2000 threshold for foreign income

I have a few questions about reporting foreign income: I am living in the UK for 5 years. I am confused about the £2000 threshold on foreign income. Does 2000 threshold still apply for an individual who has not remitted the amount to UK but not declared non-domicile status in their self-assessments. I've some rent income which is less than £2000 per year, should I declare it in my self-assessment? I've recently started to receive state pension from my past services in my origin country. My understanding is that is not declared as a taxable income (please correct me if I am wrong), does it effect this £2000 threshold?

Hi,
Please refer to guidance at Paying tax on the remittance basis (Self Assessment helpsheet HS264)
Thank you.