SA4 liu iia ff 6 black~.bE The spectral representations for arbitrary discrete parameter infinite... more SA4 liu iia ff 6 black~.bE The spectral representations for arbitrary discrete parameter infinitely divisible processes as well as for (centered) continuous parameter infinitely divisible processes, which are separabl 'e in probability, are obtained. The main tc,31s used for the proofs are (i) a "polar-factorizatior" of an arbitrary Levy measire or a separable Hilbert space, and (i i) the Wiener-type stochastic integrals of non-rartd3r functions relative to artitrary "infinitely divisible noise". S_' 0,%C' &.A At 0-ASS-PAC' 2 AGSAAC ifl'.AS.''C.A 'wo .%A 9 f: IC-. No IF:K SL'.-S Z-0C UJSES %6%01 C DES-z. , ta~.U Nri .3-. snk 22t,.
This paper contains the following three types of results: First, a 1-1 correspondence is establis... more This paper contains the following three types of results: First, a 1-1 correspondence is established between Gaussian measures on Lp, 1Gaussian measure Gaussian process zero-one law characteristic function topological support Lp space Frechet space
We provide simple and succinct solutions to two dual extremal problems in the Hardy spaces H p, a... more We provide simple and succinct solutions to two dual extremal problems in the Hardy spaces H p, and to an aspect of the linear prediction problem for a certain class of discrete and continuous parameter "'LP-harmonizable '' stochastic processes, for all 1 <p < 00. Two of the results presented appear new. The methods of proof of the rest of the results provide alternate simpler and shorter proofs for some earlier known theorems.
The Support of Gaussian Measures on Banach Spaces. [Theory of Probability and its Applications 17... more The Support of Gaussian Measures on Banach Spaces. [Theory of Probability and its Applications 17, 728 (1973)]. Balram S. Rajput.
Proceedings of the American Mathematical Society, 1977
It is shown that the topological support (supp.) of a τ \tau -regular, symmetric, infinitely divi... more It is shown that the topological support (supp.) of a τ \tau -regular, symmetric, infinitely divisible (resp. stable of any index α ∈ ( 0 , 2 ) \alpha \in (0,2) ) probability measure on a Hausdorff LCTVS E is a subgroup (resp. a subspace) of E. The part regarding the support of a stable probability measure of this theorem completes a result of A. De-Acosta [Ann. of Probability 3 (1975), 865-875], who proved a similar result for α ∈ ( 1 , 2 ) \alpha \in (1,2) , and the author [Proc. Amer. Math. Soc. 63 (1977), 306-312], who proved it for α ∈ [ 1 , 2 ) \alpha \in [1,2) . Further, it provides a complete affirmative solution to the question, raised by J. Kuelbs and V. Mandrekar [Studia Math. 50 (1974), 149-162], of whether the supp. of a symmetric stable probability measure of index α ∈ ( 0 , 1 ] \alpha \in (0,1] on a separable Hilbert space H is a subspace of H.
Journal of Mathematical Analysis and Applications, 1981
A simple necessary and sufficient condition, on a trace-class kernel K, is given in order to demo... more A simple necessary and sufficient condition, on a trace-class kernel K, is given in order to demonstrate the existence of a measurable (relative to the completed product u-algebra) Gaussian process with covariance K. Using this result, sufficient conditions are given on the means and the covariances (relative to two equivalent (-) Gaussian measures P and PA) of a process X so that the Radon-Nikodym (R-N) derivative dp,/dP is the exponential of the diagonal form in X. Analogues of the last two results in the setup of Hilbert space are also proved.
This paper contains the following three types of results: First, a 1-1 correspondence is establis... more This paper contains the following three types of results: First, a 1-1 correspondence is established between Gaussian measures on L p, 1≤ p<∞, and Gaussian processes with paths in L p. Second, a 01 law for Gaussian measures on Fréchet spaces is proved, ...
Communications on Stochastic Analysis, Jun 1, 2010
Let X = {X t } be an infinitely divisible stationary process. A good measure of the asymptotic de... more Let X = {X t } be an infinitely divisible stationary process. A good measure of the asymptotic dependence structure of X is provided by the limit of ρ X (t) as t → ∞, where ρ X (t) is equal to the joint characteristic function of (X t , X 0) minus the product of the characteristic functions of X t and X 0. An interesting case is when ρ X (t) → 0; which roughly says that, as time becomes large, the future of the random phenomenon (represented by X) is becoming independent of its past. In this paper, we study the rate of decay of ρ X (t) (as t → ∞) when X is an Ornstein-Uhlenbeck (r, α)-semi-stable process. The results obtained here generalize and complement the corresponding results for Ornstein-Uhlenbeck α-stable and Ornstein-Uhlenbeck (Gaussian) processes.
Joining of materials with the help of submerged arc welding machine is widely used in the concern... more Joining of materials with the help of submerged arc welding machine is widely used in the concerned industries. The quality of the weld is identified by several welding output parameters like weld width, reinforcement height, depth of penetration, hardness, impact strength and tensile strength etc. The input welding parameters on which outputs depend are welding current, voltage, feed rate, speed of welding, electrode extension, diameter of electrode and electrode angle etc. All the researchers try to optimize these parameters according to the concerned requirement. In this review paper it was tried to consider different research papers containing the research and analysis made on input output parameters of submerged arc welding process. This paper also focused on analysis of some research papers containing heat input rate, cooling rate of weld, micro structure of weld and heat affected zone and their effect on the performance of the joint. Total 28 research papers were selected for our consideration after a thorough investigation of several such papers. After the analysis of the research papers the future scope of research was also suggested.
In this paper we extend the definition of stochastic integrals relative to the larger class of co... more In this paper we extend the definition of stochastic integrals relative to the larger class of complex semistable and other infinitely divisible measures that was given in a previous paper. We obtain spectral representations of complex semistable and other i.d. stochastic processes.
In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or re... more In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .
SA4 liu iia ff 6 black~.bE The spectral representations for arbitrary discrete parameter infinite... more SA4 liu iia ff 6 black~.bE The spectral representations for arbitrary discrete parameter infinitely divisible processes as well as for (centered) continuous parameter infinitely divisible processes, which are separabl 'e in probability, are obtained. The main tc,31s used for the proofs are (i) a "polar-factorizatior" of an arbitrary Levy measire or a separable Hilbert space, and (i i) the Wiener-type stochastic integrals of non-rartd3r functions relative to artitrary "infinitely divisible noise". S_' 0,%C' &.A At 0-ASS-PAC' 2 AGSAAC ifl'.AS.''C.A 'wo .%A 9 f: IC-. No IF:K SL'.-S Z-0C UJSES %6%01 C DES-z. , ta~.U Nri .3-. snk 22t,.
This paper contains the following three types of results: First, a 1-1 correspondence is establis... more This paper contains the following three types of results: First, a 1-1 correspondence is established between Gaussian measures on Lp, 1Gaussian measure Gaussian process zero-one law characteristic function topological support Lp space Frechet space
We provide simple and succinct solutions to two dual extremal problems in the Hardy spaces H p, a... more We provide simple and succinct solutions to two dual extremal problems in the Hardy spaces H p, and to an aspect of the linear prediction problem for a certain class of discrete and continuous parameter "'LP-harmonizable '' stochastic processes, for all 1 <p < 00. Two of the results presented appear new. The methods of proof of the rest of the results provide alternate simpler and shorter proofs for some earlier known theorems.
The Support of Gaussian Measures on Banach Spaces. [Theory of Probability and its Applications 17... more The Support of Gaussian Measures on Banach Spaces. [Theory of Probability and its Applications 17, 728 (1973)]. Balram S. Rajput.
Proceedings of the American Mathematical Society, 1977
It is shown that the topological support (supp.) of a τ \tau -regular, symmetric, infinitely divi... more It is shown that the topological support (supp.) of a τ \tau -regular, symmetric, infinitely divisible (resp. stable of any index α ∈ ( 0 , 2 ) \alpha \in (0,2) ) probability measure on a Hausdorff LCTVS E is a subgroup (resp. a subspace) of E. The part regarding the support of a stable probability measure of this theorem completes a result of A. De-Acosta [Ann. of Probability 3 (1975), 865-875], who proved a similar result for α ∈ ( 1 , 2 ) \alpha \in (1,2) , and the author [Proc. Amer. Math. Soc. 63 (1977), 306-312], who proved it for α ∈ [ 1 , 2 ) \alpha \in [1,2) . Further, it provides a complete affirmative solution to the question, raised by J. Kuelbs and V. Mandrekar [Studia Math. 50 (1974), 149-162], of whether the supp. of a symmetric stable probability measure of index α ∈ ( 0 , 1 ] \alpha \in (0,1] on a separable Hilbert space H is a subspace of H.
Journal of Mathematical Analysis and Applications, 1981
A simple necessary and sufficient condition, on a trace-class kernel K, is given in order to demo... more A simple necessary and sufficient condition, on a trace-class kernel K, is given in order to demonstrate the existence of a measurable (relative to the completed product u-algebra) Gaussian process with covariance K. Using this result, sufficient conditions are given on the means and the covariances (relative to two equivalent (-) Gaussian measures P and PA) of a process X so that the Radon-Nikodym (R-N) derivative dp,/dP is the exponential of the diagonal form in X. Analogues of the last two results in the setup of Hilbert space are also proved.
This paper contains the following three types of results: First, a 1-1 correspondence is establis... more This paper contains the following three types of results: First, a 1-1 correspondence is established between Gaussian measures on L p, 1≤ p<∞, and Gaussian processes with paths in L p. Second, a 01 law for Gaussian measures on Fréchet spaces is proved, ...
Communications on Stochastic Analysis, Jun 1, 2010
Let X = {X t } be an infinitely divisible stationary process. A good measure of the asymptotic de... more Let X = {X t } be an infinitely divisible stationary process. A good measure of the asymptotic dependence structure of X is provided by the limit of ρ X (t) as t → ∞, where ρ X (t) is equal to the joint characteristic function of (X t , X 0) minus the product of the characteristic functions of X t and X 0. An interesting case is when ρ X (t) → 0; which roughly says that, as time becomes large, the future of the random phenomenon (represented by X) is becoming independent of its past. In this paper, we study the rate of decay of ρ X (t) (as t → ∞) when X is an Ornstein-Uhlenbeck (r, α)-semi-stable process. The results obtained here generalize and complement the corresponding results for Ornstein-Uhlenbeck α-stable and Ornstein-Uhlenbeck (Gaussian) processes.
Joining of materials with the help of submerged arc welding machine is widely used in the concern... more Joining of materials with the help of submerged arc welding machine is widely used in the concerned industries. The quality of the weld is identified by several welding output parameters like weld width, reinforcement height, depth of penetration, hardness, impact strength and tensile strength etc. The input welding parameters on which outputs depend are welding current, voltage, feed rate, speed of welding, electrode extension, diameter of electrode and electrode angle etc. All the researchers try to optimize these parameters according to the concerned requirement. In this review paper it was tried to consider different research papers containing the research and analysis made on input output parameters of submerged arc welding process. This paper also focused on analysis of some research papers containing heat input rate, cooling rate of weld, micro structure of weld and heat affected zone and their effect on the performance of the joint. Total 28 research papers were selected for our consideration after a thorough investigation of several such papers. After the analysis of the research papers the future scope of research was also suggested.
In this paper we extend the definition of stochastic integrals relative to the larger class of co... more In this paper we extend the definition of stochastic integrals relative to the larger class of complex semistable and other infinitely divisible measures that was given in a previous paper. We obtain spectral representations of complex semistable and other i.d. stochastic processes.
In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or re... more In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .
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