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By B.L.S. Prakasa Rao

This booklet provides a complete overview of effects for linked sequences and demimartingales built up to now, with distinct emphasis on demimartingales and similar strategies. Probabilistic homes of linked sequences, demimartingales and comparable approaches are mentioned within the first six chapters. functions of a few of those effects to a couple difficulties in nonparametric statistical inference for such procedures are investigated within the final 3 chapters.

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Extra resources for Associated Sequences, Demimartingales and Nonparametric Inference

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Xj ) = exp{i j and f1 (x1 , . . , xj ) = k=1 |rk |xk . j k=1 rk xk } The next result due to Newman (1984) gives sufficient conditions for a demimartingale to be a martingale with respect to the natural sequence of sub-σalgebras. 4. Let S0 = 0, and the sequence {Sn , n ≥ 1} be an L2 -demimartingale. Let Fn be the σ-algebra generated by the sequence {S1 , . . , Sn }. If the random sequence {Sn , n ≥ 1} has uncorrelated increments, that is, if Cov((Sj+1 − Sj ), (Sk+1 − Sk )) = 0, then the sequence {Sn , Fn , n ≥ 1} is a martingale.

N − 1. 9) which implies the upcrossing inequality stated in the theorem. 5 Chow Type Maximal Inequality We now derive some more maximal inequalities for demimartingales which can be used to derive strong laws of large numbers for demimartingales. The following result, due to Christofides (2000), is an analogue of the maximal inequality for submartingales proved by Chow (1960). 1. Let the sequence {Sn , n ≥ 1} be a demisubmartingale with S0 = 0. 1) j=1 where x+ = max{0, x}. Proof. Let > 0. Let A = [max1≤k≤n ck Sk ≥ ] and Aj = [ max ci Si < , cj Sj ≥ ], 1≤i

Furthermore when μ1,0 (s) and μ0,1 (s) are continuous, then the statement (iv) of the above theorem is always satisfied and the corresponding Markov Process is always associated in time. It is also true for a general birth and death process (cf. Keilson and Kester (1977) and Kirstein (1976)). Kuber and Dharmadikari (1996) discussed association in time for semi-Markov processes. Let (Ω, F, P) be a probability space and E = {0, 1, . . , k}. Define measurable functions Xn : Ω → E, Tn : Ω → R+ , n ∈ N, so that 0 = T0 ≤ T1 ≤ T2 ≤ .

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