If[0, 1], Un1 ( X1 , U1 , , . . . , Xn , Un , , Xn1 ), and P( Xn1 | X1 , U1 , , . . . , Xn , Un , ) = (.Mathematics 2021, 9,six ofIt follows from (v) that, for any Decanoyl-L-carnitine MedChemExpress measurable set B MF (X),P( 1 B| X1 , U1 , , . . . ,Xn , Un , ) = E[ (R Xn1 )( B)| X1 , U1 , , . . . , Xn , Un , ]= P ( f ( Xn1 , Un1 )) B| X1 , U1 , , . . . , Xn , Un , ;d therefore, 1 = f ( Xn1 , Un1 ) | X1 , U1 , , . . . , Xn , Un , . By Theorem eight.17 in [25], there exist random variables Xn1 and Un1 such that( 1 , X1 , U1 , , . . . , Xn , Un , , Xn1 , Un1 )= f ( Xn1 , Un1 ), X1 , U1 , , . . . , Xn , Un , , Xn1 , Un1 , and ( two , three , . . .) ( Xn1 , Un1 ) | ( X1 , U1 , . . . , . . . , Xn , Un , , 1 ). Then, in distinct, Un1 Unif[0, 1], Un1 ( X1 , U1 , , . . . , Xn , Un , , Xn1 ), anddP( Xn1 | X1 , U1 , , . . . , Xn , Un , ) = (.Moreover, 1 , f ( Xn1 , Un1 ) = f ( Xn1 , Un1 ), f ( Xn1 , Un1 ) ; thus, P 1 = f ( Xn1 , Un1 ) = P f ( Xn1 , Un1 ) = f ( Xn1 , Un1 ) = 1. By Theorem 8.12 in [25], statement (v) with n 1 is equivalent to 2 ( X1 , U1 ) | ( , . . . , 1 ) and two ( Xk1 , Uk1 ) | ( X1 , U1 , . . . , Xk , Uk , , . . . , 1 ), k = 1, . . . , n. The latter follows in the induction hypothesis since, by (iv), we’ve got ( two , . . . , 2 ) ( Xk1 , Uk1 ) | ( X1 , U1 , . . . , Xk , Uk , , . . . , 1 ) for each k = 1, . . . , n. The course of action ( Xn )n1 in Theorem 1 corresponds for the sequence of observed colors in the implied urn sampling scheme. Moreover, the replacement rule requires the form R Xn = f ( Xn , Un ), exactly where f is some measurable function, Un Unif[0, 1], and Un ( X1 , U1 , . . . , Xn-1 , Un-1 , Xn ), from which it follows that = -1 f ( Xn , Un ), andn ( i=1 f ( Xi , Ui )( . n (X) i=1 f ( Xi , Ui )(X) d(14)P( Xn1 | X1 , . . . , Xn , (Um )m1 ) =(15)As a result, the sequence (Un )n1 models the further randomness inside the reinforcement measure R. Janson [9] obtains a rather equivalent outcome; Theorem 1.3 in [9] states that any MVPP ( )n0 is usually coupled using a deterministic MVPP ( )n0 on X [0, 1] inside the sense that = , (16) where is the Lebesgue measure on [0, 1], and may be the ML-SA1 Technical Information solution measure on X [0, 1]. In our case, the MVPP defined by = and, for n 1, = -1 f ( Xn , Un ) , features a non-random replacement rule R x,u = f ( x, u) and satisfies (16) on a set of probability 1.Mathematics 2021, 9,7 of2.2. Randomly Reinforced P ya Processes It follows from (eight) that any P ya sequence generates a deterministic MVPP by means of = -1 Xn . Right here, we contemplate a randomly reinforced extension of P ya sequences within the type of an MVPP with replacement rule R x = W ( x ) x , x X, exactly where W ( x ) is often a non-negative random variable. Definition 2 (Randomly Reinforced P ya Method). We get in touch with an MVPP with parameters ( , R) a randomly reinforced P ya procedure (RRPP) if there exists KP (X, R ) such that R x = x (x ), x X, exactly where x : R MF (X) would be the map w wx . Observe that, for RRPPs, the reinforcement measure f ( x, u) in (14)15) concentrates its mass on x; therefore, we receive the following variant of the representation result in Theorem 1. Proposition 1. Let ( )n0 be an RRPP with parameters ( , ). Then, there exist a measurable function h : X [0, 1] R and a sequence (( Xn , Un ))n1 such that, utilizing Wn = h( Xn , Un ), we’ve got for every single n 1 that = -1 Wn Xn a.s., (17) exactly where X1 and, for n 1, Un Unif[0, 1], Un ( X1 , U1 , . . . , Xn-1 , Un-1 , Xn ), andP( Xn1 | X1 , W1 ,.