17 their their PRON ajrhss-1432 48 18 applications application NOUN ajrhss-1432 48 19 , , PUNCT ajrhss-1432 48 20 β€œ " PUNCT ajrhss-1432 48 21 fan fan NOUN ajrhss-1432 48 22 ” " PUNCT ajrhss-1432 48 23 , , PUNCT ajrhss-1432 48 24 tashkent tashkent NOUN ajrhss-1432 48 25 , , PUNCT ajrhss-1432 48 26 1979 1979 NUM ajrhss-1432 48 27 , , PUNCT ajrhss-1432 48 28 p. p. NOUN ajrhss-1432 48 29 15 15 NUM ajrhss-1432 48 30 ( ( PUNCT ajrhss-1432 48 31 russian russian NOUN ajrhss-1432 48 32 ) ) PUNCT ajrhss-1432 48 33 . . PUNCT ajrhss-1432 49 1 https://en.wikipedia.org/wiki/not_equal_sign https://en.wikipedia.org/wiki/not_equal_sign NOUN