Multilingual Text Classification (MLTC) is a text classification task in which documents are written each in one among a set L of natural languages, and in which all documents must be classified under the same classification scheme, irrespective of language. There are two main variants of MLTC, namely Cross-Lingual Text Classification (CLTC) and Polylingual Text Classification (PLTC). In PLTC, which is the focus of this paper, we assume (differently from CLTC) that for each language in L there is a representative set of training documents; PLTC consists of improving the accuracy of each of the |L| monolingual classifiers by also leveraging the training documents written in the other (|L| − 1) languages. The obvious solution, consisting of generating a single polylingual classifier from the juxtaposed monolingual vector spaces, is usually infeasible, since the dimensionality of the resulting vector space is roughly |L| times that of a monolingual one, and is thus often unmanageable. As a response, the use of machine translation tools or multilingual dictionaries has been proposed. However, these resources are not always available, or are not always free to use.One machine-translation-free and dictionary-free method that, to the best of our knowledge, has never been applied to PLTC before, is Random Indexing (RI). We analyse RI in terms of space and time efficiency, and propose a particular configuration of it (that we dub Lightweight Random Indexing -LRI). By running experiments on two well known public benchmarks, Reuters RCV1/RCV2 (a comparable corpus) and JRC-Acquis (a parallel one), we show LRI to outperform (both in terms of effectiveness and efficiency) a number of previously proposed machine-translation-free and dictionary-free PLTC methods that we use as baselines.