[@nevelson1973stochastic]. We explicitly evaluate the asymptotic covariance of the component wise estimate sequences at each agent. The obtained asymptotic covariance is heterogeneous in terms of scaling of the variances of the components of the parameter based on the number of agents interested in reconstructing a particular component. To the best of our knowledge, this is the first asymptotic covariance evaluation explicitly in terms of the number of agents interested in reconstructing entries of the state vector for distributed estimation of high-dimensional fields, i.e., when each node is interested only in a subset of the vector parameter.\
**Related Work:** Relevant distributed estimation literature can be classified primarily into two types. The first type includes schemes which involve single snapshot data collection followed by inter-agent fusion through consensus type protocols (see, for example,[@Mesbahi-parameter; @Giannakis-est; @Bertsekas-survey; @olfatisaberfaxmurray07; @jadbabailinmorse03]). The second type includes estimation schemes where the sensing and the processing of the information occur at the same rate and sequentially in time (see, for example [@Sayed-LMS; @Stankovic-parameter; @Giannakis-LMS; @Nedic-parameter; @nedic2015nonasymptotic; @jadbabaie2012non; @shahrampour2013exponentially; @mateos2009distributed; @mateos2012distributed; @weng2014efficient]). Representative approaches of this latter class are $\emph{consensus}+\emph{innovations}$ type ([@kar2011convergence; @sahu2016distributed]) and the *diffusion* type ([@cattivelli2010diffusion; @cattivelli2008diffusion]) algorithms. Distributed inference algorithms for random fields have been proposed in 